Analytical solutions to a class of fractional coupled nonlinear Schrödinger equations via Laplace-HPM technique

Baojian Hong; Jinghan Wang; Chen Li; Baojian Hong; Jinghan Wang; Chen Li

doi:10.3934/math.2023800

AIMS Mathematics

2023, Volume 8, Issue 7: 15670-15688. doi: 10.3934/math.2023800

Previous Article Next Article

Research article Special Issues

Analytical solutions to a class of fractional coupled nonlinear Schrödinger equations via Laplace-HPM technique

1.
Faculty of Mathematical Physics, Nanjing Institute of Technology, Nanjing 211167, China
2.
Faculty of Electric Power Engineering, Nanjing Institute of Technology, Nanjing 211167, China
3.
Faculty of Architectural Engineering, Nanjing Institute of Technology, Nanjing 211167, China

Received: 24 February 2023 Revised: 03 April 2023 Accepted: 11 April 2023 Published: 27 April 2023
MSC : 65Mxx, 34A08

In this article, a class of fractional coupled nonlinear Schrödinger equations (FCNLS) is suggested to describe the traveling waves in a fractal medium arising in ocean engineering, plasma physics and nonlinear optics. First, the modified Kudryashov method is adopted to solve exactly for solitary wave solutions. Second, an efficient and promising method is proposed for the FCNLS by coupling the Laplace transform and the Adomian polynomials with the homotopy perturbation method, and the convergence is proved. Finally, the Laplace-HPM technique is proved to be effective and reliable. Some 3D plots, 2D plots and contour plots of these exact and approximate solutions are simulated to uncover the critically important mechanism of the fractal solitary traveling waves, which shows that the efficient methods are much powerful for seeking explicit solutions of the nonlinear partial differential models arising in mathematical physics.

Keywords:

fractional coupled nonlinear Schrödinger equation,
modified Kudryashov method,
Laplace transform,
Homotopy perturbation method,
exact solutions,
approximate solutions

Citation: Baojian Hong, Jinghan Wang, Chen Li. Analytical solutions to a class of fractional coupled nonlinear Schrödinger equations via Laplace-HPM technique[J]. AIMS Mathematics, 2023, 8(7): 15670-15688. doi: 10.3934/math.2023800

Related Papers:

[1]	Nadiyah Hussain Alharthi, Abdon Atangana, Badr S. Alkahtani . Numerical analysis of some partial differential equations with fractal-fractional derivative. AIMS Mathematics, 2023, 8(1): 2240-2256. doi: 10.3934/math.2023116
[2]	Abdon Atangana, Ali Akgül . Analysis of a derivative with two variable orders. AIMS Mathematics, 2022, 7(5): 7274-7293. doi: 10.3934/math.2022406
[3]	Hasib Khan, Jehad Alzabut, Anwar Shah, Sina Etemad, Shahram Rezapour, Choonkil Park . A study on the fractal-fractional tobacco smoking model. AIMS Mathematics, 2022, 7(8): 13887-13909. doi: 10.3934/math.2022767
[4]	Khaled M. Saad, Manal Alqhtani . Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear. AIMS Mathematics, 2021, 6(4): 3788-3804. doi: 10.3934/math.2021225
[5]	Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah . Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820
[6]	Abdon Atangana, Seda İğret Araz . Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations. AIMS Mathematics, 2024, 9(3): 5763-5793. doi: 10.3934/math.2024280
[7]	Rahat Zarin, Amir Khan, Pushpendra Kumar, Usa Wannasingha Humphries . Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Mathematics, 2022, 7(10): 18897-18924. doi: 10.3934/math.20221041
[8]	Muhammad Farman, Ali Akgül, Sameh Askar, Thongchai Botmart, Aqeel Ahmad, Hijaz Ahmad . Modeling and analysis of fractional order Zika model. AIMS Mathematics, 2022, 7(3): 3912-3938. doi: 10.3934/math.2022216
[9]	Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
[10]	Muhammad Farman, Aqeel Ahmad, Ali Akgül, Muhammad Umer Saleem, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar . Dynamical behavior of tumor-immune system with fractal-fractional operator. AIMS Mathematics, 2022, 7(5): 8751-8773. doi: 10.3934/math.2022489

Abstract

1. Introduction

Chemical kinetics deals with chemistry experiments and interprets them in terms of a mathematical model. The experiments are done on chemical reactions with the passage of time. The models are differential equations for the rates at which reactants are consumed and products are produced. Chemists are able to understand how chemical reactions take place at the molecular level by combining models with investigation. Molecules react in steps to lead to the overall stoichiometric reaction which is reaction mechanism for collection of reactions. The set of reactions specifies the path (or paths) that reactant molecules take to finally arrive at the product molecules. All species in the reaction appear in at least one step and the sum of the steps gives the overall reaction. The govern the rate of the reaction which leads directly to the mechanism of differential equations ^[1]. Many processes and phenomena in chemistry generally in sciences can be designated by first-order differential equations. These equations are the most important and most frequently used to describe natural laws. The following examples are discussed: the Bouguer-Lambert-Beer law in spectroscopy, time constants of sensors, chemical reaction kinetics, radioactive decay, relaxation in nuclear magnetic resonance, and the RC constant of an electrode ^[2]. The induced kinetic differential equations of a reaction network endowed with mass action type kinetics are a system of polynomial differential equations ^[3]. We review the basic ideas of fractional differential equations and their applications on non-linear biochemical reaction models. We apply this idea to a non-linear model of enzyme inhibitor reactions ^[4].

The fractional-order, which involves integration and transect differentiation using fractional calculus is helping to better understand the explanation of real-world problems than ordinary integer order, as well as in the modeling of real phenomena due to a characterization of memory and hereditary properties in ^[5,6]. Riemann Liouville developed the concept of fractional derivative, which is based on power law, ^[7,8] offers a novel fractional derivative that makes use of the exponential kernel. Several issues include the non-singular kernel fractional derivative, which covers the trigonometric and exponential functions, and ^[9,10,11,12] illustrates some relevant techniques for epidemic models. This virus's suggested outbreak efficiently catches the timeline for the COVID-19 disease conceptual model ^[13]. In the literature, many fractional operators are employed to solve real-world issues ^[14,15].

In this paper, section 1 is introduction and section 2 consists of some basic fractional order derivative which are helpful to solve the epidemiological model. Section 3 and 4 consists of generalized from of the model, uniqueness and stability of the model. Fractal Fractional techniques with exponential decay kernel and Mittag-Leffler kernel respectively in section 5. Results and conclusion are discussed in section 6, and 7 respectively.

2. Basic definitions

Following are the basic definitions ^[7,8,14,15] used for analysis and solution of the problem.

Definition 1: Sumudu transform for any function $\phi \left(t\right)$ over a set is given as,

$A = \left\{\phi \left(t\right):there\ exist\ \mathrm{\Lambda }, {\tau }_{1}, {\tau }_{2} > 0, \left|\phi \left(t\right)\right| < \mathrm{\Lambda }exp\left(\frac{\left|t\right|}{{\tau }_{i}}\right), if\ t\in {\left(-1\right)}^{j}\times \left[0, \left.\infty \right)\right.\right\}$

is defined by

$F\left(u\right) = ST\left[\phi \left(t\right)\right] = \underset{0}{\overset{\infty }{\int }}exp\left(-t\right)\phi \left(ut\right)dt, \ \ \ \ \ \ u\in \left({-\tau }_{1}, {\tau }_{2}\right).$

Definition 2: For a function $g\left(t\right)\in {W}_{2}^{1}\left(\mathrm{0, 1}\right), b > a\ and\ \sigma \in \left(\mathrm{0, 1}\right],$ the definition of Atangana–Baleanu derivative in the Caputo sense is given by

${_0^{ABC}}{D}_{t}^{\sigma }g\left(t\right) = \frac{AB\left(\sigma \right)}{1-\sigma }\underset{0}{\overset{t}{\int }}\frac{d}{d\tau }g\left(\tau \right){M}_{\sigma }\left[-\frac{\sigma }{1-\sigma }{\left(t-\tau \right)}^{\sigma }\right]d\tau , \ \ \ \ \ \ n-1 < \sigma < n$

where

$AB\left(\sigma \right) = 1-\sigma +\frac{\sigma }{\mathrm{\Gamma }\left(\sigma \right)}.$

By using Sumudu transform (ST) for (1), we obtain

$ST\left[{_0^{ABC}}{D}_{t}^{\sigma }g\left(t\right)\right]\left(s\right) = \frac{q\left(\sigma \right)}{1-\sigma }\left\{\sigma \mathrm{\Gamma }\left(\sigma +1\right){M}_{\sigma }\left(-\frac{1}{1-\sigma }{V}^{\sigma }\right)\right\}\times \left[ST\left(g\left(t\right)\right)-g\left(0\right)\right].$

Definition 3: For a function $g\left(t\right)\in {W}_{2}^{1}\left(\mathrm{0, 1}\right), b > a\ and\ {\alpha }_{1}\in \left(\mathrm{0, 1}\right],$ the definition of Atangana–Baleanu derivative in the Caputo sense is given by

${_0^{ABC}}{D}_{t}^{{\alpha }_{1}}g\left(t\right) = \frac{AB\left({\alpha }_{1}\right)}{1-{\alpha }_{1}}\underset{0}{\overset{t}{\int }}\frac{d}{d\tau }g\left(\tau \right){E}_{{\alpha }_{1}}[-\frac{{\alpha }_{1}}{1-{\alpha }_{1}}{(t-\tau )}^{{\alpha }_{1}}]d\tau ,$

where

$AB\left({\alpha }_{1}\right) = 1-{\alpha }_{1}+\frac{{\alpha }_{1}}{\mathrm{\Gamma }\left({\alpha }_{1}\right)}.$

Definition 4: Suppose that $g\left(t\right)$ is continuous on an open interval $(a, b),$ then the fractal-fractional integral of $g\left(t\right)$ of order ${\alpha }_{1}$ having Mittag-Leffler type kernel and given by

${{}^{FFM}J}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}\left(g\left(t\right)\right) = \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{s}^{{\alpha }_{2}-1}g\left(s\right){\left(t-s\right)}^{{\alpha }_{1}}ds+\frac{{\alpha }_{2}\left(1-{\alpha }_{1}\right){t}^{{\alpha }_{2}-1}g\left(t\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}$

3. Hires problem with fractional operator

Robertson introduces this chemical process in ^[19,20]. Schafer pioneered the following chemical reactions method in 1975 ^[19,20]. It represents the high irradiance response (HIRES) of photomorphogenesis based on phytochrome. A stiff system of eight non-linear ordinary differential equations is used to create the following mathematical model.

$\begin{array}{l} {{y}_{1}}^{'} = -{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}, \\{{y}_{2}}^{'} = {M}_{1}{y}_{1}-{M}_{5}{y}_{2}, \\ {{y}_{3}}^{'} = -{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}, \\{{y}_{4}}^{'} = {M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}, \\{{y}_{5}}^{'} = -{M}_{10}{y}_{5}+{M}_{2}{y}_{6}+{M}_{2}{y}_{7}, \\{{y}_{6}}^{'} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}, \\{{y}_{7}}^{'} = {M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}, \\{{y}_{8}}^{'} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}. \end{array}$

(1)

Here ${M}_{1} = 1.7, {M}_{2} = 0.43, {M}_{3} = 8.32,$ ${M}_{4} = 0.0007, {M}_{5} = 8.75,$ ${M}_{6} = 10.03, {M}_{7} = 0.035,$ ${M}_{8} = 1.71, {M}_{9} = 1.12, {M}_{10} = 1.745,$ ${M}_{11} = 280, {M}_{12} = 0.69, {M}_{13} = 1.81.$ The initial values can be represented by $y = {\left(\mathrm{1, 0}, \mathrm{0, 0}, \mathrm{0, 0}, \mathrm{0, 0.0057}\right)}^{T}$ . By using Atangana-Baleanu in Caputo sense for system (1), we get

$\begin{array}{l} {_0^{ABC}}{D}_{t}^{\alpha }{y}_{1} = -{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{2} = {M}_{1}{y}_{1}-{M}_{5}{y}_{2}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{3} = -{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{4} = {M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{5} = -{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{6} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{7} = {M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}, \\ {_0^{ABC}}{D}_{t}^{\alpha }{y}_{8} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}. \end{array}$

(2)

Here ${}_{O}{}^{ABC}{D}_{t}^{\alpha }$ is the Atanagana-Baleanue Caputo sense fractional derivative with $0 < \alpha \le 1$ .

With given initial conditions

${y}_{i}\left(0\right)\ge 0, i = \mathrm{1, 2}, 3, \dots , 8$

(3)

Theorem 3.1: The solution of the proposed fractional-order model (1) along initial conditions is unique and bounded in R₊⁸.

Proof: In (1), we can get its existence and uniqueness on the time interval (0, ∞). Afterwards, we need to show that the non-negative region R₊⁸ is a positively invariant region. For this

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{1 = 0} = {M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{2 = 0} = {M}_{1}{y}_{1}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{3 = 0} = {M}_{2}{y}_{4}+{M}_{7}{y}_{5}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{4 = 0} = {M}_{3}{y}_{2}+{M}_{8}{y}_{3}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{5 = 0} = {M}_{2}{y}_{6}-{M}_{2}{y}_{7}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{6 = 0} = {M}_{12}{y}_{4}+{M}_{8}{y}_{5}+{M}_{12}{y}_{7}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{7 = 0} = {M}_{11}{y}_{6}{y}_{8}\ge 0,$

${_0^{ABC}}{D}_{t}^{\alpha }{y}_{1}|{y}_{8 = 0} = {M}_{13}{y}_{7}\ge 0$

If ( ${y}_{1}$ (0)), ( ${y}_{2}$ (0)), ( ${y}_{3}$ (0)), ( ${y}_{4}$ (0)), ( ${y}_{5}$ (0)), ( ${y}_{6}$ (0)), ( ${y}_{7}$ (0)), ( ${y}_{8}$ (0)) $\epsilon$ ${R}_{+}^{8}$ , then from above expression, the solution cannot escape from the hyperplane. Also on each hyperplane bounding the non-negative orthant, the vector field points into ${R}_{+}^{8}$ , i.e., the domain ${R}_{+}^{8}$ is a positively invariant set.

Now, with the help of Sumudu transform definition, we get

$\begin{array}{l} Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{1}\left(t\right)-{y}_{1}\left(0\right)\right\} = ST\left[-{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{2}\left(t\right)-{y}_{2}\left(0\right)\right\} = ST\left[{M}_{1}{y}_{1}-{M}_{5}{y}_{2}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{3}\left(t\right)-{y}_{3}\left(0\right)\right\} = ST\left[-{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{4}\left(t\right)-{y}_{4}\left(0\right)\right\} = ST\left[{M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{5}\left(t\right)-{y}_{5}\left(0\right)\right\} = ST\left[-{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{6}\left(t\right)-{y}_{6}\left(0\right)\right\} = ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{7}\left(t\right)-{y}_{7}\left(0\right)\right\} = ST\left[{M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}\right], \\ Q{E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)ST\left\{{y}_{8}\left(t\right)-{y}_{8}\left(0\right)\right\} = ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}\right]. \end{array}$

(4)

Where $Q = \frac{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right)}{1-\alpha }$

Rearranging, we get

$\begin{array}{l} ST\left({y}_{1}\left(t\right)\right) = {y}_{1}\left(0\right)+H\times ST\left[-{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}\right], \\ ST\left({y}_{2}\left(t\right)\right) = {y}_{2}\left(0\right)+H\times ST\left[{M}_{1}{y}_{1}-{M}_{5}{y}_{2}\right], \\ ST\left({y}_{3}\left(t\right)\right) = {y}_{3}\left(0\right)+H\times ST\left[-{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}\right], \\ ST\left({y}_{4}\left(t\right)\right) = {y}_{4}\left(0\right)+H\times ST\left[{M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}\right], \\ ST\left({y}_{5}\left(t\right)\right) = {y}_{5}\left(0\right)+H\times ST\left[-{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}\right], \\ ST\left({y}_{6}\left(t\right)\right) = {y}_{6}\left(0\right)+H\times ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}\right], \\ ST\left({y}_{7}\left(t\right)\right) = {y}_{7}\left(0\right)+H\times ST\left[{M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}\right], \\ ST\left({y}_{8}\left(t\right)\right) = {y}_{8}\left(0\right)+H\times ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}\right]. \end{array}$

(5)

Using the inverse Sumudu transform on both sides of the system (5), we obtain

$\begin{array}{l} {y}_{1}\left(t\right) = {y}_{1}\left(0\right)+{ST}^{-1}\left[H\times ST\left[-{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}\right]\right], \\ {y}_{2}\left(t\right) = {y}_{2}\left(0\right)+{ST}^{-1}\left[H\times ST\left[{M}_{1}{y}_{1}-{M}_{5}{y}_{2}\right]\right], \\ {y}_{3}\left(t\right) = {y}_{3}\left(0\right)+{ST}^{-1}\left[H\times ST\left[-{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}\right]\right], \\ {y}_{4}\left(t\right) = {y}_{4}\left(0\right)+{ST}^{-1}\left[H\times ST\left[{M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}\right]\right], \\ {y}_{5}\left(t\right) = {y}_{5}\left(0\right)+{ST}^{-1}\left[H\times ST\left[-{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}\right]\right], \\ {y}_{6}\left(t\right) = {y}_{6}\left(0\right)+{ST}^{-1}\left[H\times ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}\right]\right], \\ {y}_{7}\left(t\right) = {y}_{7}\left(0\right)+{ST}^{-1}\left[H\times ST\left[{M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}\right]\right], \\ {y}_{8}\left(t\right) = {y}_{8}\left(0\right)+{ST}^{-1}\left[H\times ST\left[-{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}\right]\right]. \end{array}$

(6)

We next obtain the following recursive formula.

$\begin{array}{l} {{y}_{1}}_{\left(n+1\right)}\left(t\right) = {{y}_{1}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{-{M}_{1}{{y}_{1}}_{\left(n\right)}+{M}_{2}{{y}_{2}}_{\left(n\right)}+{M}_{3}{{y}_{3}}_{\left(n\right)}+{M}_{4}\right\}\right], \\ {{y}_{2}}_{\left(n+1\right)}\left(t\right) = {{y}_{2}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{{M}_{1}{{y}_{1}}_{\left(n\right)}-{M}_{5}{{y}_{2}}_{\left(n\right)}\right\}\right], \\ {{y}_{3}}_{\left(n+1\right)}\left(t\right) = {{y}_{3}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{-{M}_{6}{{y}_{3}}_{\left(n\right)}+{M}_{2}{{y}_{4}}_{\left(n\right)}+{M}_{7}{{y}_{5}}_{\left(n\right)}\right\}\right], \\ {{y}_{4}}_{\left(n+1\right)}\left(t\right) = {{y}_{4}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{{M}_{3}{{y}_{2}}_{\left(n\right)}+{M}_{8}{{y}_{3}}_{\left(n\right)}-{M}_{9}{{y}_{4}}_{\left(n\right)}\right\}\right], \\ {{y}_{5}}_{\left(n+1\right)}\left(t\right) = {{y}_{5}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{-{M}_{10}{{y}_{5}}_{\left(n\right)}+{M}_{2}{{y}_{6}}_{\left(n\right)}-{M}_{2}{{y}_{7}}_{\left(n\right)}\right\}\right], \\ {{y}_{6}}_{\left(n+1\right)}\left(t\right) = {{y}_{6}}_{\left(n\right)}\left(0\right)+{ST}^{-1}[H\times ST\{-{M}_{11}{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}+{M}_{12}{{y}_{4}}_{\left(n\right)}+{M}_{8}{{y}_{5}}_{\left(n\right)}-{M}_{2}{{y}_{6}}_{\left(n\right)}+\\ {M}_{12}{{y}_{7}}_{\left(n\right)}\}], \\ {{y}_{7}}_{\left(n+1\right)}\left(t\right) = {{y}_{7}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{{M}_{11}{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{M}_{13}{{y}_{7}}_{\left(n\right)}\right\}\right], \\ {{y}_{8}}_{\left(n+1\right)}\left(t\right) = {{y}_{8}}_{\left(n\right)}\left(0\right)+{ST}^{-1}\left[H\times ST\left\{-{M}_{11}{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}+{M}_{13}{{y}_{7}}_{\left(n\right)}\right\}\right]. \end{array}$

(7)

Where $H = \frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}$

And the solution of system is provided by

${y}_{1}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{1}}_{\left(n\right)}\left(t\right), \ \ \ \ {y}_{2}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{2}}_{\left(n\right)}\left(t\right), \ \ \ \ {y}_{3}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{3}}_{\left(n\right)}\left(t\right),$

${y}_{4}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{4}}_{\left(n\right)}\left(t\right), \ \ \ \ {y}_{5}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{5}}_{\left(n\right)}\left(t\right), \ \ \ \ {y}_{6}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{6}}_{\left(n\right)}\left(t\right),$

${y}_{7}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{7}}_{\left(n\right)}\left(t\right), \ \ \ \ {y}_{8}\left(t\right) = \underset{n\to \infty }{\mathrm{lim}}{{y}_{8}}_{\left(n\right)}\left(t\right).$

4. Fixed-point theorem for stability analysis of iteration method

Theorem 4.1: Define K be a self-map is given by

$\begin{array}{l} K\left[ {{y_1}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_1}_{\left( {n + 1} \right)}\left( t \right) = {y_1}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ - {M_1}{y_1}_{\left( n \right)} + \\ {M_2}{y_2}_{\left( n \right)} + {M_3}{y_3}_{\left( n \right)} + {M_4}\} ], \\ K\left[ {{y_2}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_2}_{\left( {n + 1} \right)}\left( t \right) = {y_2}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ {M_1}{y_1}_{\left( n \right)}-\\ {M_5}{y_2}_{\left( n \right)}\} ], \\ K\left[ {{y_3}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_3}_{\left( {n + 1} \right)}\left( t \right) = {y_3}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ - {M_6}{y_3}_{\left( n \right)} + \\ {M_2}{y_4}_{\left( n \right)} + {M_7}{y_5}_{\left( n \right)}\} ], \\ K\left[ {{y_4}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_4}_{\left( {n + 1} \right)}\left( t \right) = {y_4}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ {M_3}{y_2}_{\left( n \right)} + {M_8}{y_3}_{\left( n \right)} - \\ {M_9}{y_4}_{\left( n \right)}\} ], \\ K\left[ {{y_5}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_5}_{\left( {n + 1} \right)}\left( t \right) = {y_5}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ - {M_{10}}{y_5}_{\left( n \right)} + \\ {M_2}{y_{6(n)}} - {M_2}{y_7}_{\left( n \right)}\} ], \\ K\left[ {{y_6}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_6}_{\left( {n + 1} \right)}\left( t \right) = {y_6}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ - {M_{11}}{y_6}_{\left( n \right)}{y_8}_{\left( n \right)} + \\ {M_{12}}{y_4}_{\left( n \right)} + {M_8}{y_5}_{\left( n \right)} - {M_2}{y_6}_{\left( n \right)} + {M_{12}}{y_7}_{\left( n \right)}\} ], \\ K\left[ {{y_7}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_7}_{\left( {n + 1} \right)}\left( t \right) = {y_7}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ {M_{11}}{y_6}_{\left( n \right)}{y_8}_{\left( n \right)} - \\ {M_{13}}{y_7}_{\left( n \right)}\} ], \\ K\left[ {{y_8}_{\left( {n + 1} \right)}\left( t \right)} \right] = {y_8}_{\left( {n + 1} \right)}\left( t \right) = {y_8}_{\left( n \right)}\left( 0 \right) + S{T^{ - 1}}[\frac{{1 - \alpha }}{{M\left( \alpha \right)\alpha \Gamma \left( {\alpha + 1} \right){E_\alpha }\left( { - \frac{1}{{1 - \alpha }}{P^\alpha }} \right)}} \times ST\{ - {M_{11}}{y_6}_{\left( n \right)}{y_8}_{\left( n \right)} + \\ {M_{13}}{y_7}_{\left( n \right)}\} ]. \end{array}$

(8)

Proof: By using triangular inequality with the definition of norms, we get

$\begin{array}{l} ||K\left[{{y}_{1}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{1}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{1}}_{\left(n\right)}\left(t\right)-{{y}_{1}}_{\left(m\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times\\ ST\left\{-{M}_{1}||{{y}_{1}}_{\left(n\right)}-{{y}_{1}}_{\left(m\right)}||+{M}_{2}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||+{M}_{3}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||+{M}_{4}\right\}], \\ ||K\left[{{y}_{2}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{2}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{2}}_{\left(n\right)}\left(t\right)-{{y}_{2}}_{\left(m\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times \\ ST\left\{{M}_{1}||{{y}_{1}}_{\left(n\right)}-{{y}_{1}}_{\left(m\right)}||-{M}_{5}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||\right\}], \\ ||K\left[{{y}_{3}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{3}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{3}}_{\left(n\right)}\left(t\right)-{{y}_{3}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times\\ ST\left\{-{M}_{6}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||+{M}_{2}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||+{M}_{7}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||\right\}], \\ ||K\left[{{y}_{4}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{4}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{4}}_{\left(n\right)}\left(t\right)-{{y}_{4}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times \\ ST\left\{{M}_{3}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||+{M}_{8}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||-{M}_{9}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||\right\}], \\ ||K\left[{{y}_{5}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{5}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{5}}_{\left(n\right)}\left(t\right)-{{y}_{5}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times \\ ST\left\{-{M}_{10}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||+{M}_{2}||{{y}_{6}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}||-{M}_{2}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\right\}], \\ ||K\left[{{y}_{6}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{6}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{6}}_{\left(n\right)}\left(t\right)-{{y}_{6}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times\\ ST\{-{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||+{M}_{12}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||+{M}_{8}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||- \\ {M}_{2}||{{y}_{6}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}||+{M}_{12}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\}], \\ ||K\left[{{y}_{7}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{7}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{7}}_{\left(n\right)}\left(t\right)-{{y}_{7}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times\\ ST\left\{{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||-{M}_{13}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\right\}], \\ ||K\left[{{y}_{8}}_{\left(n\right)}\left(t\right)\right]-K\left[{{y}_{8}}_{\left(m\right)}\left(t\right)\right]||\le ||{{y}_{8}}_{\left(n\right)}\left(t\right)-{{y}_{8}}_{\left(n\right)}\left(t\right)||+{ST}^{-1}[\frac{1-\alpha }{M\left(\alpha \right)\alpha \mathrm{\Gamma }\left(\alpha +1\right){E}_{\alpha }\left(-\frac{1}{1-\alpha }{P}^{\alpha }\right)}\times\\ ST\left\{-{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||+{M}_{13}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\right\}]. \end{array}$

Hence satisfied given conditions.

$\begin{array}{l} \theta = \left(\mathrm{0, 0}, \mathrm{0, 0}, \mathrm{0, 0}, \mathrm{0, 0}\right), \theta = \\\left\{\begin{array}{c}||{{y}_{1}}_{\left(n\right)}\left(t\right)-{{y}_{1}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{1}}_{\left(n\right)}\left(t\right)+{{y}_{1}}_{\left(m\right)}\left(t\right)\right)||-{M}_{1}||{{y}_{1}}_{\left(n\right)}-{{y}_{1}}_{\left(m\right)}||+{M}_{2}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||\\ +{M}_{3}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||+{M}_{4}\\ ||{{y}_{2}}_{\left(n\right)}\left(t\right)-{{y}_{2}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{2}}_{\left(n\right)}\left(t\right)+{{y}_{2}}_{\left(m\right)}\left(t\right)\right)||+{M}_{1}||{{y}_{1}}_{\left(n\right)}-{{y}_{1}}_{\left(m\right)}||-{M}_{5}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||\\ ||{{y}_{3}}_{\left(n\right)}\left(t\right)-{{y}_{3}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{3}}_{\left(n\right)}\left(t\right)+{{y}_{3}}_{\left(m\right)}\left(t\right)\right)||-{M}_{6}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||+{M}_{2}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||\\ +{M}_{7}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||\\ ||{{y}_{4}}_{\left(n\right)}\left(t\right)-{{y}_{4}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{4}}_{\left(n\right)}\left(t\right)+{{y}_{4}}_{\left(m\right)}\left(t\right)\right)||+{M}_{3}||{{y}_{2}}_{\left(n\right)}-{{y}_{2}}_{\left(m\right)}||+{M}_{8}||{{y}_{3}}_{\left(n\right)}-{{y}_{3}}_{\left(m\right)}||\\ -{M}_{9}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||\\ ||{{y}_{5}}_{\left(n\right)}\left(t\right)-{{y}_{5}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{5}}_{\left(n\right)}\left(t\right)+{{y}_{5}}_{\left(m\right)}\left(t\right)\right)||-{M}_{10}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||+{M}_{2}||{{y}_{6}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}||\\ -{M}_{2}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\\ ||{{y}_{6}}_{\left(n\right)}\left(t\right)-{{y}_{6}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{6}}_{\left(n\right)}\left(t\right)+{{y}_{6}}_{\left(m\right)}\left(t\right)\right)||-{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||\\ +{M}_{12}||{{y}_{4}}_{\left(n\right)}-{{y}_{4}}_{\left(m\right)}||+{M}_{8}||{{y}_{5}}_{\left(n\right)}-{{y}_{5}}_{\left(m\right)}||-{M}_{2}||{{y}_{6}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}||+{M}_{12}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\\ ||{{y}_{7}}_{\left(n\right)}\left(t\right)-{{y}_{7}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{7}}_{\left(n\right)}\left(t\right)+{{y}_{7}}_{\left(m\right)}\left(t\right)\right)||+{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||\\ -{M}_{13}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\\ ||{{y}_{8}}_{\left(n\right)}\left(t\right)-{{y}_{8}}_{\left(m\right)}\left(t\right)||\times ||-\left({{y}_{8}}_{\left(n\right)}\left(t\right)+{{y}_{8}}_{\left(m\right)}\left(t\right)\right)||-{M}_{11}||{{y}_{6}}_{\left(n\right)}{{y}_{8}}_{\left(n\right)}-{{y}_{6}}_{\left(m\right)}{{y}_{8}}_{\left(m\right)}||\\ +{M}_{13}||{{y}_{7}}_{\left(n\right)}-{{y}_{7}}_{\left(m\right)}||\end{array}\right. \end{array}$

Hence the system is stable.

Theorem 4.2: Unique singular solution with the iterative method for the special solution of system $\left(2\right).$

Proof: Considering the Hilbert space $H = {L}^{2}\left(\left(p, q\right)\times \left(0, T\right)\right)$ which can be defined as

$h:\left(p, q\right)\times \left(0, T\right)\to \mathbb{R}, \iint ghdgdh < \infty .$

For this purpose, we consider the following operator

$\theta \left(\mathrm{0, 0}, \mathrm{0, 0}, \mathrm{0, 0}, \mathrm{0, 0}\right), \theta = \left\{\begin{array}{c}-{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}, \\ {M}_{1}{y}_{1}-{M}_{5}{y}_{2}, \\ -{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}, \\ {M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}, \\ -{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}, \\ -{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}, \\ {M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}, \\ -{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}.\end{array}\right.$

By using inner product, we get

$\begin{array}{l} T(({{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}, {{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}, {{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}, {{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}, {{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}, {{y}_{6}}_{\left(61\right)}-\\ \ \ \ \ \ \ \ \ {{y}_{6}}_{\left(62\right)}, {{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}, {{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}), \left({V}_{1}, {V}_{2}, {V}_{3}, {V}_{4}, {V}_{5}, {V}_{6}, {V}_{7}, {V}_{8}\right)). \end{array}$

Where

$({{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}, {{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}, {{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)},$ ${{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}, {{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)},$ ${{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)},$ ${{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)},$ ${{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}),$ are the special solutions of the system. Taking into account the inner function and the norm, we have

$\begin{array}{l} \left\{ { - {M_1}\left( {{y_1}_{\left( {11} \right)} - {y_1}_{\left( {12} \right)}} \right) + {M_2}\left( {{y_2}_{\left( {21} \right)} - {y_2}_{\left( {22} \right)}} \right) + {M_3}\left( {{y_3}_{\left( {31} \right)} - {y_3}_{\left( {32} \right)}} \right) + {M_4}, {V_1}} \right\} \le {M_1}||{y_1}_{\left( {11} \right)} - \\ {y_1}_{\left( {12} \right)}||||{V_1}|| + {M_2}||{y_2}_{\left( {21} \right)} - {y_2}_{\left( {22} \right)}||||{V_1}|| + {M_3}||{y_3}_{\left( {31} \right)} - {y_3}_{\left( {32} \right)}||||{V_1}|| + {M_4}||{V_1}||, \\ \left\{ {{M_1}\left( {{y_1}_{\left( {11} \right)} - {y_1}_{\left( {12} \right)}} \right) - {M_5}\left( {{y_2}_{\left( {21} \right)} - {y_2}_{\left( {22} \right)}} \right), {V_2}} \right\} \le {M_1}||{y_1}_{\left( {11} \right)} - {y_1}_{\left( {12} \right)}||||{V_2}|| + {M_5}||{y_2}_{\left( {21} \right)} - \\ {y_2}_{\left( {22} \right)}||||{V_2}||, \\ \left\{ { - {M_6}\left( {{y_3}_{\left( {31} \right)} - {y_3}_{\left( {32} \right)}} \right) + {M_2}\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right) + {M_7}\left( {{y_5}_{\left( {51} \right)} - {y_5}_{\left( {52} \right)}} \right), {V_3}} \right\} \le {M_6}||({y_3}_{\left( {31} \right)} - \\ {y_3}_{\left( {32} \right)})||||{V_3}|| + {M_2}||\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right)||||{V_3}|| + {M_7}||\left( {{y_5}_{\left( {51} \right)} - {y_5}_{\left( {52} \right)}} \right)||||{V_3}||, \\ \left\{ {{M_3}\left( {{y_2}_{\left( {21} \right)} - {y_2}_{\left( {22} \right)}} \right) + {M_8}\left( {{y_3}_{\left( {31} \right)} - {y_3}_{\left( {32} \right)}} \right) - {M_9}\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right), {V_4}} \right\} \le {M_3}||({y_2}_{\left( {21} \right)} - \\ {y_2}_{\left( {22} \right)})||||{V_4}|| + {M_8}||\left( {{y_3}_{\left( {31} \right)} - {y_3}_{\left( {32} \right)}} \right)||||{V_4}|| + {M_9}||\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right)||||{V_4}||, \\ \left\{ { - {M_{10}}\left( {{y_5}_{\left( {51} \right)} - {y_5}_{\left( {52} \right)}} \right) + {M_2}\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right) - {M_2}\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right), {V_5}} \right\} \le {M_{10}}||({y_5}_{\left( {51} \right)} - \\ {y_5}_{\left( {52} \right)})||||{V_5}|| + {M_2}||\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right)||||{V_5}|| + {M_2}||\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right)||||{V_5}||, \\ \{ - {M_{11}}\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right)\left( {{y_8}_{\left( {81} \right)} - {y_8}_{\left( {82} \right)}} \right) + {M_{12}}\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right) + {M_8}\left( {{y_5}_{\left( {51} \right)} - {y_5}_{\left( {52} \right)}} \right) - \\ {M_2}\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right) + {M_{12}}\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right), {V_6}\} \le {M_{11}}||\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right)||||({y_8}_{\left( {81} \right)} - \\ {y_8}_{\left( {82} \right)})||||{V_6}|| + {M_{12}}||\left( {{y_4}_{\left( {41} \right)} - {y_4}_{\left( {42} \right)}} \right)||||{V_6}|| + {M_8}||\left( {{y_5}_{\left( {51} \right)} - {y_5}_{\left( {52} \right)}} \right)||||{V_6}|| + {M_2}||({y_6}_{\left( {61} \right)} - \\ {y_6}_{\left( {62} \right)})||||{V_6}|| + {M_{12}}||\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right)||||{V_6}||, \\ \left\{ {{M_{11}}\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right)\left( {{y_8}_{\left( {81} \right)} - {y_8}_{\left( {82} \right)}} \right) - {M_{13}}\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right), {V_7}} \right\} \le {M_{11}}||({y_6}_{(61)} - \\ {y_6}_{\left( {62} \right)})||||\left( {{y_8}_{\left( {81} \right)} - {y_8}_{\left( {82} \right)}} \right)||||{V_7}|| + {M_{13}}||\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right)||||{V_7}||, \\ \left\{ { - {M_{11}}\left( {{y_6}_{\left( {61} \right)} - {y_6}_{\left( {62} \right)}} \right)\left( {{y_8}_{\left( {81} \right)} - {y_8}_{\left( {82} \right)}} \right) + {M_{13}}\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right), {V_8}} \right\} \le {M_{11}}||({y_6}_{(61)} - \\ {y_6}_{\left( {62} \right)})||||\left( {{y_8}_{\left( {81} \right)} - {y_8}_{\left( {82} \right)}} \right)||||{V_8}|| + {M_{13}}||\left( {{y_7}_{\left( {71} \right)} - {y_7}_{\left( {72} \right)}} \right)||||{V_8}||. \end{array}$

In the case for large number ${e}_{1}, {e}_{2}, {e}_{3}, {e}_{4}, {e}_{5}, {e}_{6}, {e}_{7}and{e}_{8},$ both solutions happen to be converged to the exact solution. Employing the topology concept, we can obtain eight positive very small parameters $\left({\chi }_{{e}_{1}}, {\chi }_{{e}_{2}}, {\chi }_{{e}_{3}}, {\chi }_{{e}_{4}}, {\chi }_{{e}_{5}}, {\chi }_{{e}_{6}}, {\chi }_{{e}_{7}}and{\chi }_{{e}_{8}}\right).$

$||{y}_{1}-{{y}_{1}}_{\left(11\right)}||, ||{y}_{1}-{{y}_{1}}_{\left(12\right)}||\le \frac{{\chi }_{{e}_{1}}}{\varpi }, ||{y}_{2}-{{y}_{2}}_{\left(21\right)}||, ||{y}_{2}-{{y}_{2}}_{\left(22\right)}||\le \frac{{\chi }_{{e}_{2}}}{\varsigma },$

$||{y}_{3}-{{y}_{3}}_{\left(31\right)}||, ||{y}_{3}-{{y}_{3}}_{\left(32\right)}||\le \frac{{\chi }_{{e}_{3}}}{\upsilon }, ||{y}_{4}-{{y}_{4}}_{\left(41\right)}||, ||{y}_{4}-{{y}_{4}}_{\left(42\right)}||\le \frac{{\chi }_{{e}_{4}}}{\kappa },$

$||{y}_{5}-{{y}_{5}}_{\left(51\right)}||, ||{y}_{5}-{{y}_{5}}_{\left(52\right)}||\le \frac{{\chi }_{{e}_{5}}}{\varrho }, ||{y}_{6}-{{y}_{6}}_{\left(61\right)}||, ||{y}_{6}-{{y}_{6}}_{\left(62\right)}||\le \frac{{\chi }_{{e}_{6}}}{\zeta },$

$||{y}_{7}-{{y}_{7}}_{\left(71\right)}||, ||{y}_{7}-{{y}_{7}}_{\left(72\right)}||\le \frac{{\chi }_{{e}_{7}}}{\nu }, ||{y}_{8}-{{y}_{8}}_{\left(81\right)}||, ||{y}_{8}-{{y}_{8}}_{\left(82\right)}||\le \frac{{\chi }_{{e}_{8}}}{\varepsilon }.$

Where

$\varpi = 8\left({M}_{1}||{{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}||+{M}_{2}||{{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}||+{M}_{3}||{{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}||+{M}_{4}\right)||{V}_{1}||$

$\varsigma = 8\left({M}_{1}||{{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}||+{M}_{5}||{{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}||\right)||{V}_{2}||$

$\upsilon = 8\left({M}_{6}||\left({{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}\right)||+{M}_{2}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||+{M}_{7}||\left({{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}\right)||\right)||{V}_{3}||$

$\kappa = 8\left({M}_{3}||\left({{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}\right)||+{M}_{8}||\left({{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}\right)||+{M}_{9}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||\right)||{V}_{4}||$

$\varrho = 8\left({M}_{10}||\left({{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}\right)||+{M}_{2}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||+{M}_{2}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||\right)||{V}_{5}||$

$\begin{array}{l} \zeta = 8({M}_{11}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)||+{M}_{12}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||+{M}_{8}||({{y}_{5}}_{\left(51\right)}-\\ {{y}_{5}}_{\left(52\right)})||+{M}_{2}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||+{M}_{12}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||)||{V}_{6}|| \end{array}$

$\nu = 8\left({M}_{11}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)||+{M}_{13}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||\right)||{V}_{7}||$

$\varepsilon = 8\left({M}_{11}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)||+{M}_{13}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||\right)||{V}_{8}||$

But, it is obvious that

$\left({M}_{1}||{{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}||+{M}_{2}||{{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}||+{M}_{3}||{{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}||+{M}_{4}\right)\ne 0$

$\left({M}_{1}||{{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}||+{M}_{5}||{{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}||\right)\ne 0$

$\left({M}_{6}||\left({{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}\right)||+{M}_{2}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||+{M}_{7}||\left({{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}\right)||\right)\ne 0$

$\left({M}_{3}||\left({{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}\right)||+{M}_{8}||\left({{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}\right)||+{M}_{9}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||\right)\ne 0$

$\left({M}_{10}||\left({{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}\right)||+{M}_{2}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||+{M}_{2}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||\right)\ne 0$

$\begin{array}{l} ({M}_{11}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)||+{M}_{12}||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)||+{M}_{8}||({{y}_{5}}_{\left(51\right)}-\\ {{y}_{5}}_{\left(52\right)})||+{M}_{2}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||+{M}_{12}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||)\ne 0 \end{array}$

$\left({M}_{11}||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)||||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)||+{M}_{13}||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)||\right)\ne 0$

where $||{V}_{1}||, ||{V}_{2}||, ||{V}_{3}||, ||{V}_{4}||, ||{V}_{5}||, ||{V}_{6}||, ||{V}_{7}||, ||{V}_{8}||\ne 0$ .

Therefore, we have

$||{{y}_{1}}_{\left(11\right)}-{{y}_{1}}_{\left(12\right)}|| = 0, ||{{y}_{2}}_{\left(21\right)}-{{y}_{2}}_{\left(22\right)}|| = 0, ||{{y}_{3}}_{\left(31\right)}-{{y}_{3}}_{\left(32\right)}|| = 0,$

$||\left({{y}_{4}}_{\left(41\right)}-{{y}_{4}}_{\left(42\right)}\right)|| = 0, ||\left({{y}_{5}}_{\left(51\right)}-{{y}_{5}}_{\left(52\right)}\right)|| = 0, ||\left({{y}_{6}}_{\left(61\right)}-{{y}_{6}}_{\left(62\right)}\right)|| = 0,$

$||\left({{y}_{7}}_{\left(71\right)}-{{y}_{7}}_{\left(72\right)}\right)|| = 0, ||\left({{y}_{8}}_{\left(81\right)}-{{y}_{8}}_{\left(82\right)}\right)|| = 0.$

Which yields that

$\begin{array}{*{20}{c}} { {{y}_{1}}_{\left(11\right)} = {{y}_{1}}_{\left(12\right)}, {{y}_{2}}_{\left(21\right)} = {{y}_{2}}_{\left(22\right)}, {{y}_{3}}_{\left(31\right)} = {{y}_{3}}_{\left(32\right)}, {{y}_{4}}_{\left(41\right)} = {{y}_{4}}_{\left(42\right)}, {{y}_{5}}_{\left(51\right)} = {{y}_{5}}_{\left(52\right)},{{y}_{6}}_{\left(61\right)} = }\\ {{{y}_{6}}_{\left(62\right)}, {{y}_{7}}_{\left(71\right)} = {{y}_{7}}_{\left(72\right)}, {{y}_{8}}_{\left(81\right)} = {{y}_{8}}_{\left(82\right)} } \end{array}$

This completes the proof of uniqueness.

An operator $B:Z\to Z$ can be defined as:

$B\left(\varphi \right)\left(t\right) = \varphi \left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}(1-{\alpha }_{1})}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, \varphi \left(t\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(t, \varphi \left(t\right)\right)d\lambda$

(10)

If $£ \left(t, \varphi \left(t\right)\right)$ satisfies the Lipschitz condition and the following extension then

● For every $\varphi \in Z$ there exists constants ${L}_{£ } > 0$ and ${M}_{£ }$ such that

$\left|£ \left(t, \varphi \left(t\right)\right)|\le {L}_{£ }\right|\varphi \left(t\right)|+{M}_{£ }$

(11)

● For every $\varphi, \overline{\varphi }\in Z$ , there exists a constant ${M}_{£ } > 0$ such that

$\left|£ \left(t, \varphi \left(t\right)\right)-£ \left(t, \overline{\varphi }\left(t\right)\right)\left|\right|\le {M}_{£ }\right|\varphi \left(t\right)-\overline{\varphi }\left(t\right)|$

(12)

Theorem 4.2: If the condition of (11) holds then for the function $£ :[0, T]\times Z\to R$ there exists at least one solution for the (1).

Proof: Since $£$ in (10) is continuous function, so $B$ is also a continuous. Assume $M = \left\{\varphi \in \left|\left|\varphi \right|\right|\le R, R > 0\right\}$ , then for $\varphi \in Z,$ we have

$\begin{array}{l} B\left(\varphi \right)\left(t\right) = {max}_{t\in \left[0, T\right]}|\varphi \left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, \varphi \left(t\right)\right)\\ \;\;\;\;\;\;\;\;\;\;+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(t, \varphi \left(t\right)\right)d\lambda \end{array}$

$\begin{array}{l} \le |\varphi \left(0\right)+\frac{\mathsf{μ} {T}^{\mathsf{μ} -1}(1-{\alpha }_{1})}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}\left({L}_{£ }\left|\right|\varphi \left(t\right)\left|\right|+{M}_{£ }\right)\\ \;\;\;\;\;\;\;\;\;\;+{max}_{t\in [0, T]}\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(t, \varphi \left(t\right)\right)d\lambda | \end{array}$

$\begin{array}{l} \le \varphi \left(0\right)+\frac{\mathsf{μ} {T}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}\left({L}_{£ }\left|\left|\varphi \left(t\right)\right|\right|+{M}_{£ }\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{\left({L}_{£ }\left|\left|\varphi \left(t\right)\right|\right|+{M}_{£ }\right)T}^{\mathsf{μ} +{\alpha }_{1-1}}M\left(\mathsf{μ} , {\alpha }_{1}\right)\\ \le R. \end{array}$

Hence, $B$ is uniformly bounded, and $M\left(\mathsf{μ}, {\alpha }_{1}\right)$ is a beta function. For equicontinuity of $B$ , we take ${t}_{1} < {t}_{2}\le T,$ then consider

$B\left(\varphi \right)\left({t}_{2}\right)-B\left(\varphi \right)\left({t}_{1}\right) = |\frac{\mathsf{μ} {{t}_{2}}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left({t}_{2}, \varphi \left({t}_{2}\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}$

$\begin{array}{l} \underset{0}{\overset{{t}_{2}}{\int }}{\lambda }^{\mathsf{μ} -1}{\left({t}_{2}-\lambda \right)}^{\mathsf{μ} -1}£ \left(t, \varphi \left(t\right)\right)d\lambda -\frac{\mathsf{μ} {{t}_{1}}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left({t}_{1}, \varphi \left({t}_{1}\right)\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{{t}_{2}}{\int }}{\lambda }^{\mathsf{μ} -1}{\left({t}_{1}-\lambda \right)}^{\mathsf{μ} -1}£ \left(t, \varphi \left(t\right)\right)d\lambda | \end{array}$

$\le \frac{\mathsf{μ} {{t}_{2}}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}\left({L}_{£ }\left|\varphi \left(t\right)\right|+{M}_{£ }\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{\left({L}_{£ }\left|\varphi \left(t\right)\right|+{M}_{£ }\right){t}_{2}}^{\mathsf{μ} +{\alpha }_{1-1}}M\left(\mathsf{μ} , {\alpha }_{1}\right)$

$\frac{\mathsf{μ} {{t}_{1}}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}\left({L}_{£ }\left|\varphi \left(t\right)\right|+{M}_{£ }\right)-\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{\left({L}_{£ }\left|\varphi \left(t\right)\right|+{M}_{£ }\right){t}_{1}}^{\mathsf{μ} +{\alpha }_{1-1}}M\left(\mathsf{μ} , {\alpha }_{1}\right)$

If ${t}_{1}\to {t}_{2}$ then $\left|\left|B\left(\varphi \right)\left({t}_{2}\right)-B\left(\varphi \right)\left({t}_{1}\right)\to 0\right|\right|$ Consequently $\left|\left|B\left(\varphi \right)\left({t}_{2}\right)-B\left(\varphi \right)\left({t}_{1}\right)\to 0\right|\right|, as{t}_{1}\to {t}_{2}.$ Hence $B$ is equicontinous. Thus, by Arzela-Ascoli theorem $B$ is completely continuous. Consequently, by the result of Schauder's fixed point, it has at least one solution.

Theorem 4.3: If $\eta = \frac{\mathsf{μ} {T}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{T}^{\mathsf{μ} +{\alpha }_{1}-1}M\left(\mathsf{μ}, {\alpha }_{1}\right){M}_{£ } < 1$ and the condition (12) holds, then $\eta$ has a unique solution.

Proof: For $\varphi, \overline{\varphi }\in Z$ , we have

$\left|B\left(\varphi \right)-B\left(\overline{\varphi }\right)\right| = {max}_{t\in \left[0, T\right]}|\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}\left[£ \left(t, \varphi \left(t\right)\right)|-£ \left(t, \overline{\varphi }\left(t\right)\right)\right]$

$+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}\left[£ \left(t, \varphi \left(t\right)\right)|-£ \left(t, \overline{\varphi }\left(t\right)\right)\right]d\lambda |$

$\le [\frac{\mathsf{μ} {T}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{T}^{\mathsf{μ} +{\alpha }_{1}-1}M\left(\mathsf{μ} , {\alpha }_{1}\right)]\left|\left|B\left(\varphi \right)-B\left(\overline{\varphi }\right)\right|\right|$

$\le \eta \left|\left|B\left(\varphi \right)-B\left(\overline{\varphi }\right)\right|\right|.$

Hence, $B$ is a contraction. So, by the principle of Banach contraction, it has a unique solution.

Ulam-Hyres stability

The proposed model is Ulam-Hyres stable if there exists ${B}_{\mathsf{μ}, {\alpha }_{1}}\ge 0$ such that for every $\varepsilon > 0$ and for every $\varphi \in (L\left[0, T\right], R)$ satisfies the following inequality ${{}^{FFM}J}_{0, t}^{\mathsf{μ}, {\alpha }_{1}}\left(\varphi \left(t\right)\right)-£ \left(t, \varphi \left(t\right)\right)\le \varepsilon, t\in \left[0, T\right]$ such that $\left|\varphi \left(t\right)-£ \left(t\right)\right|\le {B}_{\mathsf{μ}, {\alpha }_{1}}\varepsilon, t\in \left[0, T\right]$ .

Suppose a perturbation $\omega \in L\left[0, T\right], R$ then $\omega \left(0\right) = 0$ and

● For every $\varepsilon > 0 \exists \omega \left(t\right)\le \varepsilon |$

● ${{}_{0}{}^{FFM}J}_{t}^{\mathsf{μ}, {\alpha }_{1}}\left(\varphi \left(t\right)\right) = £ \left(t, \varphi \left(t\right)\right)+\omega \left(t\right).$

Lemma 4.4: The solution of the perturbed model ${{}_{0}{}^{FFM}J}_{t}^{\mathsf{μ}, {\alpha }_{1}}\left(\varphi \left(t\right)\right) = £ \left(t, \varphi \left(t\right)\right)+\omega \left(t\right), \varphi \left(0\right) = {\varphi }_{0}$ fulfills the relation

$\begin{array}{l} B\left(t\right)-[\varphi \left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}(1-{\alpha }_{1})}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, \varphi \left(t\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(\lambda , \varphi \left(\lambda \right)\right)d\lambda \\ \ \ \ \ \ \ \ \ \le {{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}\varepsilon \end{array}$

Where ${{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}\varepsilon = \frac{\mathsf{μ} {T}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{T}^{\mathsf{μ} +{\alpha }_{1}-1}M\left(\mathsf{μ}, {\alpha }_{1}\right).$

Lemma 4.5: By using condition (12) with lemma (4.4), proposed model is Ulam-Hyres stable if $\eta < 1.$

Proof: Suppose ${\alpha }_{1}\in Z$ be a solution and $\varphi \in Z$ be any solution of (1), then

$\begin{array}{l} |\varphi \left(t\right)-{\alpha }_{1}\left(t\right)| = |\varphi \left(t\right)-[{\alpha }_{1}\left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, {\alpha }_{1}\left(t\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t}{\lambda }^{\mathsf{μ} -1}{(1-}\\ \lambda )^{\mathsf{μ} -1}£ \left(\lambda , {\alpha }_{1}\left(\lambda \right)\right)d\lambda ]| \end{array}$

$\begin{array}{l} \le \left|\varphi \left(t\right)-\left[\varphi \left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, \varphi \left(t\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(\lambda , \varphi \left(\lambda \right)\right)d\lambda \right]\right|\\ \ \ \ \ \ \ \ \ \ \ \ \ +|\varphi \left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, \varphi \left(t\right)\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\underset{0}{\overset{t}{\int }}{\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(\lambda , \varphi \left(\lambda \right)\right)d\lambda | \end{array}$

$- |{\alpha }_{1}\left(0\right)+\frac{\mathsf{μ} {t}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}£ \left(t, {\alpha }_{1}\left(t\right)\right)+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\lambda }^{\mathsf{μ} -1}{\left(1-\lambda \right)}^{\mathsf{μ} -1}£ \left(\lambda , {\alpha }_{1}\left(\lambda \right)\right)d\lambda |$

$\le {{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}\varepsilon +(\frac{\mathsf{μ} {T}^{\mathsf{μ} -1}\left(1-{\alpha }_{1}\right)}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)}+\frac{\mathsf{μ} {\alpha }_{1}}{\mathrm{A}\mathrm{B}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}{T}^{\mathsf{μ} +{\alpha }_{1}-1}){L}_{£ }|\varphi \left(t\right)-{\alpha }_{1}\left(t\right)|$

$\le {{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}\varepsilon +\eta \left|\varphi \left(t\right)-{\alpha }_{1}\left(t\right)\right|.$

Consequently,

$\left|\right|\varphi -{\alpha }_{1}\left|\right|\le {{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}\varepsilon +\eta \left|\left|\varphi \left(t\right)-{\alpha }_{1}\left(t\right)\right|\right|.$

So, we can write it as

$\left|\left|\varphi -{\alpha }_{1}\right|\right|\le {B}_{{\alpha }_{1}, \mathsf{μ} }\varepsilon ,$

Where ${B}_{{\alpha }_{1}, \mathsf{μ} }\varepsilon = \frac{{{\alpha }_{1}}_{{\alpha }_{1}, \mathsf{μ} }^{*}}{1-\eta }.$ Hence the solution is Ulam-Hyres stable.

5. Fractal-fractional operator for hires problem

In this section, we present the Hires problem model $\left(1\right)$ using fractal-fractional Atangana-Baleanu derivative. We have

$\begin{array}{l} {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{1} = -{M}_{1}{y}_{1}+{M}_{2}{y}_{2}+{M}_{3}{y}_{3}+{M}_{4}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{2} = {M}_{1}{y}_{1}-{M}_{5}{y}_{2}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{3} = -{M}_{6}{y}_{3}+{M}_{2}{y}_{4}+{M}_{7}{y}_{5}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{4} = {M}_{3}{y}_{2}+{M}_{8}{y}_{3}-{M}_{9}{y}_{4}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{5} = -{M}_{10}{y}_{5}+{M}_{2}{y}_{6}-{M}_{2}{y}_{7}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{6} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{12}{y}_{4}+{M}_{8}{y}_{5}-{M}_{2}{y}_{6}+{M}_{12}{y}_{7}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{7} = {M}_{11}{y}_{6}{y}_{8}-{M}_{13}{y}_{7}, \\ {}^{FF}{D}_{0, t}^{{\alpha }_{1}, {\alpha }_{2}}{y}_{8} = -{M}_{11}{y}_{6}{y}_{8}+{M}_{13}{y}_{7}. \end{array}$

(13)

With initial conditions

$\begin{array}{l} {y}_{1}\left(0\right) = {{y}_{1}}_{\left(0\right)}, {y}_{2}\left(0\right) = {{y}_{2}}_{\left(0\right)}, {y}_{3}\left(0\right) = {{y}_{3}}_{\left(0\right)}, {y}_{4}\left(0\right) = {{y}_{4}}_{\left(0\right)}, \\ {y}_{5}\left(0\right) = {{y}_{5}}_{\left(0\right)}, {y}_{6}\left(0\right) = {{y}_{6}}_{\left(0\right)}, {y}_{7}\left(0\right) = {{y}_{7}}_{\left(0\right)}, {y}_{8}\left(0\right) = {{y}_{8}}_{\left(0\right)}. \end{array}$

We present the numerical algorithm for the fractal-fractional Hires problem model (13). The following is obtained by integrating the system (13).

$\begin{array}{l} {y}_{1}\left(t\right)-{y}_{1}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{1}{y}_{1}\left(t\right)+{M}_{2}{y}_{2}\left(t\right)+{M}_{3}{y}_{3}\left(t\right)+{M}_{4}\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\left\{-{M}_{1}{y}_{1}\left(\tau \right)+{M}_{2}{y}_{2}\left(\tau \right)+{M}_{3}{y}_{3}\left(\tau \right)+{M}_{4}\right\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{2}\left(t\right)-{y}_{2}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{1}{y}_{1}\left(t\right)-{M}_{5}{y}_{2}\left(t\right)\right\}+\frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\{\{{M}_{1}{y}_{1}\left(\tau \right)-\\ {M}_{5}{y}_{2}\left(\tau \right)\}\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{3}\left(t\right)-{y}_{3}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{6}{y}_{3}\left(t\right)+{M}_{2}{y}_{4}\left(t\right)+{M}_{7}{y}_{5}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\left\{-{M}_{6}{y}_{3}\left(\tau \right)+{M}_{2}{y}_{4}\left(\tau \right)+{M}_{7}{y}_{5}\left(\tau \right)\right\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{4}\left(t\right)-{y}_{4}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{3}{y}_{2}\left(t\right)+{M}_{8}{y}_{3}\left(t\right)-{M}_{9}{y}_{4}\left(t\right)\right\}+\frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\{{M}_{3}{y}_{2}\left(\tau \right)+\\ {M}_{8}{y}_{3}\left(\tau \right)-{M}_{9}{y}_{4}\left(\tau \right)\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{5}\left(t\right)-{y}_{5}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{10}{y}_{5}\left(t\right)+{M}_{2}{y}_{6}\left(t\right)-{M}_{2}{y}_{7}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\left\{\left\{-{M}_{10}{y}_{5}\left(\tau \right)+{M}_{2}{y}_{6}\left(\tau \right)-{M}_{2}{y}_{7}\left(\tau \right)\right\}\right\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{6}\left(t\right)-{y}_{6}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{12}{y}_{4}\left(t\right)+{M}_{8}{y}_{5}\left(t\right)-{M}_{2}{y}_{6}\left(t\right)+\\ {M}_{12}{y}_{7}\left(t\right)\}+\frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\{-{M}_{11}{y}_{6}\left(\tau \right){y}_{8}\left(\tau \right)+{M}_{12}{y}_{4}\left(\tau \right)+{M}_{8}{y}_{5}\left(\tau \right)-{M}_{2}{y}_{6}\left(\tau \right)+\\ {M}_{12}{y}_{7}\left(\tau \right)\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{7}\left(t\right)-{y}_{7}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)-{M}_{13}{y}_{7}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\left\{{M}_{11}{y}_{6}\left(\tau \right){y}_{8}\left(\tau \right)-{M}_{13}{y}_{7}\left(\tau \right)\right\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{8}\left(t\right)-{y}_{8}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{13}{y}_{7}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} {\tau }^{{\alpha }_{2}-1}\left\{\left\{-{M}_{11}{y}_{6}\left(\tau \right){y}_{8}\left(\tau \right)+{M}_{13}{y}_{7}\left(\tau \right)\right\}\right\}{\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \end{array}$

(14)

Let

$k\left(t, {y}_{1}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{1}{y}_{1}\left(t\right)+{M}_{2}{y}_{2}\left(t\right)+{M}_{3}{y}_{3}\left(t\right)+{M}_{4}\right\},$

$k\left(t, {y}_{2}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{1}{y}_{1}\left(t\right)-{M}_{5}{y}_{2}\left(t\right)\right\},$

$k\left(t, {y}_{3}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{6}{y}_{3}\left(t\right)+{M}_{2}{y}_{4}\left(t\right)+{M}_{7}{y}_{5}\left(t\right)\right\},$

$k\left(t, {y}_{4}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{3}{y}_{2}\left(t\right)+{M}_{8}{y}_{3}\left(t\right)-{M}_{9}{y}_{4}\left(t\right)\right\},$

$k\left(t, {y}_{5}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{10}{y}_{5}\left(t\right)+{M}_{2}{y}_{6}\left(t\right)-{M}_{2}{y}_{7}\left(t\right)\right\},$

$k\left(t, {y}_{6}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{12}{y}_{4}\left(t\right)+{M}_{8}{y}_{5}\left(t\right)-{M}_{2}{y}_{6}\left(t\right)+{M}_{12}{y}_{7}\left(t\right)\right\},$

$k\left(t, {y}_{7}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)-{M}_{13}{y}_{7}\left(t\right)\right\},$

$k\left(t, {y}_{8}\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{13}{y}_{7}\left(t\right)\right\}.$

Then system (14) becomes

$\begin{array}{l} {y}_{1}\left(t\right)-{y}_{1}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{1}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{1}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{2}\left(t\right)-{y}_{2}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{2}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{2}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{3}\left(t\right)-{y}_{3}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{3}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{3}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{4}\left(t\right)-{y}_{4}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{4}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{4}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{5}\left(t\right)-{y}_{5}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{5}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{5}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{6}\left(t\right)-{y}_{6}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{6}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{6}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{7}\left(t\right)-{y}_{7}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{7}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{7}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{8}\left(t\right)-{y}_{8}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left(t, {y}_{8}\left(t\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{t} k\left(\tau , {y}_{8}\left(\tau \right)\right){\left(t-\tau \right)}^{{\alpha }_{1}-1}d\tau , \end{array}$

(15)

At ${t}_{n+1} = \left(n+1\right)\Delta t,$ we have

$\begin{array}{l} {y}_{1}\left({t}_{n+1}\right)-{y}_{1}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{1}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{2}\left({t}_{n+1}\right)-{y}_{2}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{2}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{2}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{3}\left({t}_{n+1}\right)-{y}_{3}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{3}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{3}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{4}\left({t}_{n+1}\right)-{y}_{4}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{4}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{4}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{5}\left({t}_{n+1}\right)-{y}_{5}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{5}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{5}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{6}\left({t}_{n+1}\right)-{y}_{6}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{6}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{6}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{7}\left({t}_{n+1}\right)-{y}_{7}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{7}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{7}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{8}\left({t}_{n+1}\right)-{y}_{8}\left(0\right) = \frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{8}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\int_{0}^{{t}_{n+1}} k\left(\tau , {y}_{8}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau . \end{array}$

(16)

Also, we have

$\begin{array}{l} {y}_{1}\left({t}_{n+1}\right) = {y}_{1}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{1}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{2}\left({t}_{n+1}\right) = {y}_{2}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{2}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{2}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{3}\left({t}_{n+1}\right) = {y}_{3}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{3}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{3}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{4}\left({t}_{n+1}\right) = {y}_{4}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{4}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{4}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{5}\left({t}_{n+1}\right) = {y}_{5}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{5}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{5}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{6}\left({t}_{n+1}\right) = {y}_{6}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{6}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{6}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{7}\left({t}_{n+1}\right) = {y}_{7}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{7}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{7}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \\ {y}_{8}\left({t}_{n+1}\right) = {y}_{8}\left(0\right)+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{8}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 0}^{n}\int_{{t}_{j}}^{{t}_{j+1}} k\left(\tau , {y}_{8}\left(\tau \right)\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau . \end{array}$

(17)

In general, approximating the function $k\left(\tau, y\left(\tau \right)\right),$ using the Newton polynomial, we have

$\begin{array}{l} {P}_{n}\left(\tau \right) = \frac{k\left({t}_{n}, y\left({t}_{n}\right)\right)}{{t}_{n}-{t}_{n-1}}\left(\tau -{t}_{n-1}\right)+\frac{k\left({t}_{n-1}, y\left({t}_{n-1}\right)\right)}{{t}_{n}-{t}_{n-1}}\left(\tau -{t}_{n}\right) = \frac{k\left({t}_{n}, y\left({t}_{n}\right)\right)}{h}\left(\tau -{t}_{n-1}\right)\frac{k\left({t}_{n-1}, y\left({t}_{n-1}\right)\right)}{h}\left(\tau -\\ {t}_{n}\right). \end{array}$

(18)

Using Eq (18) into system (17) we have

(19)

Rearranging the above equation, we have

(20)

Writing further system (20) we have

$\begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\int_{{t}_{j}}^{{t}_{j+1}} {\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}\frac{k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{\Delta t}\int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau +\\ \frac{{\alpha }_{1}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}\right)}\sum _{j = 2}^{n}\frac{k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)}{{2\left(\Delta t\right)}^{2}}\int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right)\left(\tau -{t}_{j-1}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau , \end{array}$

(21)

Now, calculating the integrals in system (21) we get

$\int_{{t}_{j}}^{{t}_{j+1}} {\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}}}{{\alpha }_{1}}\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right],$

$\begin{array}{l} \int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}+1}}{{\alpha }_{1}\left({\alpha }_{1}+1\right)}[{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-{\left(n-j+1\right)}^{{\alpha }_{1}}(n-\\ j+3+3{\alpha }_{1})], \end{array}$

$\begin{array}{l} \int_{{t}_{j}}^{{t}_{j+1}} \left(\tau -{t}_{j-2}\right)\left(\tau -{t}_{j-1}\right){\left({t}_{n+1}-\tau \right)}^{{\alpha }_{1}-1}d\tau = \frac{{\left(\Delta t\right)}^{{\alpha }_{1}+2}}{{\alpha }_{1}\left({\alpha }_{1}+1\right)\left({\alpha }_{1}+2\right)}[{\left(n-j+1\right)}^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\\ \left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\right\}-{\left(n-j\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+\left(5{\alpha }_{1}+10\right)\left(n-j\right)+6{{\alpha }_{1}}^{2}+\\ 18{\alpha }_{1}+12\}]. \end{array}$

Inserting them into system (21) we get

$\begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}k\left({t}_{n}, {y}_{1}\left({t}_{n}\right)\right)+\frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+1\right)}\sum _{j = 2}^{n}k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right]+\\ \frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+2\right)}\sum _{j = 2}^{n}\left[k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)-k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\right][{(n-j+1)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-(n-j+\\ 1)^{{\alpha }_{1}}\left(n-j+3+3{\alpha }_{1}\right)]+\frac{{\alpha }_{1}{\left(\Delta t\right)}^{{\alpha }_{1}}}{2\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+3\right)}\sum _{j = 2}^{n}\left[k\left({t}_{j}, {{y}_{1}}^{j}\right)-2k\left({t}_{j-1}, {{y}_{1}}^{j-1}\right)+k\left({t}_{j-2}, {{y}_{1}}^{j-2}\right)\right][(n-\\ j+1)^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\right\}-{\left(n-j\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+(5{\alpha }_{1}+\\ 10)\left(n-j\right)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}] \end{array}$

(22)

Finally, we have the following approximation:

$\begin{array}{l} {{y}_{1}}^{n+1} = {{y}_{1}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{1}{y}_{1}(t)+{M}_{2}{y}_{2}(t)+{M}_{3}{y}_{3}(t)+{M}_{4}\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+{M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-1}+{M}_{2}{{y}_{2}}^{j-1}+{M}_{3}{{y}_{3}}^{j-1}+{M}_{4}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+\\ {M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j}+{M}_{2}{{y}_{2}}^{j}+{M}_{3}{{y}_{3}}^{j}+{M}_{4}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-1}+{M}_{2}{{y}_{2}}^{j-1}+\\ {M}_{3}{{y}_{3}}^{j-1}+{M}_{4}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{1}{{y}_{1}}^{j-2}+{M}_{2}{{y}_{2}}^{j-2}+{M}_{3}{{y}_{3}}^{j-2}+{M}_{4}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+\\ (3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+\\ 18{\alpha }_{1}+12\}], \\ {{y}_{2}}^{n+1} = {{y}_{2}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{1}{y}_{1}(t)-{M}_{5}{y}_{2}(t)\}+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-\\ {M}_{5}{{y}_{2}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-1}-{M}_{5}{{y}_{2}}^{j-1}\}-\\ {t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-{M}_{5}{{y}_{2}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+\\ 3{\alpha }_{1})]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j}-{M}_{5}{{y}_{2}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-1}-{M}_{5}{{y}_{2}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{1}{{y}_{1}}^{j-2}-{M}_{5}{{y}_{2}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-\\ {(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{3}}^{n+1} = {{y}_{3}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{6}{y}_{3}(t)+{M}_{2}{y}_{4}(t)+{M}_{7}{y}_{5}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+{M}_{7}{{y}_{5}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-1}+{M}_{2}{{y}_{4}}^{j-1}+{M}_{7}{{y}_{5}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+\\ {M}_{7}{{y}_{5}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j}+{M}_{2}{{y}_{4}}^{j}+\\ {M}_{7}{{y}_{5}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-1}+{M}_{2}{{y}_{4}}^{j-1}+{M}_{7}{{y}_{5}}^{j-1}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{6}{{y}_{3}}^{j-2}+{M}_{2}{{y}_{4}}^{j-2}+\\ {M}_{7}{{y}_{5}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-\\ j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{4}}^{n+1} = {{y}_{4}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{3}{y}_{2}(t)+{M}_{8}{y}_{3}(t)-{M}_{9}{y}_{4}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-{M}_{9}{{y}_{4}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-1}+{M}_{8}{{y}_{3}}^{j-1}-{M}_{9}{{y}_{4}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-\\ {M}_{9}{{y}_{4}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j}+{M}_{8}{{y}_{3}}^{j}-{M}_{9}{{y}_{4}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-1}+{M}_{8}{{y}_{3}}^{j-1}-{M}_{9}{{y}_{4}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{3}{{y}_{2}}^{j-2}+{M}_{8}{{y}_{3}}^{j-2}-{M}_{9}{{y}_{4}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \\ {{y}_{5}}^{n+1} = {{y}_{5}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{10}{y}_{5}(t)+{M}_{2}{y}_{6}(t)-{M}_{2}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+{M}_{2}{{y}_{6}}^{j-2}-{M}_{2}{{y}_{7}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-1}+{M}_{2}{{y}_{6}}^{j-1}-{M}_{2}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+{M}_{2}{{y}_{6}}^{j-2}-\\ {M}_{2}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j}+{M}_{2}{{y}_{6}}^{j}-\\ {M}_{2}{{y}_{7}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-1}+{M}_{2}{{y}_{6}}^{j-1}-{M}_{2}{{y}_{7}}^{j-1}\}+{t}^{{\alpha }_{2}-1}\{-{M}_{10}{{y}_{5}}^{j-2}+\\ {M}_{2}{{y}_{6}}^{j-2}-{M}_{2}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+\\ 10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+\\ 10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+\\ 12\}], \end{array}$

(23)

$\begin{array}{l} {{y}_{6}}^{n+1} = {{y}_{6}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{-{M}_{11}{y}_{6}(t){y}_{8}(t)+{M}_{12}{y}_{4}(t)+{M}_{8}{y}_{5}(t)-{M}_{2}{y}_{6}(t)+{M}_{12}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+{M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}[(n-\\ j+1)^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{12}{{y}_{4}}^{j-1}+{M}_{8}{{y}_{5}}^{j-1}-\\ {M}_{2}{{y}_{6}}^{j-1}+{M}_{12}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+\\ {M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}+{M}_{12}{{y}_{4}}^{j}+{M}_{8}{{y}_{5}}^{j}-{M}_{2}{{y}_{6}}^{j}+{M}_{12}{{y}_{7}}^{j}\}-\\ 2{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{12}{{y}_{4}}^{j-1}+{M}_{8}{{y}_{5}}^{j-1}-{M}_{2}{{y}_{6}}^{j-1}+{M}_{12}{{y}_{7}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{12}{{y}_{4}}^{j-2}+{M}_{8}{{y}_{5}}^{j-2}-{M}_{2}{{y}_{6}}^{j-2}+{M}_{12}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2(n-\\ j)^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+\\ 6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \end{array}$

$\begin{array}{l} {{y}_{7}}^{n+1} = {{y}_{7}}^{0}+\frac{(1-{\alpha }_{1})}{\mathrm{C}({\alpha }_{1})}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\{{M}_{11}{y}_{6}(t){y}_{8}(t)-{M}_{13}{y}_{7}(t)\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+1)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}[{(n-j+1)}^{{\alpha }_{1}}-{(n-j)}^{{\alpha }_{1}}]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+2)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}-{M}_{13}{{y}_{7}}^{j-1}\}-{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}][(n-\\ j+1)^{{\alpha }_{1}}(n-j+3+2{\alpha }_{1})-{(n-j+1)}^{{\alpha }_{1}}(n-j+3+3{\alpha }_{1})]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{(\Delta t)}^{{\alpha }_{1}}}{2\mathrm{C}({\alpha }_{1})\mathrm{\Gamma }({\alpha }_{1}+3)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}-{M}_{13}{{y}_{7}}^{j}\}-2{t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}-{M}_{13}{{y}_{7}}^{j-1}\}+\\ {t}^{{\alpha }_{2}-1}\{{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}-{M}_{13}{{y}_{7}}^{j-2}\}][{(n-j+1)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(3{\alpha }_{1}+10)(n-j)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{(n-j)}^{{\alpha }_{1}}\{2{(n-j)}^{2}+(5{\alpha }_{1}+10)(n-j)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\}], \end{array}$

$\begin{array}{l} {{y}_{8}}^{n+1} = {{y}_{8}}^{0}+\frac{\left(1-{\alpha }_{1}\right)}{\mathrm{C}\left({\alpha }_{1}\right)}{\alpha }_{2}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{y}_{6}\left(t\right){y}_{8}\left(t\right)+{M}_{13}{y}_{7}\left(t\right)\right\}+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+1\right)}\sum _{j = 2}^{n}{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{13}{{y}_{7}}^{j-2}\right\}\left[{\left(n-j+1\right)}^{{\alpha }_{1}}-{\left(n-j\right)}^{{\alpha }_{1}}\right]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+2\right)}\sum _{j = 2}^{n}[{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{13}{{y}_{7}}^{j-1}\right\}-{t}^{{\alpha }_{2}-1}\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+\\ {M}_{13}{{y}_{7}}^{j-2}\}]\left[{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+2{\alpha }_{1}\right)-{\left(n-j+1\right)}^{{\alpha }_{1}}\left(n-j+3+3{\alpha }_{1}\right)\right]+\\ \frac{{\alpha }_{1}{\alpha }_{2}{\left(\Delta t\right)}^{{\alpha }_{1}}}{2\mathrm{C}\left({\alpha }_{1}\right)\mathrm{\Gamma }\left({\alpha }_{1}+3\right)}\sum _{j = 2}^{n}\left[{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j}{{y}_{8}}^{j}+{M}_{13}{{y}_{7}}^{j}\right\}-2{t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-1}{{y}_{8}}^{j-1}+{M}_{13}{{y}_{7}}^{j-1}\right\}+\\ {t}^{{\alpha }_{2}-1}\left\{-{M}_{11}{{y}_{6}}^{j-2}{{y}_{8}}^{j-2}+{M}_{13}{{y}_{7}}^{j-2}\right\}\right][{\left(n-j+1\right)}^{{\alpha }_{1}}\{2{\left(n-j\right)}^{2}+\left(3{\alpha }_{1}+10\right)\left(n-j\right)+2{{\alpha }_{1}}^{2}+\\ 9{\alpha }_{1}+12\}-{\left(n-j\right)}^{{\alpha }_{1}}\left\{2{\left(n-j\right)}^{2}+\left(5{\alpha }_{1}+10\right)\left(n-j\right)+6{{\alpha }_{1}}^{2}+18{\alpha }_{1}+12\right\}]. \end{array}$

6. Results and discussions

A fractional-order model is proposed for analysis and simulation, to observe the concentration of chemicals in chemistry kinematics problems with a stiff differential equation. For this purpose, we used ABC with Mittage-Lefffier law, Atangana-Tufik scheme, and fractal fractional derivative for hires problem with given initial conditions. Details of parameters values of real data are also given in ^[18,19] which will consider for simulation analysis for the proposed study. Solution of compartment shows in to with fractional fractal operator at different order. Effect of fraction order can easily be observed in simulation of the compartments having a concentration of chemical reaction with stiff differential equations. The concentration ${y}_{1}$ and ${y}_{8}$ of the chemical species start decreasing by decreasing fractional values respectively while concentration ${y}_{2}, {y}_{3}$ , ${y}_{4}$ , ${y}_{5}$ , ${y}_{6}$ and ${y}_{7}$ of the chemical species start increasing by decreasing fractional values. These concentrations of chemical species converge to our desired value according to steady state by decreasing the fractional values which shows that it provides us appropriate results at non integer value. We can get better concentration of the components by using the fractional derivative which are very important for chemical problem to check the actual behavior of the concentration of the chemical with smallest changes in derivative with respect to time. It is also very important for solutions of nonlinear problems which are commonly used researcher and scientist in kinetics chemistry.

Figure 1. Simulation of

${y}_{1}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 2. Simulation of

${y}_{2}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 3. Simulation of

${y}_{3}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 4. Simulation of

${y}_{4}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 5. Simulation of

${y}_{5}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 6. Simulation of

${y}_{6}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 7. Simulation of

${y}_{7}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

Figure 8. Simulation of

${y}_{8}\left(t\right)$ with fractal fractional derivative.

DownLoad: Full-Size Img PowerPoint

7. Conclusions

We examine the hires problems with stiff systems of nonlinear ordinary equations that rely on the concentration of chemical reaction of components in this study. The advanced techniques of fractional operator have been implemented for initial value problem arising from chemical reactions composed of large systems of stiff ordinary differential equations. The arbitrary derivative of fractional order has been taken with Atangana-Toufik scheme and fractal fractional derivative. Solutions have been obtained efficiently within limited time which shows the actual behavior of kinetic chemical reactions. Existence and uniqueness of results have been verified by fixed point theorem. Simulations are carried out for different fractional values. New chemical reactions can be done with the help of these analyses. These concepts are very important to use for real life problems like Brine tank cascade, Recycled Brine tank cascade, pond pollution, home heating and biomass transfer problem.

Acknowledgments

Research Supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

No conflict of interest.

References

[1]	R. F. Zhang, S. Bilige, Bilinear, neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation, Nonlinear Dyn., 95 (2019), 3041–3048. https://doi.org/10.1007/s11071-018-04739-z
[2]	M. S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE Trans. Ind. Electron., 55 (2008), 4094–4101. https://doi.org/10.1109/TIE.2008.925774 doi: 10.1109/TIE.2008.925774
[3]	A. Almutairi, H. El-Metwally, M. A. Sohaly, I. M. Elbaz, Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology, Adv. Differ. Equ., 2021 (2021), 1–32. https://doi.org/10.1186/s13662-021-03344-6 doi: 10.1186/s13662-021-03344-6
[4]	J. H. He, Seeing with a single scale is always unbelieving: from magic to two-scale fractal, Therm. Sci., 25 (2021), 1217–1219. https://doi.org/10.2298/TSCI2102217H doi: 10.2298/TSCI2102217H
[5]	P. Korn, A regularity-aware algorithm for variational data assimilation of an idealized coupled atmosphere–ocean model, J. Sci. Comput., 79 (2019), 748–786. https://doi.org/10.1007/s10915-018-0871-y doi: 10.1007/s10915-018-0871-y
[6]	A. Yokus, H. M. Baskonus, Dynamics of traveling wave solutions arising in fiber optic communication of some nonlinear models, Soft Comput., 26 (2022), 13605–13614. https://doi.org/10.1007/s00500-022-07320-4 doi: 10.1007/s00500-022-07320-4
[7]	H. G. Abdelwahed, E. K. El-Shewy, M. A. E. Abdelrahman, A. F. Alsarhana, On the physical nonlinear (n+1)-dimensional Schrödinger equation applications, Results Phys., 21 (2021), 103798. https://doi.org/10.1016/j.rinp.2020.103798 doi: 10.1016/j.rinp.2020.103798
[8]	M. E. Samei, L. Karimi, M. K. A. Kaabar, To investigate a class of multi-singular pointwise defined fractional q-integro-differential equation with applications, AIMS Math., 7 (2022), 7781–7816. https://doi.org/10.3934/math.2022437 doi: 10.3934/math.2022437
[9]	C. H. Gu, Soliton theory and its applications, Springer-Verlag Berlin and Heidelberg GmbH & Co. K, Berlin, 1995. https://doi.org/10.1007/978-3-662-03102-5
[10]	D. C. Lu, B. J. Hong, L. X. Tian, Bäcklund transformation and n-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients, Int. J. Nonlinear Sci., 2 (2006), 3–10.
[11]	V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer Berlin, Heidelberg, 1991.
[12]	K. L. Geng, D. S. Mou, C. Q. Dai, Nondegenerate solitons of 2-coupled mixed derivative nonlinear Schrödinger equations, Nonlinear Dyn., 111 (2023), 603–617. https://doi.org/10.1007/s11071-022-07833-5 doi: 10.1007/s11071-022-07833-5
[13]	D. C. Lu, B. J. Hong, L. X. Tian, New explicit exact solutions for the generalized coupled Hirota–Satsuma KdV system, Comput. Math. Appl., 53 (2007), 1181–1190. https://doi.org/10.1016/j.camwa.2006.08.047 doi: 10.1016/j.camwa.2006.08.047
[14]	B. J. Hong, New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation, Appl. Math. Comput., 215 (2009), 2908–2913. https://doi.org/10.1016/j.amc.2009.09.035 doi: 10.1016/j.amc.2009.09.035
[15]	P. R. Kundu, M. R. A. Fahim, M. E. lslam, M. A. Akbar, The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis, Heliyon, 7 (2021), e06459. https://doi.org/10.1016/j.heliyon.2021.e06459
[16]	J. J. Fang, D. S. Mou, H. C. Zhang, Y. Y. Wang, Discrete fractional soliton dynamics of the fractional Ablowitz–Ladik model, Optik, 228 (2021), 166186. https://doi.org/10.1016/j.ijleo.2020.166186 doi: 10.1016/j.ijleo.2020.166186
[17]	H. F. Ismael, H. Bulut, H. M. Baskonus, Optical soliton solutions to the Fokas-Lenells equation via sine-Gordon expansion method and (m+(G'/G))-expansion method, Pramana, 94 (2020), 1–9. https://doi.org/10.1007/s12043-019-1897-x doi: 10.1007/s12043-019-1897-x
[18]	Y. Fang, G. Z. Wu, Y. Y. Wang, C. Q. Dai, Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN, Nonlinear Dyn., 105 (2021), 603–616. https://doi.org/10.1007/s11071-021-06550-9 doi: 10.1007/s11071-021-06550-9
[19]	S. T. Mohyud-Din, S. Bibi, Exact solutions for nonlinear fractional differential equations using G'/G²-expansion method, Alex. Eng. J., 57 (2018), 1003–1008. https://doi.org/10.1016/j.aej.2017.01.035 doi: 10.1016/j.aej.2017.01.035
[20]	A. M. Elsherbeny, R. El-Barkouky, H. M. Ahmed, R. M. El-Hassani, A. H. Arnous, Optical solitons and another solutions for Radhakrishnan-Kundu-Laksmannan equation by using improved modified extended tanh-function method, Opt. Quant. Electron., 53 (2021), 1–15. https://doi.org/10.1007/s11082-021-03382-0 doi: 10.1007/s11082-021-03382-0
[21]	H. Durur, A Kurt, O. Tasbozan, New travelling wave solutions for KdV6 equation using sub equation method, Appl. Math. Nonlinear Sci., 5 (2020), 455–460. https://doi.org/10.2478/amns.2020.1.00043 doi: 10.2478/amns.2020.1.00043
[22]	W. B. Bo, R. R. Wang, Y. Fang, Y. Y. Wang, C. Q. Dai, Prediction and dynamical evolution of multipole soliton families in fractional Schrödinger equation with the PT-symmetric potential and saturable nonlinearity, Nonlinear Dyn., 111 (2023), 1577–1588. https://doi.org/10.1007/s11071-022-07884-8 doi: 10.1007/s11071-022-07884-8
[23]	R. R. Wang, Y. Y. Wang, C. Q. Dai, Influence of higher-order nonlinear effects on optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser, Opt. Laser Technol., 152 (2022), 108103. http://dx.doi.org/10.1016/j.optlastec.2022.108103 doi: 10.1016/j.optlastec.2022.108103
[24]	A. M. Nass, Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay, Appl. Math. Comput., 347 (2019), 370–380. https://doi.org/10.1016/j.amc.2018.11.002 doi: 10.1016/j.amc.2018.11.002
[25]	J. S. Zhang, R. Qin, Y. Yu, J. Zhu, Y. Yu, Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem, Comput. Math. Appl., 138 (2023), 23–36. https://doi.org/10.1016/j.camwa.2023.02.023 doi: 10.1016/j.camwa.2023.02.023
[26]	S. O. Abdulla, S. T. Abdulazeez, M. Modanli, Comparison of third-order fractional partial differential equation based on the fractional operators using the explicit finite difference method, Alex. Eng. J., 70 (2023), 37–44. https://doi.org/10.1016/j.aej.2023.02.032 doi: 10.1016/j.aej.2023.02.032
[27]	A. H. Salas, Computing solutions to a forced KdV equation, Nonlinear Anal., Real World Appl., 12 (2011), 1314–1320. https://doi.org/10.1016/j.nonrwa.2010.09.028 doi: 10.1016/j.nonrwa.2010.09.028
[28]	L. N. Song, W. G. Wang, A new improved Adomian decomposition method and its application to fractional differential equations, Appl. Math. Model., 37 (2013), 1590–1598. https://doi.org/10.1016/j.apm.2012.03.016 doi: 10.1016/j.apm.2012.03.016
[29]	B. J. Hong, D. C. Lu, Modified fractional variational iteration method for solving the generalized time-space fractional Schrödinger equation, Sci. World J., 2014 (2014), 1–7. https://doi.org/10.1155/2014/964643 doi: 10.1155/2014/964643
[30]	M. Nadeem, J. H. He, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J. Math. Chem., 59 (2021), 1234–1245. https://doi.org/10.1007/s10910-021-01236-4 doi: 10.1007/s10910-021-01236-4
[31]	G. V. Bhaskar, S. M. R. Bhamidimarri, Approximate analytical solutions for a biofilm reactor model with Monod kinetics and product inhibition, Can. J. Chem. Eng., 69 (1991), 544–547. https://doi.org/10.1002/cjce.5450690220 doi: 10.1002/cjce.5450690220
[32]	J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35 (2000), 37–43. https://doi.org/10.1016/S0020-7462(98)00085-7 doi: 10.1016/S0020-7462(98)00085-7
[33]	J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3 doi: 10.1016/S0045-7825(99)00018-3
[34]	E. K. Jaradat, O. Alomari, M. Abudayah, A. A. M. Al-Faqih, An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method, Adv. Math. Phys., 2018 (2018), 1–11. https://doi.org/10.1155/2018/6765021
[35]	B. J. Hong, D. C. Lu, W. Chen, Exact and approximate solutions for the fractional Schrödinger equation with variable coefficients, Adv. Differ. Equ., 2019 (2019), 1–10. https://doi.org/10.1186/s13662-019-2313-z doi: 10.1186/s13662-019-2313-z
[36]	A. Burqan, A. El-Ajou, R. Saadeh, M. Al-Smadi, A new efficient technique using Laplace transforms and smooth expansions to construct a series solution to the time-fractional Navier-Stokes equations, Alex. Eng. J., 61 (2022), 1069–1077. https://doi.org/10.1016/j.aej.2021.07.020
[37]	C. Burgos, J. C. Cortés, L. Villafuerte, R. J. Villanueva, Solving random fractional second-order linear equations via the mean square Laplace transform: theory and statistical computing, Appl. Math. Comput., 418 (2022), 126846. https://doi.org/10.1016/j.amc.2021.126846 doi: 10.1016/j.amc.2021.126846
[38]	S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), 3154–3163. https://doi.org/10.1016/j.apm.2013.11.035 doi: 10.1016/j.apm.2013.11.035
[39]	S. Arbabi, M. Najafi, Exact solitary wave solutions of the complex nonlinear Schrödinger equations, Optik, 127 (2016), 4682–4688. https://doi.org/10.1016/j.ijleo.2016.02.008 doi: 10.1016/j.ijleo.2016.02.008
[40]	B. J. Hong, Exact solutions for the conformable fractional coupled nonlinear Schrödinger equations with variable coefficients, J. Low Freq. Noise, V. A., 41 (2022), 1–14. https://doi.org/10.1177/14613484221135478 doi: 10.1177/14613484221135478
[41]	B. J. Hong, Abundant explicit solutions for the M-fractional generalized coupled nonlinear Schrödinger KdV equations, J. Low Freq. Noise, V. A., 42 (2023), 1–20. https://doi.org/10.1177/14613484221148411 doi: 10.1177/14613484221148411
[42]	K. Hosseini, A. Bekir, R. Ansari, New exact solutions of the conformable time-fractional Cahn-Allen and Cahn-Hilliard equations using the modified Kudryashov method, Optik, 132 (2017), 203–209. https://doi.org/10.1016/j.ijleo.2016.12.032 doi: 10.1016/j.ijleo.2016.12.032
[43]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent: part Ⅱ, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[44]	Y. Asıf, D. Hülya, D. Kaya, H. Ahmad, T. A. Nofal, Numerical comparison of Caputo and Conformable derivatives of time fractional Burgers-Fisher equation, Results Phys., 25 (2021), 104247. https://doi.org/10.1016/j.rinp.2021.104247 doi: 10.1016/j.rinp.2021.104247
[45]	M. Hadjer, M. Faycal, M. Ahcene, Solution of Sakata-Taketani equation via the Caputo and Riemann-Liouville fractional derivatives, Rep. Math. Phys., 89 (2022), 359–370. https://doi.org/10.1016/S0034-4877(22)00038-6 doi: 10.1016/S0034-4877(22)00038-6
[46]	R. W. Boyd, Nonlinear optics, Academic Press, 2020.
[47]	M. Lakestani, J. Manafian, Analytical treatments of the space-time fractional coupled nonlinear Schrödinger equations, Opt. Quant. Electron., 396 (2018), 1–33. https://doi.org/10.1007/s11082-018-1615-9 doi: 10.1007/s11082-018-1615-9
[48]	T. Han, Z. Li, X. Zhang, Bifurcation and new exact traveling wave solutions to time-space coupled fractional nonlinear Schrödinger equation, Phys. Lett. A, 395 (2021), 127217. https://doi.org/10.1016/j.physleta.2021.127217 doi: 10.1016/j.physleta.2021.127217
[49]	B. H. Wang, P. H. Lu, C. Q. Dai, Y. X. Chen, Vector optical soliton and periodic solutions of a coupled fractional nonlinear Schrödinger equation, Results Phys., 17 (2020), 103036. https://doi.org/10.1016/j.rinp.2020.103036 doi: 10.1016/j.rinp.2020.103036
[50]	M. Eslami, Exact traveling wave solutions to the fractional coupled nonlinear Schrödinger equations, Appl. Math. Comput., 285 (2016), 141–148. https://doi.org/10.1016/j.amc.2016.03.032 doi: 10.1016/j.amc.2016.03.032
[51]	P. F. Dai, Q. B. Wu, An efficient block Gauss–Seidel iteration method for the space fractional coupled nonlinear Schrödinger equations, Appl. Math. Lett., 117 (2021), 107116. https://doi.org/10.1016/j.aml.2021.107116 doi: 10.1016/j.aml.2021.107116
[52]	C. R. Menyuk, Stability of solitons in birefringent optical fibers. Ⅱ. Arbitrary amplitudes, J. Opt. Soc. Am. B, 5 (1988), 392–402. https://doi.org/10.1364/JOSAB.5.000392 doi: 10.1364/JOSAB.5.000392
[53]	J. Q. Gu, A. Akbulut, M. Kaplan, M. K. A. Kaabar, X. G. Yue, A novel investigation of exact solutions of the coupled nonlinear Schrödinger equations arising in ocean engineering, plasma waves, and nonlinear optics, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.06.014
[54]	S. Alshammari, N. Iqba, M. Yar, Analytical investigation of nonlinear fractional Harry Dym and Rosenau-Hyman equation via a novel transform, J. Funct. Spaces, 2022 (2022), 8736030. https://doi.org/10.1155/2022/8736030 doi: 10.1155/2022/8736030
[55]	J.Singh, D. Kumar, S. Kuma, New treatment of fractional Fornberg-Whitham equation via Laplace transform, Ain Shams Eng. J., 4 (2013), 557–562. https://doi.org/10.1016/j.asej.2012.11.009 doi: 10.1016/j.asej.2012.11.009
[56]	R. A. Khan, Y. J. Li, F. Jarad, Exact analytical solutions of fractional order telegraph equations via triple Laplace transform, Discrete Cont. Dyn. Syst.-S, 14 (2021), 2387–2397. http://dx.doi.org/10.3934/dcdss.2020427 doi: 10.3934/dcdss.2020427
[57]	J. H. He, Recent development of the homotopy perturbation method, Topol. Methods Nonlinear Anal., 31 (2008), 205–209.
[58]	J. H. He, M. L. Jiao, C. H. He, Homotopy perturbation method for fractal Duffing oscillator with arbitrary conditions, Fractals, 30 (2022), 1–10. https://doi.org/10.1142/S0218348X22501651 doi: 10.1142/S0218348X22501651

This article has been cited by:

1.	Hira Soomro, Nooraini Zainuddin, Hanita Daud, Joshua Sunday, Noraini Jamaludin, Abdullah Abdullah, Mulono Apriyanto, Evizal Abdul Kadir, Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems, 2022, 12, 2076-3417, 4484, 10.3390/app12094484
2.	Zeeshan Ali, Faranak Rabiei, Kamyar Hosseini, A fractal–fractional-order modified Predator–Prey mathematical model with immigrations, 2023, 207, 03784754, 466, 10.1016/j.matcom.2023.01.006
3.	Sümeyra Uçar, Analysis of hepatitis B disease with fractal–fractional Caputo derivative using real data from Turkey, 2023, 419, 03770427, 114692, 10.1016/j.cam.2022.114692
4.	Mohammad Partohaghighi, Ali Akgül, Rubayyi T. Alqahtani, New Type Modelling of the Circumscribed Self-Excited Spherical Attractor, 2022, 10, 2227-7390, 732, 10.3390/math10050732
5.	G.M. Vijayalakshmi, Roselyn Besi. P, A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis, 2022, 8, 26667207, 100144, 10.1016/j.rico.2022.100144
6.	SHAIMAA A. M. ABDELMOHSEN, SHABIR AHMAD, MANSOUR F. YASSEN, SAEED AHMED ASIRI, ABDELBACKI M. M. ASHRAF, SAYED SAIFULLAH, FAHD JARAD, NUMERICAL ANALYSIS FOR HIDDEN CHAOTIC BEHAVIOR OF A COUPLED MEMRISTIVE DYNAMICAL SYSTEM VIA FRACTAL–FRACTIONAL OPERATOR BASED ON NEWTON POLYNOMIAL INTERPOLATION, 2023, 31, 0218-348X, 10.1142/S0218348X2340087X
7.	D.A. Tverdyi, R.I. Parovik, A.R. Hayotov, A.K. Boltaev, Распараллеливание численного алгоритма решения задачи Коши для нелинейного дифференциального уравнения дробного переменного порядка с помощью технологии OpenMP, 2023, 20796641, 87, 10.26117/2079-6641-2023-43-2-87-110
8.	Khalid Hattaf, A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels, 2023, 7, 2504-3110, 395, 10.3390/fractalfract7050395
9.	Ali Raza, Ovidiu V. Stadoleanu, Ahmed M. Abed, Ali Hasan Ali, Mohammed Sallah, Heat transfer model analysis of fractional Jeffery-type hybrid nanofluid dripping through a poured microchannel, 2024, 22, 26662027, 100656, 10.1016/j.ijft.2024.100656
10.	Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen, New solutions of time-space fractional coupled Schrödinger systems, 2023, 8, 2473-6988, 27033, 10.3934/math.20231383
11.	Muhammad Shahzad, Soma Mustafa, Sarbaz H A Khoshnaw, Inference of complex reaction mechanisms applying model reduction techniques, 2024, 99, 0031-8949, 045242, 10.1088/1402-4896/ad3291
12.	B. El Ansari, E. H. El Kinani, A. Ouhadan, Symmetry analysis of the time fractional potential-KdV equation, 2025, 44, 2238-3603, 10.1007/s40314-024-02991-1
13.	Muhammad Farman, Changjin Xu, Perwasha Abbas, Aceng Sambas, Faisal Sultan, Kottakkaran Sooppy Nisar, Stability and chemical modeling of quantifying disparities in atmospheric analysis with sustainable fractal fractional approach, 2025, 142, 10075704, 108525, 10.1016/j.cnsns.2024.108525
14.	Hira Khan, Gauhar Rahman, Muhammad Samraiz, Kamal Shah, Thabet Abdeljawad, On Generalized Fractal-Fractional Derivative and Integral Operators Associated with Generalized Mittag-Leffler Function, 2025, 24058440, e42144, 10.1016/j.heliyon.2025.e42144
15.	Emmanuel Kengne, Ahmed Lakhssassi, Dynamics of stochastic nonlinear waves in fractional complex media, 2025, 542, 03759601, 130423, 10.1016/j.physleta.2025.130423

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)