Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Ruonan Liu; Tomás Caraballo; Ruonan Liu; Tomás Caraballo

doi:10.3934/math.2024390

AIMS Mathematics

2024, Volume 9, Issue 4: 8020-8042. doi: 10.3934/math.2024390

Previous Article Next Article

Research article

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Ruonan Liu ¹,
Tomás Caraballo ^{2
,
,}

1.
School of Mathematics and Statistics, Xuzhou University of Technology, Jiangsu, 221008, China
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain

Received: 25 January 2024 Revised: 14 February 2024 Accepted: 20 February 2024 Published: 26 February 2024
MSC : 35F05, 60H15

In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.

Keywords:

Citation: Ruonan Liu, Tomás Caraballo. Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional[J]. AIMS Mathematics, 2024, 9(4): 8020-8042. doi: 10.3934/math.2024390

Related Papers:

[1]	Osama Ala'yed, Belal Batiha, Diala Alghazo, Firas Ghanim . Cubic B-Spline method for the solution of the quadratic Riccati differential equation. AIMS Mathematics, 2023, 8(4): 9576-9584. doi: 10.3934/math.2023483
[2]	Azhar Iqbal, Abdullah M. Alsharif, Sahar Albosaily . Numerical study of non-linear waves for one-dimensional planar, cylindrical and spherical flow using B-spline finite element method. AIMS Mathematics, 2022, 7(8): 15417-15435. doi: 10.3934/math.2022844
[3]	Seherish Naz Khalid Ali Khan, Md Yushalify Misro . Hybrid B-spline collocation method with particle swarm optimization for solving linear differential problems. AIMS Mathematics, 2025, 10(3): 5399-5420. doi: 10.3934/math.2025249
[4]	Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed . On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925
[5]	Yan Wu, Chun-Gang Zhu . Generating bicubic B-spline surfaces by a sixth order PDE. AIMS Mathematics, 2021, 6(2): 1677-1694. doi: 10.3934/math.2021099
[6]	Mohammad Tamsir, Neeraj Dhiman, Deependra Nigam, Anand Chauhan . Approximation of Caputo time-fractional diffusion equation using redefined cubic exponential B-spline collocation technique. AIMS Mathematics, 2021, 6(4): 3805-3820. doi: 10.3934/math.2021226
[7]	Rabia Noureen, Muhammad Nawaz Naeem, Dumitru Baleanu, Pshtiwan Othman Mohammed, Musawa Yahya Almusawa . Application of trigonometric B-spline functions for solving Caputo time fractional gas dynamics equation. AIMS Mathematics, 2023, 8(11): 25343-25370. doi: 10.3934/math.20231293
[8]	Osama Ala'yed, Ahmad Qazza, Rania Saadeh, Osama Alkhazaleh . A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications. AIMS Mathematics, 2024, 9(2): 4665-4683. doi: 10.3934/math.2024225
[9]	Mostafa M. A. Khater, A. El-Sayed Ahmed . Strong Langmuir turbulence dynamics through the trigonometric quintic and exponential B-spline schemes. AIMS Mathematics, 2021, 6(6): 5896-5908. doi: 10.3934/math.2021349
[10]	Kai Wang, Guicang Zhang . Curve construction based on quartic Bernstein-like basis. AIMS Mathematics, 2020, 5(5): 5344-5363. doi: 10.3934/math.2020343

Abstract

1. Introduction

The study of nonlinear systems of singular initial value problems has recently attracted many mathematicians and physicists ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. One of the systems in this category is the following Lane-Emden system of the form:

$\frac{{d}^{2}{\omega }_{1}\left(\tau \right)}{d{\tau }^{2}}+\frac{{\delta }_{1}}{\tau }\frac{d{\omega }_{1}\left(\tau \right)}{d\tau }+{{\mathrm{\hslash }}}_{1}\left({\omega }_{1}\right(\tau ),{\omega }_{2}(\tau \left)\right) = {{\mathrm{\aleph }}}_{1}\left(\tau \right),$

(1)

subject to

${\omega }_{1}\left(0\right) = {\varepsilon }_{1},{{\omega }^{{\mathrm{{\text{'}}}}}}_{1}\left(0\right) = 0,$

(2)

${\omega }_{2}\left(0\right) = {\vartheta }_{1},{{\omega }^{{\mathrm{{\text{'}}}}}}_{2}\left(0\right) = 0,$

where ${{\mathrm{\aleph }}}_{1},{{\mathrm{\aleph }}}_{2}$ are source functions, $\tau \in \left[{\mathrm{0,1}}\right],$ ${\delta }_{1},{\delta }_{2},{\varepsilon }_{1},$ and ${\vartheta }_{1}$ are real constants. The Lane-Emden equation, which was first investigated by astrophysicists Jonathan Homer Lane and Robert Emden, comes in various scientific applications in two kinds. For ${\mathrm{\hslash }}\left(\omega \left(\tau \right)\right) = {\omega }^{\gamma }\left(\tau \right)$ , (1) is called the Lane-Emden equation of index $\gamma$ , or of the first kind, that is a fundamental equation in the theory of star structure ^{[13,14,15,16,17,18,19]}. It represents the temperature variant of a spherical gas cloud subject to the principles of thermodynamics and its molecules' mutual attraction, see ^{[20,21,22,23,24,25,26]} and references therein. In astrophysics, the Lane-Emden equation represents Poisson's equation for the gravitational potential of a spherically symmetric, polytropic fluid, and self-gravitating at hydrostatic equilibrium ^[27]. Moreover, an important area of application for this type of equation is the study of species' diffusive transit and chemical reactions inside porous catalyst particles ^[27]. However, for ${\mathrm{\hslash }}\left(\omega \left(\tau \right)\right) = {e}^{\omega \left(\tau \right)}$ , (1) is called the Lane-Emden equation of the second kind that describes the dimensionless density distribution in a sphere of isothermal gas ^[28]. The singular behavior that arises at $\tau = 0$ is the primary difficulty of the Lane-Emden equations.

Recently, modeling a variety of physical and chemical phenomena, including chemical reactions, population evolution, and pattern formation leads to the system of Lane-Emden equations ^[7]. Therefore, numerous approaches have been proposed for the solutions of scalar and system of Lane-Emden equations ^{[1,2,10,11,12,12,13,14,15,17,18,42,43,44,45,46,47]}.

Spline methods employing piecewise polynomial functions have been demonstrated to be convenient methods for obtaining numerical solutions to many challenging models in science, engineering, and mathematics due to their simplicity of implementation and efficiency ^{[29,30,31,32,33,34]}. One of the well-known spline methods is the so-called B-spline (the "B" stands for basis) functions, which were first proposed by Schoenberg in 1946. The B-spline functions ^[35,36] have recently been a valuable tool in numerical computation, approximation theory, and image processing as they have various useful properties such as numerical stability of computations, local effects of coefficient changes, and built-in smoothness between neighboring polynomial pieces. The degrees of B-spline and the collocation points are the main factors that play a significant role in the execution of the technique and affect the outcomes to be achieved up to a required level of accuracy.

One of the most efficient and versatile techniques for obtaining approximate solutions is the cubic B-spline method (CBSM). The CBSM is a third-order piecewise polynomial constructed from a combination of recursive formulas referred to as the cubic B-spline basis. The derivation of the B-spline basis and the construction of the B-spline function are thoroughly discussed in ^[37,38]. In recent years, the CBSM has been successfully applied to various mathematical problems ^[39]. This demonstrates the effectiveness and usefulness of spline approaches through their numerous successful implementations. Therefore, this paper investigates the approximate solution of systems of the Lane-Emden equations using the CBSM.

This paper is organized as follows, in the next section, we present the basic preliminaries of the method. A short summary of cubic B-Spline method is presented in Section 3. We show the convergence analysis in Section 4. Finally, we present some numerical examples in Section 5.

2. Preliminaries

In this section, we introduce some basic facts regarding cubic B-spline approximation. Assume that the interval ${\mathrm{\Gamma }} = [\alpha ,\beta ]$ can be divided into $k$ subintervals [ ${\tau }_{i},{\tau }_{i+1}$ ] via

${\tau }_{i} = \alpha +i{\mathrm{\Lambda }},i = 0,{\mathrm{ }}\cdots ,k ,$

where

${\mathrm{\Lambda }} = ({\mathrm{\beta }}-{\mathrm{\alpha }})/k .$

The linear space of the cubic spline over the given partition is

${{{{\rm M}}}}_{3}\left(I\right) = \{\mu \left(\tau \right)\in {C}^{2}\left(I\right):{\mu \left(\tau \right)|}_{{I}_{i}}\in {P}_{3},i = 0,{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k-1\},$

where ${\mu \left(\tau \right)|}_{{I}_{i}}$ indicates the restriction of $\mu \left(\tau \right)$ to ${I}_{i}$ and ${P}_{3}$ indicates the set of cubic polynomials in one-variable. The dimension of linear space ${{{{\rm M}}}}_{3}\left(I\right)$ is $(k+3)$ . Extend ${\mathrm{\Gamma }} = [\alpha ,\beta ]$ to

$\stackrel{-}{{\mathrm{\Gamma }}} = [\alpha -3{\mathrm{\Lambda }},\beta +3{\mathrm{\Lambda }}]$

with the equidistant knots

${\tau }_{i} = \alpha +i{\mathrm{\Lambda }},i = -3,{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k+3 .$

The cubic B-spline function

${{{{\rm K}}}}_{i}\left(\tau \right),i = -1,{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k+1 ,$

is given by ^[40]

${{\rm K}_{{\text{i}}}}\left( \tau \right) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{{(\tau - {\tau _i})}^3}}}{{6{\Lambda ^3}}}, \tau \in \left[ {{\tau _i},{\tau _{i + 1}}} \right],} \\ {\frac{{{{(\tau - {\tau _i})}^3}}}{{6{\Lambda ^3}}} - 2\frac{{{{\left( {\tau - {\tau _{i + 1}}} \right)}^3}}}{{3{\Lambda ^3}}},\; \tau \in \left[ {{\tau _{i + 1}},{\tau _{i + 2}}} \right],} \\ {\frac{{{{({\tau _{i + 4}} - \tau )}^3}}}{{6{\Lambda ^3}}} - 2\frac{{{{\left( {{\tau _{i + 3}} - \tau } \right)}^3}}}{{3{\Lambda ^3}}}, \tau \in \left[ {{\tau _{i + 2}},{\tau _{i + 3}}} \right],} \\ {\frac{{{{({\tau _{i + 4}} - \tau )}^3}}}{{6{\Lambda ^3}}}, \tau \in \left[ {{\tau _{i + 3}},{\tau _{i + 4}}} \right],} \\ {0, {{\text{else}}}.} \end{array}} \right.$

The ${{{{\rm K}}}}_{i}\left(\tau \right),i = -1,{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k+1$ , form basis splines of ${{{{\rm M}}}}_{3}\left(I\right)$ . The values of ${{{{\rm K}}}}_{i}\left(\tau \right),{{{{\rm K}}}}_{i}^{{\mathrm{{\text{'}}}}}\left(\tau \right)$ and ${{{{\rm K}}}}_{i}^{{\mathrm{{\text{'}}}}{\mathrm{{\text{'}}}}}\left(\tau \right)$ at the knots are recorded in Table 1.

Table 1. The values of

${{{{\rm K}}}}_{i}\left(\tau \right),{{{{\rm K}}}}_{i}^{{\text{'}}}\left(\tau \right)$ and

${{{{\rm K}}}}_{i}^{{\text{'}}{\text{'}}}\left(\tau \right)$ at the knots.

	${\tau }_{i-1}$	${\tau }_{i}$	${\tau }_{i+1}$	else
${{{{\rm K}}}}_{i}\left(\tau \right)$	$\frac{1}{6}$	$\frac{4}{6}$	$\frac{1}{6}$	0
${{{{\rm K}}}}_{i}^{{\text{'}}}\left(\tau \right)$	$\frac{1}{2{\mathrm{\Lambda }}}$	$0$	$-\frac{1}{2{\mathrm{\Lambda }}}$	0
${{{{\rm K}}}}_{i}^{{\text{'}}{\text{'}}}\left(\tau \right)$	$\frac{1}{{{\mathrm{\Lambda }}}^{2}}$	$\frac{-2}{{{\mathrm{\Lambda }}}^{2}}$	$\frac{1}{{{\mathrm{\Lambda }}}^{2}}$	0

| Show Table

DownLoad: CSV

For a sufficiently smooth function $\rho \left(\tau \right)$ , there is a unique cubic spline $\mu \left(\tau \right)\in {{{{\rm M}}}}_{3}\left(I\right)$ fulfilling the interpolation conditions

$\mu \left({\tau }_{i}\right) = \rho \left({\tau }_{i}\right),i = 0,{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k ,$

and

$\mu {\mathrm{{\text{'}}}}\left(\alpha \right) = \rho {\mathrm{{\text{'}}}}\left(\alpha \right),$

such that

$\mu \left(\tau \right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{\rm K}}}}_{i}\left(\tau \right),$

(3)

where ${{\mathtt{λ}}}_{i}{\mathrm{{\text{'}}}}$ s are constants to be estimated.

Using (3), we get

$\mu \left({\tau }_{j}\right) = \sum \limits_{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{\rm K}}}}_{i}\left({\tau }_{j}\right) = \frac{{{\mathtt{λ}}}_{j-1}+4{{\mathtt{λ}}}_{j}+{{\mathtt{λ}}}_{j+1}}{6},$

(4)

${\mu }^{{\mathrm{{\text{'}}}}}\left({\tau }_{j}\right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{\rm K}}}{\mathrm{{\text{'}}}}}_{i}\left({\tau }_{j}\right) = \frac{{{\mathtt{λ}}}_{j+1}-{{\mathtt{λ}}}_{j-1}}{2{\mathrm{\Lambda }}},$

(5)

${\mu }^{{\mathrm{{\text{'}}}}{\mathrm{{\text{'}}}}}\left({\tau }_{j}\right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{{\rm K}}}}^{{\mathrm{{\text{'}}}}{\mathrm{{\text{'}}}}}}_{i}\left({\tau }_{j}\right) = \frac{{{\mathtt{λ}}}_{j-1}-2{{\mathtt{λ}}}_{j}+{{\mathtt{λ}}}_{j+1}}{{{\mathrm{\Lambda }}}^{2}}.$

(6)

Equations (4)–(6) are the most important relations in deriving the CBSM.

3. Cubic B-spline method

In this section, we present the cubic B-spline method for (1) and (2). Let

${\mu }_{1}\left(\tau \right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{\rm K}}}}_{i}\left(\tau \right)$

and

${\mu }_{2}\left(\tau \right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{η}}}_{i}{{{\rm K}}}}_{i}\left(\tau \right)$

denote the approximate solutions of system (1) and (2) of ${\omega }_{1}\left(\tau \right)$ and ${\omega }_{2}\left(\tau \right)$ , respectively. To overcome the singularity of (1) at $\tau = 0$ , we apply the L'Hôpital's rule to the second term as $\tau$ approaches zero, to achieve

$\left\{\begin{array}{c}(1+{\delta }_{1})\frac{{d}^{2}{\omega }_{1}\left(\tau \right)}{d{\tau }^{2}}+{{\mathrm{\hslash }}}_{1}\left({\omega }_{1}\right(\tau ),{\omega }_{2}(\tau \left)\right) = {{\mathrm{\aleph }}}_{1}\left(\tau \right),\\ (1+{\delta }_{2})\frac{{d}^{2}{\omega }_{2}\left(\tau \right)}{d{\tau }^{2}}+{{\mathrm{\hslash }}}_{2}\left({\omega }_{1}\right(\tau ),{\omega }_{2}(\tau \left)\right) = {{\mathrm{\aleph }}}_{2}\left(\tau \right),\end{array}\right.{\mathrm{f}}{\mathrm{o}}{\mathrm{r}}\;\tau = 0,\\\left\{\begin{array}{c}\frac{{d}^{2}{\omega }_{1}\left(\tau \right)}{d{\tau }^{2}}+\frac{{\delta }_{1}}{\tau }\frac{d{\omega }_{1}\left(\tau \right)}{d\tau }+{{\mathrm{\hslash }}}_{1}\left({\omega }_{1}\right(\tau ),{\omega }_{2}(\tau \left)\right) = {{\mathrm{\aleph }}}_{1}\left(\tau \right),\\ \frac{{d}^{2}{\omega }_{2}\left(\tau \right)}{d{\tau }^{2}}+\frac{{\delta }_{2}}{\tau }\frac{d{\omega }_{2}\left(\tau \right)}{d\tau }+{{\mathrm{\hslash }}}_{2}\left({\omega }_{1}\right(\tau ),{\omega }_{2}(\tau \left)\right) = {{\mathrm{\aleph }}}_{2}\left(\tau \right),\end{array}\right.{\mathrm{f}}{\mathrm{o}}{\mathrm{r}}\;\tau \ne 0.$

(7)

By discretizing (7), we get

$\left\{\begin{array}{c}(1+{\delta }_{1})\frac{{d}^{2}{\omega }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}+{{\mathrm{\hslash }}}_{1}\left({\omega }_{1}\right({\tau }_{0}),{\omega }_{2}({\tau }_{0}\left)\right) = {{\mathrm{\aleph }}}_{1}\left({\tau }_{0}\right),\\ (1+{\delta }_{2})\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}+{{\mathrm{\hslash }}}_{2}\left({\omega }_{1}\right({\tau }_{0}),{\omega }_{2}({\tau }_{0}\left)\right) = {{\mathrm{\aleph }}}_{2}\left({\tau }_{0}\right),\end{array}\right.\\\left\{\begin{array}{c}\frac{{d}^{2}{\omega }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}+\frac{{\delta }_{1}}{{\tau }_{j}}\frac{d{\omega }_{1}\left({\tau }_{j}\right)}{d\tau }+{{\mathrm{\hslash }}}_{1}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right) = {{\mathrm{\aleph }}}_{1}\left({\tau }_{j}\right),\\ \frac{{d}^{2}{\omega }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}+\frac{{\delta }_{2}}{{\tau }_{j}}\frac{d{\omega }_{2}\left({\tau }_{j}\right)}{d\tau }+{{\mathrm{\hslash }}}_{2}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right) = {{\mathrm{\aleph }}}_{2}\left({\tau }_{j}\right),\end{array}\right.$

(8)

where $j = 1,\cdots ,k$ . By using (2)–(5), (8) becomes

$\left\{\begin{array}{c}\left(1+{\delta }_{1}\right)\left(\frac{{{\mathtt{λ}}}_{-1}-2{{\mathtt{λ}}}_{0}+{{\mathtt{λ}}}_{1}}{{{\mathrm{\Lambda }}}^{2}}\right)+{{\mathrm{\hslash }}}_{1}({\varepsilon }_{1},{\vartheta }_{1}) = {{\mathrm{\aleph }}}_{1}\left({\tau }_{0}\right),\\ \left(1+{\delta }_{2}\right)\left(\frac{{{\mathtt{η}}}_{-1}-2{{\mathtt{η}}}_{0}+{{\mathtt{η}}}_{1}}{{{\mathrm{\Lambda }}}^{2}}\right)+{{\mathrm{\hslash }}}_{2}({\varepsilon }_{1},{\vartheta }_{1}) = {{\mathrm{\aleph }}}_{2}\left({\tau }_{0}\right),\end{array}\right.$

$\left\{\begin{array}{c}\begin{array}{c}\left(\frac{{\lambda }_{i-1}-2{\lambda }_{i}+{\lambda }_{i+1}}{{\Lambda }^{2}}\right)+\frac{{\delta }_{1}}{{\tau }_{j}}\left(\frac{{\lambda }_{j+1}-{\lambda }_{j-1}}{2\Lambda }\right)+{\hslash }_{1}\left(\frac{{\lambda }_{i-1}+4{\lambda }_{i}+{\lambda }_{i+1}}{6}{\text{,}}\frac{{\eta }_{i-1}+4{\eta }_{i}+{\eta }_{i+1}}{6}\right)\\ = {\aleph }_{1}\left({\tau }_{j}\right),\hspace{0.33em}j = 1,\cdots ,k,\end{array}\\ \begin{array}{c}\left(\frac{{\eta }_{i-1}-2{\eta }_{i}+{\eta }_{i+1}}{{\Lambda }^{2}}\right)+\frac{{\delta }_{2}}{{\tau }_{j}}\left(\frac{{\eta }_{j+1}-{\eta }_{j-1}}{2\Lambda }\right)+{\hslash }_{2}\left(\frac{{\lambda }_{i-1}+4{\lambda }_{i}+{\lambda }_{i+1}}{6}{\text{,}}\frac{{\eta }_{i-1}+4{\eta }_{i}+{\eta }_{i+1}}{6}\right)\\ = {\aleph }_{2}\left({\tau }_{j}\right),\hspace{0.33em}\hspace{0.33em}j = 1,\cdots ,k.\end{array}\end{array}\right.$

(9)

The initial conditions (2) as well provide the following four equations

${\omega }_{1}\left(0\right) = {\varepsilon }_{1} = \frac{{{\mathtt{λ}}}_{-1}+4{{\mathtt{λ}}}_{0}+{{\mathtt{λ}}}_{1}}{6},$

(10)

${\omega {\text{'}}}_{1}\left(0\right) = 0 = \frac{{{\mathtt{λ}}}_{1}-{{\mathtt{λ}}}_{-1}}{2{\mathrm{\Lambda }}},$

(11)

${\omega }_{2}\left(0\right) = {\vartheta }_{1} = \frac{{{\mathtt{η}}}_{-1}+4{{\mathtt{η}}}_{0}+{{\mathtt{η}}}_{1}}{6},$

(12)

${\omega {\text{'}}{\text{'}}}_{2}\left(0\right) = 0 = \frac{{{\mathtt{η}}}_{1}-{{\mathtt{η}}}_{-1}}{2{\mathrm{\Lambda }}}.$

(13)

Equations (9)–(13) give us $2(k+3)$ nonlinear equations with ${{\mathtt{λ}}}_{i}$ and ${{\mathtt{η}}}_{i},$ $i = -1,\dots ,k+1$ as unknowns. Upon solving this system, we determine the coefficients of

${\mu }_{1}\left(\tau \right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{λ}}}_{i}{{{\rm K}}}}_{i}\left(\tau \right)$

and

${\mu }_{2}\left(\tau \right) = \sum\limits _{i = -1}^{k+1}{{{\mathtt{η}}}_{i}{{{\rm K}}}}_{i}\left(\tau \right).$

4. Convergence analysis

In this section, we analyze the convergence for the proposed method. For this purpose, we assume ${\omega }_{1}\left(\tau \right),{\omega }_{2}\left(\tau \right)\in {C}^{5}\left[{\mathrm{0,1}}\right].$ From (3)–(5), we get ^[39]

${\mathrm{\Lambda }}\left[{\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i-1}\right)+4{\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right)+{\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i+1}\right)\right] = 3\left[{\omega }_{j}\left({\tau }_{i+1}\right)-{\omega }_{j}\left({\tau }_{i-1}\right)\right],$

(14)

${{\mathrm{\Lambda }}}^{2}{\mu }_{j}^{{\mathrm{{\text{'}}}}{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right) = 6\left[{\mu }_{j}\left({\tau }_{i+1}\right)-{\mu }_{j}\left({\tau }_{i}\right)\right]-2{\mathrm{\Lambda }}\left[2{\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right)+{\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i+1}\right)\right],$

(15)

$j = {\mathrm{1,2}}.$ Using the shifting operator, $E\left({\mu }_{j}\left({\tau }_{i}\right)\right) = {\mu }_{j}\left({\tau }_{i+1}\right),$ (14) may be written as

$\frac{{\mathrm{\Lambda }}}{6}\left({E}^{-1}+4+E\right){\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right) = \frac{1}{2}\left(E-{E}^{-1}\right){\omega }_{j}\left({\tau }_{i}\right),$

(16)

$j = {\mathrm{1,2}}.$ Since $E = {e}^{{\mathrm{\Lambda }}D}$ and $D\equiv d/d\tau$ , one can get

${e}^{{\mathrm{\Lambda }}D}+{e}^{-{\mathrm{\Lambda }}D} = 2\sum\limits _{k = 0}^{{\mathrm{\infty }}}\frac{{\left({\mathrm{\Lambda }}D\right)}^{2k}}{\left(2k\right)!},\\{e}^{{\mathrm{\Lambda }}D}-{e}^{-{\mathrm{\Lambda }}D} = 2\sum\limits _{k = 0}^{{\mathrm{\infty }}}\frac{{\left({\mathrm{\Lambda }}D\right)}^{2k+1}}{\left(2k+1\right)!}.$

(17)

Therefore, using (17), (16) can be expressed as

$\left(1+\frac{1}{3}\sum\limits _{k = 1}^{{\mathrm{\infty }}}\frac{{\left({\mathrm{\Lambda }}D\right)}^{2k}}{\left(2k\right)!}\right){\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right) = \left(\sum\limits _{k = 0}^{{\mathrm{\infty }}}\frac{{\left({\mathrm{\Lambda }}D\right)}^{2k+1}}{(2k+1)!}\right){\omega }_{j}\left({\tau }_{i}\right).$

(18)

Simplifying (18) gives

${\mu }_{j}^{{\text{'}}}\left({\tau }_{i}\right) = \left(\sum\limits _{k = 0}^{\infty }\frac{{\left(\Lambda D\right)}^{2k+1}}{(2k+1)!}\right){\left(1+\frac{1}{3}\sum\limits _{k = 1}^{\infty }\frac{{\left(\Lambda D\right)}^{2k}}{\left(2k\right)!}\right)}^{-1}{\omega }_{j}\left({\tau }_{i}\right)\\{\text{ = (D+}}\frac{{{\mathrm{\Lambda }}}^{2}{{\mathrm{D}}}^{3}}{3!}+\frac{{{\mathrm{\Lambda }}}^{4}{{\mathrm{D}}}^{5}}{5!}+\cdots )\left(1-\left(\frac{{{\mathrm{\Lambda }}}^{2}{{\mathrm{D}}}^{2}}{6}+\frac{{{\mathrm{\Lambda }}}^{4}{{\mathrm{D}}}^{4}}{72}+\cdots \right)\right.\\\left.+{\left(\frac{{\Lambda }^{2}{D}^{2}}{6}+\frac{{\Lambda }^{4}{D}^{4}}{72}+\cdots \right)}^{2}+\cdots \right){\omega }_{j}\left({\tau }_{i}\right)\\ = D\left(1-\frac{{\Lambda }^{4}{D}^{4}}{180}+\frac{{\Lambda }^{6}{D}^{6}}{1512}-\cdots \right){\omega }_{j}\left({\tau }_{i}\right).$

(19)

Therefore,

${\mu }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right) = {\omega }_{j}^{{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right)-\frac{{{\mathrm{\Lambda }}}^{4}}{180}{\omega }_{j}^{\left(5\right)}\left({\tau }_{i}\right)+\cdots .$

(20)

Similarly, (15) gives

${\mu }_{j}^{{\mathrm{{\text{'}}}}{\mathrm{{\text{'}}}}}\left({\tau }_{i}\right) = {\omega }_{j}^{{\text{'}}{\text{'}}}\left({\tau }_{i}\right)-\frac{1}{12}{{\mathrm{\Lambda }}}^{2}{\omega }_{j}^{\left(4\right)}\left({\tau }_{i}\right)+\frac{1}{360}{{\mathrm{\Lambda }}}^{4}{\omega }_{j}^{\left(6\right)}\left({\tau }_{i}\right)+O\left({{\mathrm{\Lambda }}}^{6}\right),$

(21)

At this point, the error functions ${e}_{j}\left(\tau \right),j = {\mathrm{1,2}},$ are stated as follows:

${e}_{1}\left({\tau }_{i}\right) = {\aleph }_{1}\left({\tau }_{j}\right)-\frac{{d}^{2}{\omega }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{\delta }_{1}}{{\tau }_{j}}\frac{d{\omega }_{1}\left({\tau }_{j}\right)}{d\tau }-{\hslash }_{1}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right)\\ = \frac{{d}^{2}{\mu }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}+\frac{{\delta }_{1}}{{\tau }_{j}}\frac{d{\mu }_{1}\left({\tau }_{j}\right)}{d\tau }+{\hslash }_{1}\left({\mu }_{1}\left({\tau }_{j}\right),{\mu }_{2}\left({\tau }_{j}\right)\right)\\ -\frac{{d}^{2}{\omega }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{\delta }_{1}}{{\tau }_{j}}\frac{d{\omega }_{1}\left({\tau }_{j}\right)}{d\tau }-{\hslash }_{1}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right)\\ = \left[\frac{{d}^{2}{\mu }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{d}^{2}{\omega }_{1}\left({\tau }_{j}\right)}{d{\tau }^{2}}\right]+\frac{{\delta }_{1}}{{\tau }_{j}}\left[\frac{d{\mu }_{1}\left({\tau }_{j}\right)}{d\tau }-\frac{d{\omega }_{1}\left({\tau }_{j}\right)}{d\tau }\right],\\{e}_{2}\left({\tau }_{i}\right) = {\aleph }_{2}\left({\tau }_{j}\right)-\frac{{d}^{2}{\omega }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{\delta }_{2}}{{\tau }_{j}}\frac{d{\omega }_{2}\left({\tau }_{j}\right)}{d\tau }-{\hslash }_{2}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right)\\ = \frac{{d}^{2}{\mu }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}+\frac{{\delta }_{2}}{{\tau }_{j}}\frac{d{\mu }_{2}\left({\tau }_{j}\right)}{d\tau }+{\hslash }_{2}\left({\mu }_{1}\left({\tau }_{j}\right),{\mu }_{2}\left({\tau }_{j}\right)\right)\\-\frac{{d}^{2}{\omega }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{\delta }_{2}}{{\tau }_{j}}\frac{d{\omega }_{2}\left({\tau }_{j}\right)}{d\tau }-{\hslash }_{2}\left({\omega }_{1}\right({\tau }_{j}),{\omega }_{2}({\tau }_{j}\left)\right)\\ = \left[\frac{{d}^{2}{\mu }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}-\frac{{d}^{2}{\omega }_{2}\left({\tau }_{j}\right)}{d{\tau }^{2}}\right]+\frac{{\delta }_{2}}{{\tau }_{j}}\left[\frac{d{\mu }_{2}\left({\tau }_{j}\right)}{d\tau }-\frac{d{\omega }_{2}\left({\tau }_{j}\right)}{d\tau }\right],$

(22)

where $j = 1,{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }}.{\mathrm{ }},k$ . Substitute (20) and (21) in (22) yields

${‖{e}_{1}\left({\tau }_{i}\right)‖}_{{\mathrm{\infty }}} = O\left({{\mathrm{\Lambda }}}^{2}\right),$

${‖{e}_{2}\left({\tau }_{i}\right)‖}_{{\mathrm{\infty }}} = O\left({{\mathrm{\Lambda }}}^{2}\right).$

(23)

For i = 0, we get

${e}_{1}\left({\tau }_{0}\right) = {\aleph }_{1}\left({\tau }_{0}\right)-\left(1+{\delta }_{1}\right)\frac{{d}^{2}{\omega }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}-{\hslash }_{1}\left({\omega }_{1}\left({\tau }_{0}\right),{\omega }_{2}\left({\tau }_{0}\right)\right)\\ = \left(1+{\delta }_{1}\right)\frac{{d}^{2}{\mu }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}+{\hslash }_{1}\left({\mu }_{1}\left({\tau }_{0}\right),{\mu }_{2}\left({\tau }_{0}\right)\right)\\-\left(1+{\delta }_{1}\right)\frac{{d}^{2}{\omega }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}-{\hslash }_{1}({\omega }_{1}\left({\tau }_{0}\right),{\omega }_{2}\left({\tau }_{0}\right)\\ = \left(1+{\delta }_{1}\right)\left[\frac{{d}^{2}{\mu }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}-\frac{{d}^{2}{\omega }_{1}\left({\tau }_{0}\right)}{d{\tau }^{2}}\right],\\{e}_{2}\left({\tau }_{0}\right) = {\aleph }_{2}\left({\tau }_{0}\right)-(1+{\delta }_{2})\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}-{\hslash }_{2}({\omega }_{1}\left({\tau }_{0}\right),{\omega }_{2}\left({\tau }_{0}\right)) \\ = \left(1+{\delta }_{2}\right)\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}+{\hslash }_{2}\left({\mu }_{1}\left({\tau }_{0}\right),{\mu }_{2}\left({\tau }_{0}\right)\right)\\ -\left(1+{\delta }_{2}\right)\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}-{\hslash }_{1}({\omega }_{1}\left({\tau }_{0}\right),{\omega }_{2}\left({\tau }_{0}\right)\\ = (1+{\delta }_{2})\left[\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}-\frac{{d}^{2}{\omega }_{2}\left({\tau }_{0}\right)}{d{\tau }^{2}}\right].$

(24)

Using (21) in (24), we have

${‖{e}_{1}\left({\tau }_{0}\right)‖}_{{\mathrm{\infty }}} = O\left({{\mathrm{\Lambda }}}^{2}\right),\\{‖{e}_{2}\left({\tau }_{0}\right)‖}_{{\mathrm{\infty }}} = O\left({{\mathrm{\Lambda }}}^{2}\right).$

(25)

Therefore, from (23) and (25), the truncation error for the considered system is $O\left({{\mathrm{\Lambda }}}^{2}\right)$ .

5. Numerical results

In this section, we present the numerical solution to (1) and (2) using the cubic B-spline technique. Several problems are examined to prove the accuracy and efficiency of the proposed method using the absolute errors between the approximate solutions and the exact solutions ( $\left|{e}_{j}\left(\tau \right)\right|)$ for various k. All the results are calculated by using MATHEMATICA 12.

Problem 1. Consider the following system ^[24]

$\frac{{d}^{2}{\omega }_{1}\left(\tau \right)}{d{\tau }^{2}}+\frac{3}{\tau }\frac{d{\omega }_{1}\left(\tau \right)}{d\tau }-4({\omega }_{1}\left(\tau \right)+{\omega }_{2}\left(\tau \right)) = 0,\\\frac{{d}^{2}{\omega }_{2}\left(\tau \right)}{d{\tau }^{2}}+\frac{2}{\tau }\frac{d{\omega }_{2}\left(\tau \right)}{d\tau }+3({\omega }_{1}\left(\tau \right)+{\omega }_{2}\left(\tau \right)) = 0,$

(26)

subject to

${\omega }_{1}\left(0\right) = 1,{\omega }_{1}^{{\text{'}}}\left(0\right) = 0,\\{\omega }_{2}\left(0\right) = 1,{\omega }_{2}^{{\text{'}}}\left(0\right) = 0,$

(27)

where the exact solutions are ${\omega }_{1}\left(\tau \right) = 1+{\tau }^{2}$ and ${\omega }_{2}\left(\tau \right) = 1-{\tau }^{2}$ . In this example, we use ${\mathrm{\Lambda }} = 0.1$ . The numerical results and the exact solution at the grid points are listed in Table 2. We can conclude from Table 2 that the obtained numerical results are in excellent agreement with the exact solutions. We note that for this problem, our results are exact and the achieved errors are only due to round-off calculations. The CPU time for this problem, with ${\mathrm{\Lambda }} = 0.1$ , is 0.0156s.

Table 2. Numerical results for Problem 1.

$\tau$	${\omega }_{1}\left(\tau \right)$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$	${\omega }_{2}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$
$0.0$	1	$1$	$0$	$1$	$1$	$2.22045\times {10}^{-16}$
$0.1$	1.01	$1.01$	$0$	0.99	0.99	$2.22045\times {10}^{-16}$
$0.2$	1.04	$1.04$	$0$	0.96	0.96	$2.22045\times {10}^{-16}$
$0.3$	1.09	$1.09$	$2.22045\times {10}^{-16}$	0.91	0.91	$2.22045\times {10}^{-16}$
$0.4$	1.16	1.16	$2.22045\times {10}^{-16}$	0.84	0.84	$3.33067\times {10}^{-16}$
$0.5$	1.25	1.25	$2.22045\times {10}^{-16}$	0.75	0.75	$3.33067\times {10}^{-16}$
$0.6$	1.36	1.36	$4.44089\times {10}^{-16}$	0.64	0.64	$3.33067\times {10}^{-16}$
$0.7$	1.49	1.49	$2.22045\times {10}^{-16}$	0.51	0.51	$4.44089\times {10}^{-16}$
$0.8$	1.61	1.61	$0$	0.36	0.36	$4.44089\times {10}^{-16}$
$0.9$	1.81	1.81	$2.22045\times {10}^{-16}$	0.19	0.19	$5.55112\times {10}^{-16}$
$1.0$	2	2	$0$	0	5.64363 $\times {10}^{-16}$	$5.64363\times {10}^{-16}$

| Show Table

DownLoad: CSV

Problem 2. Consider the following system ^[27]

${\omega }_{1}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{2}{\tau }{\omega }_{1}^{{\text{'}}}\left(\tau \right)-\left(4{\tau }^{2}+6\right){\omega }_{1}\left(\tau \right)+{\omega }_{2}\left(\tau \right) = {\tau }^{4}-{\tau }^{3},\\{\omega }_{2}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{8}{\tau }{\omega }_{2}^{{\text{'}}}\left(\tau \right)+{\omega }_{1}\left(\tau \right)+\tau {\omega }_{2}\left(\tau \right) = {e}^{{\tau }^{2}}+{\tau }^{5}-{\tau }^{4}+44{\tau }^{2}-30\tau ,$

(28)

subject to

${\omega }_{1}\left(0\right) = 1,{\omega }_{1}^{{\text{'}}}\left(0\right) = 0,\\{\omega }_{2}\left(0\right) = 0,{\omega }_{2}^{{\text{'}}}\left(0\right) = 0,$

(29)

where the exact solutions are

${\omega }_{1}\left(\tau \right) = {e}^{{\tau }^{2}}$

and

${\omega }_{2}\left(\tau \right) = {\tau }^{4}-{\tau }^{3}.$

The obtained numerical and exact solutions, with different values of ${\mathrm{\Lambda }}$ , are depicted in Figure 1. The absolute errors between exact and numerical results, with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ , are presented in Tables 3 and 4. In Table 5, we compare the maximum absolute error of CBSM with those of ^[27]. Our results seem to be better than those of ^[27]. The CPU time for this problem, with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ , are 0.0156s and 0.0625s respectively.

Figure 1. Absolute error functions for Problem 2.

DownLoad: Full-Size Img PowerPoint

Table 3. Numerical result for approximate solution of

${\omega }_{1}\left(\tau \right)$ in Problem 2.

$\tau$	${\omega }_{1}\left(\tau \right)$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$
$0.0$	$1$	$1$	$0$	$1$	$0$
$0.1$	$1.01005$	$1.01008$	$3.38838\times {10}^{-5}$	$1.01005$	$1.71898\times {10}^{-7}$
$0.2$	$1.04081$	$1.04091$	$9.51843\times {10}^{-5}$	$1.04081$	$7.1747\times {10}^{-7}$
$0.3$	1.09417	$1.09438$	$2.01533\times {10}^{-4}$	$1.09417$	$1.75608\times {10}^{-6}$
$0.4$	1.17351	1.17389	$3.80997\times {10}^{-4}$	$1.17351$	$3.5098\times {10}^{-6}$
$0.5$	1.28403	1.2847	$6.71612\times {10}^{-4}$	$1.28403$	$6.3527\times {10}^{-6}$
$0.6$	1.43333	1.43446	$1.13524\times {10}^{-3}$	$1.43334$	$1.08922\times {10}^{-5}$
$0.7$	1.63232	1.63419	$1.87147\times {10}^{-3}$	$1.63233$	$1.81055\times {10}^{-5}$
$0.8$	1.89648	1.89952	$3.04087\times {10}^{-3}$	$1.89651$	$2.9567\times {10}^{-5}$
$0.9$	2.24791	2.25281	$4.90399\times {10}^{-3}$	$2.24796$	$4.78283\times {10}^{-5}$
$1.0$	2.71828	2.72617	$7.88713\times {10}^{-3}$	$2.71836$	$7.70601\times {10}^{-5}$

| Show Table

DownLoad: CSV

Table 4. Numerical result for approximate solution of

${\omega }_{2}\left(\tau \right)$ in Problem 2.

$\tau$	${\omega }_{2}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$
0.0	0	$5.99863\times {10}^{-31}$	$5.99863\times {10}^{-31}$	0	0
0.1	$-0.0009$	$-0.000853346$	$4.66538\times {10}^{-5}$	$-0.000899887$	$1.13186\times {10}^{-7}$
0.2	$-0.0064$	$-0.0063379$	$6.21025\times {10}^{-5}$	$-0.00639955$	$4.45865\times {10}^{-7}$
$0.3$	-0.0189	$-0.0187802$	$1.19801\times {10}^{-4}$	$-0.01889900$	$9.98298\times {10}^{-7}$
$0.4$	-0.0384	$-0.0382022$	$1.97763\times {10}^{-4}$	$-0.0383982$	$1.76662\times {10}^{-6}$
$0.5$	-0.0625	$-0.0622052$	$2.94843\times {10}^{-4}$	$-0.0624973$	$2.74383\times {10}^{-6}$
$0.6$	-0.0864	$-0.0859881$	$4.11936\times {10}^{-4}$	$-0.0863961$	$3.91802\times {10}^{-6}$
$0.7$	-0.1029	$-0.102353$	$5.46598\times {10}^{-4}$	$-0.102895$	$5.2695\times {10}^{-6}$
$0.8$	-0.1024	$-0.101704$	$6.95542\times {10}^{-4}$	$-0.102393$	$6.76623\times {10}^{-6}$
$0.9$	-0.0729	$-0.0720466$	$8.5344\times {10}^{-4}$	$-0.0728916$	$8.35631\times {10}^{-6}$
$1.0$	0	$0.00101164$	$1.01164\times {10}^{-3}$	$9.95549\times {10}^{-6}$	$9.95549\times {10}^{-6}$

| Show Table

DownLoad: CSV

Table 5. Comparison of maximum absolute error of Problem 2.

	${\mu }_{1}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$
CBSM $({\mathrm{\Lambda }}=0.1)$	$7.88\times {10}^{-3}$	$1.01\times {10}^{-3}$
CBSM $({\mathrm{\Lambda }}=0.01)$	$7.70\times {10}^{-5}$	$9.95\times {10}^{-6}$
^[27] (N=5)	$3.14\times {10}^{-2}$	$4.19\times {10}^{-5}$
^[27] (N=6)	$6.11\times {10}^{-4}$	$7.22\times {10}^{-6}$

| Show Table

DownLoad: CSV

Problem 3. Consider the following system ^[24,41]

${\omega }_{1}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{5}{\tau }{\omega }_{1}^{{\text{'}}}\left(\tau \right)+8({e}^{{\omega }_{1}\left(\tau \right)}+2{e}^{-\frac{{\omega }_{2}\left(\tau \right)}{2}}) = 0,\\ {\omega }_{2}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{3}{\tau }{\omega }_{2}^{{\text{'}}}\left(\tau \right)-8({e}^{\frac{{\omega }_{1}\left(\tau \right)}{2}}+{e}^{-{\omega }_{2}\left(\tau \right)}) = 0,$

(30)

subject to

${\omega }_{1}\left(0\right) = 1-2{\mathrm{ln}}\left(2\right),{\omega }_{1}^{{\text{'}}}\left(0\right) = 0,\\{\omega }_{2}\left(0\right) = 1+2{\mathrm{ln}}\left(2\right),{\omega }_{2}^{{\text{'}}}\left(0\right) = 0,$

(31)

where the exact solutions are

${\omega }_{1}\left(\tau \right) = 1-2{\mathrm{ln}}({\tau }^{2}+2)$

and

${\omega }_{2}\left(\tau \right) = 1+2{\mathrm{ln}}({\tau }^{2}+2).$

We depicted our numerical and exact solutions with different values of ${\mathrm{\Lambda }}$ in Figure 2. The absolute errors between exact and numerical results are reported in Tables 6 and 7. Table 8 compares the maximum absolute error of CBSM with those of the method in ^[41]. It appears that our findings are superior to the outcomes presented in ^[41]. The CPU time for this problem, with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ , are 0.0156s and 0.0781s respectively.

Figure 2. Absolute error functions for Problem 3.

DownLoad: Full-Size Img PowerPoint

Table 6. Numerical result for approximate solution of

${\omega }_{1}\left(\tau \right)$ in Problem 3.

$\tau$	${\omega }_{1}\left(\tau \right)$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$
$0.0$	$-0.386294$	$-0.386294$	$2.22045\times {10}^{-16}$	$-0.386294$	$-2.22045\times {10}^{-16}$
$0.1$	$-0.396269$	$-0.396257$	$1.28893\times {10}^{-5}$	$-0.396269$	$4.13534\times {10}^{-8}$
$0.2$	$-0.4259$	$-0.425878$	$2.11827\times {10}^{-5}$	$-0.425899$	$1.5319\times {10}^{-7}$
$0.3$	$-0.474328$	$-0.474293$	$3.503\times {10}^{-5}$	$-0.474328$	$3.07878\times {10}^{-7}$
$0.4$	$-0.540216$	$-0.540166$	$5.02533\times {10}^{-5}$	$-0.540216$	$4.67063\times {10}^{-7}$
$0.5$	$-0.62186$	$-0.621799$	$6.18095\times {10}^{-5}$	$-0.62186$	$5.91857\times {10}^{-7}$
$0.6$	$-0.717323$	$-0.717256$	$6.68475\times {10}^{-5}$	$-0.717323$	$6.51126\times {10}^{-7}$
$0.7$	$-0.824565$	$-0.824502$	$6.3593\times {10}^{-5}$	$-0.824565$	$6.26415\times {10}^{-7}$
$0.8$	$-0.941558$	$-0.941506$	$5.16047\times {10}^{-5}$	$-0.941557$	$5.13027\times {10}^{-7}$
$0.9$	$-1.06637$	$-1.06634$	$3.15988\times {10}^{-5}$	$-1.06637$	$3.18094\times {10}^{-7}$
$1.0$	$-1.19722$	$-1.19722$	$5.10219\times {10}^{-6}$	$-1.19722$	$5.69956\times {10}^{-8}$

| Show Table

DownLoad: CSV

Table 7. Numerical result for approximate solution of

${\omega }_{2}\left(\tau \right)$ in Problem 3.

$\tau$	${\omega }_{2}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$
$0.0$	2.38629	2.38629	$2.22045\times {10}^{-16}$	2.38629	$2.22045\times {10}^{-16}$
$0.1$	2.39627	2.39625	$1.47619\times {10}^{-5}$	2.39627	$6.20966\times {10}^{-8}$
$0.2$	2.4259	2.42587	$3.17866\times {10}^{-5}$	2.42599	$2.32511\times {10}^{-7}$
$0.3$	2.47433	2.47427	$5.42035\times {10}^{-5}$	2.47433	$4.75075\times {10}^{-7}$
$0.4$	2.54022	2.54014	$7.94034\times {10}^{-5}$	2.54022	$7.3881\times {10}^{-7}$
$0.5$	2.62186	2.62176	$1.01507\times {10}^{-4}$	2.62186	$9.70554\times {10}^{-7}$
$0.6$	2.71732	2.71721	$1.16022\times {10}^{-4}$	2.71732	$1.12546\times {10}^{-6}$
$0.7$	2.82457	2.82445	$1.1998\times {10}^{-4}$	2.82456	$1.17361\times {10}^{-6}$
$0.8$	2.94156	2.94145	$1.121\times {10}^{-4}$	2.94156	$1.10206\times {10}^{-6}$
$0.9$	3.06637	3.06628	$9.26097\times {10}^{-5}$	3.06637	$9.13128\times {10}^{-7}$
$1.0$	3.19722	3.19716	$6.2868\times {10}^{-5}$	3.19722	$6.20546\times {10}^{-7}$

| Show Table

DownLoad: CSV

Table 8. Comparison of maximum absolute error of Problem 3.

	${\mu }_{1}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$
CBSM $({\mathrm{\Lambda }}=0.1)$	$6.68\times {10}^{-5}$	$1.19\times {10}^{-4}$
CBSM $({\mathrm{\Lambda }}=0.01)$	$6.51\times {10}^{-7}$	$1.17\times {10}^{-6}$
^[41] (j=3)	$1.47\times {10}^{-3}$	$1.64\times {10}^{-3}$
^[41] (j=4)	$3.67\times {10}^{-4}$	$4.10\times {10}^{-4}$

| Show Table

DownLoad: CSV

Problem 4. Consider the following system of LEE ^{[24,27,40,41]}

${\omega }_{1}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{1}{\tau }{\omega }_{1}^{{\text{'}}}\left(\tau \right)-{\omega }_{2}^{3}\left(\tau \right)({\omega }_{1}^{2}+1) = 0,\\{\omega }_{2}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{3}{\tau }{\omega }_{2}^{{\text{'}}}\left(\tau \right)+{\omega }_{2}^{5}\left(\tau \right)({\omega }_{1}^{2}+3) = 0,$

(32)

subject to

${\omega }_{1}\left(0\right) = 1,{{\omega }^{{\text{'}}}}_{1}\left(0\right) = 0,\\{\omega }_{2}\left(0\right) = 1,{{\omega }^{{\text{'}}}}_{2}\left(0\right) = 0,$

(33)

where the exact solutions are

${\omega }_{1}\left(\tau \right) = \sqrt{{1+\tau }^{2}}$

and

${\omega }_{2}\left(\tau \right) = \frac{1}{\sqrt{{1+\tau }^{2}}}.$

The achieved numerical results with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ are exposed in Tables 9 and 10. Table 11 shows a comparison between the maximum absolute error of CBSM and that of the approaches discussed in ^[27,40,41]. Based on the results, it seems that our research outperforms the results reported in ^[27,40,41]. Absolute errors for different values of ${\mathrm{\Lambda }}$ are exposed in Figure 3. The CPU time for this problem, with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ , are 0.0156s and 0.0781s respectively.

Table 9. Numerical result for approximate solution of

${\omega }_{1}\left(\tau \right)$ in Problem 4.

$\tau$	${\omega }_{1}\left(\tau \right)$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$
$0.0$	1	1	0	1	0
$0.1$	1.004988	1.004978	$9.41481\times {10}^{-6}$	1.004987	$6.21775\times {10}^{-8}$
$0.2$	1.019804	1.019776	$2.80913\times {10}^{-5}$	1.019804	$2.29506\times {10}^{-7}$
$0.3$	1.044031	1.043979	$5.15984\times {10}^{-5}$	1.04403	$4.61458\times {10}^{-7}$
$0.4$	1.077033	1.076957	$7.60906\times {10}^{-5}$	1.077032	$7.07716\times {10}^{-7}$
$0.5$	1.118034	1.117936	$9.76652\times {10}^{-5}$	1.118033	$9.25606\times {10}^{-7}$
$0.6$	1.16619	1.166076	$1.13971\times {10}^{-4}$	1.166189	$1.09013\times {10}^{-6}$
$0.7$	1.220656	1.220531	$1.24491\times {10}^{-4}$	1.220654	$1.19564\times {10}^{-6}$
$0.8$	1.280625	1.280495	$1.30176\times {10}^{-4}$	1.280624	$1.25163\times {10}^{-6}$
$0.9$	1.345362	1.34523	$1.32854\times {10}^{-4}$	1.345361	$1.27653\times {10}^{-6}$
$1.0$	1.414214	1.414079	$1.34662\times {10}^{-4}$	1.414212	$1.29187\times {10}^{-6}$

| Show Table

DownLoad: CSV

Table 10. Numerical result for approximate solution of

${\omega }_{2}\left(\tau \right)$ in Problem 4.

$\tau$	${\omega }_{2}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$
$0.0$	1	1	0	1	$2.22045\times {10}^{-16}$
$0.1$	0.995037	0.995059	$2.1554\times {10}^{-5}$	0.995037	$9.03189\times {10}^{-8}$
$0.2$	0.980581	0.980623	$4.20048\times {10}^{-5}$	0.980581	$3.08008\times {10}^{-7}$
$0.3$	0.957826	0.957887	$6.03609\times {10}^{-5}$	0.957827	$5.34936\times {10}^{-7}$
$0.4$	0.928477	0.928545	$6.87324\times {10}^{-5}$	0.928477	$6.48155\times {10}^{-7}$
$0.5$	0.894427	0.894487	$5.93414\times {10}^{-5}$	0.894428	$5.76216\times {10}^{-7}$
$0.6$	0.857493	0.857525	$3.17569\times {10}^{-5}$	0.857493	$3.15637\times {10}^{-7}$
$0.7$	0.819232	0.819223	$9.41889\times {10}^{-6}$	0.819232	$8.61722\times {10}^{-8}$
$0.8$	0.780869	0.780811	$5.74405\times {10}^{-5}$	0.780868	$5.60283\times {10}^{-7}$
$0.9$	0.743294	0.743188	$1.05817\times {10}^{-4}$	0.743293	$1.04065\times {10}^{-6}$
$1.0$	0.707107	0.706957	$1.4962\times {10}^{-4}$	0.707105	$1.4772\times {10}^{-6}$

| Show Table

DownLoad: CSV

Table 11. Comparison of maximum absolute error of Problem 4.

	${\mu }_{1}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$
CBSM $({\mathrm{\Lambda }}=0.1)$	$1.34\times {10}^{-4}$	$1.49\times {10}^{-4}$
CBSM $({\mathrm{\Lambda }}=0.01)$	$1.29\times {10}^{-6}$	$1.47\times {10}^{-6}$
^[27] (N=4)	$6.44\times {10}^{-4}$	$8.87\times {10}^{-4}$
^[27] (N=5)	$7.46\times {10}^{-5}$	$6.08\times {10}^{-5}$
^[40] (n=4)	$4.86\times {10}^{-4}$	$1.45\times {10}^{-3}$
^[41] (j=3)	$2.76\times {10}^{-5}$	$1.31\times {10}^{-4}$
^[41] (j=4)	$6.87\times {10}^{-6}$	$3.28\times {10}^{-5}$

| Show Table

DownLoad: CSV

Figure 3. Absolute error functions for Problem 4.

DownLoad: Full-Size Img PowerPoint

Problem 5. Consider the following system of LEE ^[24,40]

${\omega }_{1}^{{\text{'}}{\text{'}}}+\frac{8}{\tau }{\omega }_{1}^{{\text{'}}}\left(\tau \right)+(18{\omega }_{1}\left(\tau \right)-4{\omega }_{1}\left(\tau \right){\mathrm{ln}}{\omega }_{2}\left(\tau \right)) = 0,\\{\omega }_{2}^{{\text{'}}{\text{'}}}\left(\tau \right)+\frac{4}{\tau }{\omega }_{2}^{{\text{'}}}\left(\tau \right)+(4{\omega }_{2}\left(\tau \right){\mathrm{ln}}{\omega }_{1}\left(\tau \right)-10{\omega }_{2}\left(\tau \right)) = 0,$

(34)

subject to

${\omega }_{1}\left(0\right) = 1,{\omega }_{1}^{{\text{'}}}\left(0\right) = 0, \\ {\omega }_{2}\left(0\right) = 1,{\omega }_{2}^{{\text{'}}}\left(0\right) = 0,$

(35)

where the exact solutions are ${\omega }_{1}\left(\tau \right) = {e}^{-{\tau }^{2}}$ and ${\omega }_{2}\left(\tau \right) = {e}^{{\tau }^{2}}$ .

Figure 4 represents the plot of our numerical and exact solutions for Problem 5 with different values of ${\mathrm{\Lambda }}$ . The absolute errors for ${\mathrm{\Lambda }} = 0.1$ and $0.01$ are presented in Tables 12 and 13. Table 14 displays a comparison between the maximum absolute error of CBSM and that of the approaches discussed in ^[40,41]. Our results indicate that they are better than the results reported in ^[40,41]. The CPU time for this problem, with ${\mathrm{\Lambda }} = 0.1$ and $0.01$ , are 0.0156s and 0.0625s respectively. From all these tables and figures, we can observe that our numerical results are in good agreement with the exact ones. Moreover, it appears from our findings that the cubic B-spline method exhibits more accurate results than some existing methods.

Figure 4. Absolute error functions for Problem 5.

DownLoad: Full-Size Img PowerPoint

Table 12. Numerical result for approximate solution of

${\omega }_{1}\left(\tau \right)$ in Problem 5.

$\tau$	${\omega }_{1}\left(\tau \right)$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$	${\mu }_{1}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{1}\left(\tau \right)\right\|$
$0.0$	1	1	0	1	0
$0.1$	0.99005	0.990073	$2.29673\times {10}^{-5}$	0.99005	$5.53154\times {10}^{-8}$
$0.2$	0.960789	0.960817	$2.7658"\times {10}^{-5}$	0.96079	$2.03451\times {10}^{-7}$
$0.3$	0.913931	0.913979	$4.76609\times {10}^{-5}$	0.913932	$4.05169\times {10}^{-7}$
$0.4$	0.852144	0.85221	$6.64108\times {10}^{-5}$	0.852144	$6.04412\times {10}^{-7}$
$0.5$	0.778801	0.778879	$7.81482\times {10}^{-5}$	0.778802	$7.42993\times {10}^{-7}$
$0.6$	0.697676	0.697756	$7.9594\times {10}^{-5}$	0.697677	$7.74528\times {10}^{-7}$
$0.7$	0.612626	0.612695	$6.83427\times {10}^{-5}$	0.612627	$6.75274\times {10}^{-7}$
$0.8$	0.527292	0.527337	$4.49264\times {10}^{-5}$	0.527293	$4.49447\times {10}^{-7}$
$0.9$	0.444858	0.444871	$1.24858\times {10}^{-5}$	0.444858	$1.28099\times {10}^{-7}$
$1.0$	0.367879	0.367856	$2.39084\times {10}^{-5}$	0.367879	$2.37741\times {10}^{-7}$

| Show Table

DownLoad: CSV

Table 13. Numerical result for approximate solution of

${\omega }_{2}\left(\tau \right)$ in Problem 5.

$\tau$	${\omega }_{2}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.1$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$	${\mu }_{2}\left(\tau \right)$ ( ${\mathrm{\Lambda }}=0.01$ )	$\left\|{e}_{2}\left(\tau \right)\right\|$
$0.0$	1	1	0	1	0
$0.1$	1.01005	1.010078	$2.82181\times {10}^{-5}$	1.01005	$1.0345\times {10}^{-7}$
$0.2$	1.040811	1.040871	$6.04278\times {10}^{-5}$	1.040811	$4.35512\times {10}^{-7}$
$0.3$	1.094174	1.094299	$1.24976\times {10}^{-4}$	1.094175	$1.0806\times {10}^{-6}$
$0.4$	1.173511	1.173751	$2.40286\times {10}^{-4}$	1.173513	$2.19844\times {10}^{-6}$
$0.5$	1.284025	1.284458	$4.32197\times {10}^{-4}$	1.284029	$4.06313\times {10}^{-6}$
$0.6$	1.433329	1.434077	$7.47197\times {10}^{-4}$	1.433337	$7.12898\times {10}^{-6}$
$0.7$	1.632316	1.633578	$1.26164\times {10}^{-3}$	1.632328	$1.21425\times {10}^{-5}$
$0.8$	1.896481	1.898582	$2.1011\times {10}^{-3}$	1.896501	$1.21425\times {10}^{-5}$
$0.9$	2.247908	2.251381	$3.47326\times {10}^{-3}$	2.247942	$3.3724\times {10}^{-5}$
$1.0$	2.718282	2.724006	$5.72394\times {10}^{-3}$	2.718338	$5.56973\times {10}^{-5}$

| Show Table

DownLoad: CSV

Table 14. Comparison of maximum absolute error of Problem 5.

	${\mu }_{1}\left(\tau \right)$	${\mu }_{2}\left(\tau \right)$
CBSM $({\mathrm{\Lambda }}=0.1)$	$7.95\times {10}^{-5}$	$5.72\times {10}^{-3}$
CBSM $({\mathrm{\Lambda }}=0.01)$	$7.74\times {10}^{-7}$	$5.56\times {10}^{-5}$
^[40] (n=4)	$2.99\times {10}^{-3}$	$9.00\times {10}^{-3}$
^[41] (j=3)	$4.54\times {10}^{-4}$	$5.05\times {10}^{-4}$
^[41] (j=4)	$1.88\times {10}^{-4}$	$1.26\times {10}^{-4}$

| Show Table

DownLoad: CSV

6. Conclusions

The system of Lane-Emden type equations describes a variety of phenomena in theoretical physics, star structure, and astrophysics. In this study, we introduce and examine the use of the cubic B-spline method for studying the solution of singular and nonlinear systems of Lane-Emden equations. To address the singularity that occurs at τ = 0, we use L'Hôpital's rule. We also evaluate the accuracy and validity of the proposed technique, demonstrating its success in solving the considered system. The presented test problems have shown the simplicity and applicability of the proposed method. We provide tabular and graphical representations to confirm its effectiveness, observing that our numerical solutions are in good agreement with the exact solutions. It is observed that our numerical solutions are in good agreement with the exact ones. Furthermore, we show that by decreasing the mesh size, the numerical results converge to the analytical solution, which confirms the convergence of the algorithm. It is noteworthy that the CPU time of the proposed method for each evaluated problem is under 1 second.

Acknowledgments

The authors express their gratitude to the dear referees, who wish to remain anonymous and the editor for their helpful suggestions, which improved the final version of this paper.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, Basel, 2000.
[2]	M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619–4627. https://doi.org/10.1016/S0362-546X(97)00169-7 doi: 10.1016/S0362-546X(97)00169-7
[3]	M. Chipot, J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447–467. https://doi.org/10.1051/M2AN/1992260304471 doi: 10.1051/M2AN/1992260304471
[4]	B. Lovat, Etudes de quelques problèmes paraboliques non locaux, PhD Thesis, Uniersité de Metz, 1995.
[5]	Y. Shi, X. H. Yang, Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation, Electron. Res. Arch., 32 (2024), 1471–1497. https://doi.org/10.3934/era.2024068 doi: 10.3934/era.2024068
[6]	J. W. Wang, X. X. J, H. X. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
[7]	J. W. Wang, X. Xiao, X. H. Yang, H. X. Zhang, A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers' type nonlinearity, J. Appl. Math. Comput., 70 (2024), 489–511. https://doi.org/10.1007/s12190-023-01975-4 doi: 10.1007/s12190-023-01975-4
[8]	X. Wang, X. H. Yang, Z. Y. Zhou, X. Wang, X. J. Yang, Z. Y. Zhou, Pointwise-in-time-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients, Commun. Anal. Mech., 16 (2024), 53–70. https://doi.org/10.3934/cam.2024003 doi: 10.3934/cam.2024003
[9]	C. J. Li, H. X. Zhang, X. H. Yang, A new-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation, Commun. Anal. Mech., 16 (2024), 147–168. https://doi.org/10.3934/cam.2024007 doi: 10.3934/cam.2024007
[10]	J. H. Xu, Z. C. Zhang, T. Caraballo, Mild solutions to time fractional stochastic 2D-Stokes equations with bounded and unbounded delay, J. Dynam. Differential Equations, 34 (2022), 583–603. https://doi.org/10.1007/s10884-019-09809-3 doi: 10.1007/s10884-019-09809-3
[11]	H. X. Zhang, X. X. Jiang, F. R. Wang, X. H. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation. J. Appl. Math. Comput., (2024). https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
[12]	R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105–2137. https://doi.org/10.3934/dcds.2013.33.2105 doi: 10.3934/dcds.2013.33.2105
[13]	R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831–855. https://doi.org/10.1017/S0308210512001783 doi: 10.1017/S0308210512001783
[14]	J. H. Xu, Z. C. Zhang, T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505–546. https://doi.org/10.1016/j.jde.2020.07.037 doi: 10.1016/j.jde.2020.07.037
[15]	M. Fardi, M. A. Zaky, A. S. Hendy, Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian, Math. Comput. Simulation, 206 (2023), 614–635. https://doi.org/10.1016/j.matcom.2022.12.009 doi: 10.1016/j.matcom.2022.12.009
[16]	M. Fardi, S. K. Q. Al-Omari, S. Araci, A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation, Adv. Contin. Discrete Models, (2022), Paper No. 54, 14. https://doi.org/10.1186/s13662-022-03726-4
[17]	S. Mohammadi, M. Fardi, M. Ghasemi, A numerical investigation with energy-preservation for nonlinear space-fractional Klein-Gordon-Schrödinger system, Comput. Appl. Math., 42 (2023), Paper No. 356, 27. https://doi.org/10.1007/s40314-023-02495-4 doi: 10.1007/s40314-023-02495-4
[18]	S. Mohammadi, M. Ghasemi, M. Fardi, A fast Fourier spectral exponential time-differencing method for solving the time-fractional mobile-immobile advection-dispersion equation, Comput. Appl. Math., 41 (2022), Paper No. 264, 26. https://doi.org/10.1007/s40314-022-01970-8 doi: 10.1007/s40314-022-01970-8
[19]	H. L, J. G. Qi, B. X. Wang, M. J. Zhang, Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683–706. https://doi.org/10.3934/dcds.2019028 doi: 10.3934/dcds.2019028
[20]	B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544–1583. https://doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[21]	B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269–300. https://doi.org/10.3934/dcds.2014.34.269 doi: 10.3934/dcds.2014.34.269
[22]	H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307–341. https://doi.org/10.1007/BF02219225 doi: 10.1007/BF02219225
[23]	H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Th. Rel. Fields, 100 (1994), 365–393. https://doi.org/10.1007/BF01193705 doi: 10.1007/BF01193705
[24]	B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.
[25]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[26]	B. X. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60–82. https://doi.org/10.1016/j.na.2017.04.006 doi: 10.1016/j.na.2017.04.006
[27]	J. H. Xu, T. Caraballo, Dynamics of stochastic nonlocal reaction-diffusion equations driven by multiplicative noise, Anal. Appl., 21 (2023), 597–633. https://doi.org/10.1142/S0219530522500075 doi: 10.1142/S0219530522500075
[28]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
[29]	J. H. Xu, T. Caraballo, Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approximations, SIAM J. Math. Anal., 54 (2022), 2792–2844. https://doi.org/10.1137/21M1412645 doi: 10.1137/21M1412645

This article has been cited by:

1.	Osama Ala'yed, Ahmad Qazza, Rania Saadeh, Osama Alkhazaleh, A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications, 2024, 9, 2473-6988, 4665, 10.3934/math.2024225
2.	Oday Hazaimah, Polytropic Dynamical Systems with Time Singularity, 2024, 2581-8147, 721, 10.34198/ejms.14424.721746
3.	Ahmad El-Ajou, Mohammed Shqair, Ibrahim Ghabar, Aliaa Burqan, Rania Saadeh, A solution for the neutron diffusion equation in the spherical and hemispherical reactors using the residual power series, 2023, 11, 2296-424X, 10.3389/fphy.2023.1229142
4.	Richard Olu Awonusika, Oluwaseun Biodun Onuoha, Analytical method for systems of nonlinear singular boundary value problems, 2024, 11, 26668181, 100762, 10.1016/j.padiff.2024.100762
5.	Ajjanna Roja, Rania Saadeh, Raman Kumar, Ahmad Qazza, Umair Khan, Anuar Ishak, El-Sayed M. Sherif, Ioan Pop, Ramification of Hall effects in a non-Newtonian model past an inclined microchannel with slip and convective boundary conditions, 2024, 34, 1617-8106, 10.1515/arh-2024-0010
6.	Saeid Abbasbandy, A hybrid quantum-spectral-successive linearization method for general Lane–Emden type equations, 2024, 1598-5865, 10.1007/s12190-024-02290-2
7.	Rania Saadeh, Iqbal M. Batiha, Ahmad Qazza, Iqbal H. Jebri, Ali Elrashidi, Shaher Momani, 2024, Controlling DC Motor Speed with FoPID-Controllers, 979-8-3315-4001-2, 1, 10.1109/ACIT62805.2024.10877226
8.	Ahmad Qazza, Issam Bendib, Raed Hatamleh, Rania Saadeh, Adel Ouannas, Dynamics of the Gierer–Meinhardt reaction–diffusion system: Insights into finite-time stability and control strategies, 2025, 14, 26668181, 101142, 10.1016/j.padiff.2025.101142

Reader Comments

Your name:*

Email:*
© 2024 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(981) PDF downloads(57) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Cubic B-spline method

4. Convergence analysis

5. Numerical results

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Cubic B-spline method

4. Convergence analysis

5. Numerical results

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog