Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

1.   Introduction

Fractional calculus is a study of the fractional order of integral and derivative operators. Several experts define the fractional integral and derivative as excellent for modelling the memory and heredity features of diverse substances or systems and other real world challenges. The use of fractional differentiation for the mathematical modelling of real world physical models has been extended in recent years, e.g., the modelling of national disasters, the fluid dynamic traffic model with fractional derivatives, the measurement of viscoelastic material properties, oil pollution, etc. In recent years, scientists have drawn the attention of many applications related to integro-differential equations by improving their results in modelling real-world problems, as seen in Abdou et al. ^[1,2], Jalili et al. ^[3,4,5], Kilbas et al. ^[6], Miller and Ross ^[7], Podulbny ^[8], Zhou et al. ^[8,9,10], and Trujillo ^[11]. Additionally, in recent years, qualitative evaluations of fractional calculus have drawn the interest of several scientists, for example, Tunç et al. ^[12,13,14], Bohner et al. ^[15], and Haibu et al. ^[16]. Moreover, there are only a few techniques for the approximate solution of fractional integro-differential equations. Some of these methods are: the Adomian decomposition method (ADM), illustrated by Mittal et al. ^[17], the fractional differential transform method (FDTM), shown by Davoud et al. ^[18], the collocation method by Yang et al. ^[19], and the Sumudo transform method (STM), illustrated by Amer et al. ^[20].

Furthermore, in the previous years, numerous authors examined the existence of solutions of abstract fractional integro-differential equations.

Recently, Baleanu et al. ^[21] studied the FPDE

Using fixed-point methods, explored the existence and uniqueness of a solution for the nonlinear fractional boundary value problem raised by Devi and Sreedhar ^[22] utilised the monotonic iterative technique to the Caputo fractional integro-differential equation of the type

Dong et al. ^[23] showed the existence and uniqueness of solutions via Banach and Schauder fixed point techniques for the issue presented by

Benchohra et al. ^[24] examined existence and stability of solutions for a class of boundary value issue for implicit Caputo fractional differential equations of the type

Hussain ^[25] developed some additional requirements for the existence and uniqueness of solutions for a Caputo fractional Volterra integro-differential equations with nonlocal conditions of the type

Moreover, Abdou et al. ^[26] employed the semi-group technique to investigate the existence and uniqueness of solutions for fractional and partial integro differential equations of heat type in Banach space E provided by

Motivated by the references ^{[21,22,23,24,25,26,31,32]} and as a generalized case of the previous Eqs (1.1)–(1.6), we explore the following nonlinear fractional integro-differential equation NFIDEq

with initial condition

where x ∈ [a, b], t ∈ I = [0, T], $D_t^\alpha$ and $D_x^\beta$ be standard Caputo fractional derivatives with orders α, β respectively such that $n - 1 < \beta < \alpha < n$, $n$∈$N$ and $u(x, t) \in \, \, {C_E}(E \times [0, T]),$ where C_E (E × [0, T]) be a dual Banach space, the functions $f(x, t), {\text{ g(x, t)}}$ are continuous functions identified as

Using the previous information, Eq (1.7) can be considered a new modification. The goal of this article is to provide novel results related to the existence and uniqueness solution of NFIDEq and also give a numerical solution using the approach of the Bernoulli matrix. The results will be helpful to researchers working on fractional calculus, especially the solvability study of NFIDEq, and they provide some new improvements on the topic.

The outline of this article is organized as follows: essential topics are discussed in Section 2. In Section 3, we formulate the sufficient conditions for the existence and uniqueness of a solution to Eq (1.7). The Bernoulli matrix approach BMA method is used in Section 4 to obtain the numerical solution of Eq (1.7), and the convergence analysis is also proven. Afterward, in Section 5, we explain numerical examples related to what we introduced in Section 4 to demonstrate the preciseness of the method and also compute the absolute error of the case. Finally, in Section 6, a conclusion is given.

2.   Basic concepts

Definition 2.1. (Odibat et al. ^[27]). Riemann-Liouville fractional integral and derivative operator of order $\alpha \in \left(CorR\right)$ is respectively defined by:

and

Where $n\, = \, [R(\alpha)\, + \, 1\, ], \, \, R(\alpha)$ indicates the integer part of $\alpha$, while Caputo fractional derivative of order $\alpha \in \left(NorR\right)$ is defined by

or

Proposition 2.1. (Burton et al. ^[28]). Let $R(\alpha), \, R(\beta)\, > \, 0$, $x \in [a, b]$ and $f(x)\, \in \, \, C[a, b],$ then we have

$({\rm{i}})\ \ \, \left({{\text{I}}_{{\text{a}} + }^\alpha {\text{ I}}_{{\text{a}} + }^\beta {\text{ f}}} \right){\text{(x) }} = {\text{ }}\left({{\text{I}}_{{\text{a}} + }^{\alpha + \beta }{\text{ f}}} \right){\text{ (x), }}\, \, \, \,$

$({\rm{ii}}) \ \ \left({{\text{D}}_{{\text{a}} + }^\beta {\text{ I}}_{{\text{a}} + }^\alpha {\text{ f}}} \right){\text{(x) }} = {\text{ }}\left({{\text{I}}_{{\text{a}} + }^{\alpha - \beta }{\text{ f}}} \right){\text{ (x), }}$ for $R(\alpha) > \, R(\beta),$

$({\rm{iii}}) \ \ \left({{\text{D}}_{{\text{a}} + }^\alpha {\text{ I}}_{{\text{a}} + }^\alpha {\text{ f}}} \right){\text{(x) }} = {\text{ f(x), }}$

$({\rm{iv}})\ \ \left({{\text{I}}_{{\text{a}} + }^\alpha {\text{ D}}_{{\text{a}} + }^\alpha {\text{ f}}} \right){\text{(x) }} = {\text{ f(x) - }}\sum\limits_{{\text{k}} = {\text{1}}}^{\text{n}} {{\text{ }}\frac{{{\text{f}}_{{\text{n - }}\alpha }^{{\text{(n - k)}}}{\text{ (a)}}}}{{\Gamma {\text{ }}\left({\alpha {\text{ - k }} + {\text{ 1}}} \right)}}} {\text{ }}{({\text{x - a)}}^{\alpha {\text{ - k}}}}.$

Lemma 2.1. (Burton et al. ^[28]). Let $n - 1 < \beta < \alpha < n$, for $x \in [a, b]$, if we have ${\text{y}} \in {\text{ }}{{\text{C}}_{\text{E}}}{\text{ [a, b]}}$ and $\left({{\text{D}}_{{\text{a}} + }^\alpha {\text{ y}}} \right){\text{ (x) }} \in {\text{ }}{{\text{C}}_{\text{E}}}{\text{ [a, b]}}$, then $\left({{\text{D}}_{{\text{a}} + }^\beta {\text{ y}}} \right){\text{(x) }} \in {\text{ }}{{\text{C}}_{\text{E}}}{\text{ [a, b]}}{\text{.}}$

Lemma 2.2. (Karthikeyan et al. ^[29]). Let $n \in {N_0}$, the space ${C^n}[a, b]$ is composed of continuous functions, which are represented in the form

Definition 2.2. (Boas et al. ^[35]). Bernoulli polynomials of order m can be stated as

Where ${B_i}\, = \, {B_i}(0)\, , \, \, i\, = \, 0, \, 1, \, 2\, , \, ...\, m$, are Bernoulli numbers.

Proposition 2.2. (Boas et al. ^[35]). The standard Bernoulli polynomials are commonly identified by the following relation

Proposition 2.3. (Samadi et al. ^[36]). Bernoulli polynomials have a full basis on the interval [0, 1].

Definition 2.3. (Samadi et al. ^[36]). Legendre Gauss quadrature formula can be specifically defined as

where ${t}_{i}fori = \mathrm{0, 1}, 2, ...N,$ are the roots of the (𝑁+1) Legendre polynomial 𝑃_𝑁+1(𝑡) in the interval (-1, 1) where ${p_n}(t) = \, \frac{1}{{{2^n}n!}}\frac{{{d^n}}}{{d{t^n}}}{({t^2} - 1)^n}$ and ${\omega _{\text{i}}}\,$ be the corresponding weights such that ${{\omega }_{\text{i}}}={}^{1}\!\!\diagup\!\!{}_{\left( 1-\text{t}_{\text{i}}^{2} \right)}\;\text{P}_{\text{N}+1}^{\backslash }\left( {{\text{t}}_{\text{i}}} \right) .$

Theorem 2.1. (Odibat et al. ^[27]). Let X be a Banach space, $S = \{ \, s(t)\}$ be a family of continuous mappings $s:\, \, J \to \, X.$ If $S$ is uniformly bounded and equicontinuous, and for any ${t^*} \in J,$ the set ${S^*}$ is relatively compact, then there exists a uniformly convergent function sequence in $S$.

Theorem 2.2. (Odibat et al. ^[27]). Let $X$ be a Banach space, $K$ a convex subset of $X$, ${A}$ an open set of $K.$ Suppose that $T:\, \, \overline A \to \, K$ is a continuous and compact operator where $\overline A$ is closure of ${A}$. Then either $T$ has a fixed point in $\overline A$, or $\exists \, \, v \in \, \partial A$ such that $v = \lambda \, Tv$ for $\lambda \in \, (0, 1).$

Theorem 2.3. (Karthikeyan et al. ^[29]). Every contraction mapping on a Banach space admits a unique fixed point.

3.   Existence and uniqueness solution of NFIDEq

To verify the existence and uniqueness of the solution of Eq (1.7), we first assume the following conditions:

C₁: $f(x, t)$ is a continuous function and $\exists$${P}_{1}, {P}_{2}\in {R}^{+}$ such that

$\left| {\, f\, \left({s, t, u, v} \right)\, } \right| \leqslant \, {P_1}\, \left({\, \left| u \right|\, + \, \left| {\, v} \right|\, } \right) + \, {P_2}\, ,$ where $s \in [a, b], \, \, t \in \, I$ and $u, v\in R$;

C₂: $g(x, t)$ is a continuous function and $\exists$${q}_{1}, {q}_{2}\in {R}^{+}$ such that

$\left| {\, g\, \left({{s_1}, t, {s_2}, u} \right)\, } \right| \leqslant \, {q_1}\, \left| {\, u\, } \right| + {q_2},$ where ${s}_{1}, {s}_{2}\in [a, b], t\in I, u\in R$;

C₃: there exist ${N}_{1}\in {R}^{+}$ such that

$\left|f\left(s, t, u, v\right)-f\left(s, t, w, z\right)\right|\le {N}_{1}\left(\left|u-w\right|+\left|v-z\right|\right),$ where $u, v, w, z\in R$, $s \in [a, b], \, \, t \in \, I;$

C₄: there exist ${N}_{2}\in {R}^{+}$ such that

$\left| {\, g\, \left({{s_1}, t, {s_2}, u} \right)\, - g\, \left({{s_1}, t, {s_2}, v} \right)\, \, } \right| \leqslant \, {N_2}\, \left| {\, u - v\, } \right|,$ where $u, v\, \in {R}$, ${N_2}\, = \, \left| {{\text{ N (}}{{\text{s}}_{\text{1}}}{\text{, t, }}{{\text{s}}_{\text{2}}}{\text{, }}\nu){\text{ }}} \right|.$

Before proving the theory of existence and uniqueness of the solution, we must prove the following principle lemma.

Lemma 3.1. If $u \in \, \, {C_E}(E \times [0, T])$, then $u(x, t)$ can be written in the following form $u(x, t)\, = \, \frac{1}{{\Gamma (\beta \, )}}\, \int\limits_{{t_0}}^t {\, \frac{{y(x, s)}}{{{{(t - s)}^{1 - \beta }}}}\, \, ds\, \, + \, \sum\limits_{m = 0}^{n - 1} {\frac{{{u^{(m)}}(x, {t_0})}}{{m!}}\, {{(t - {t_0})}^m}\, } }$, where; $y(x, t) \in \, {C_E}(E\, \times \, \, [0, T])$ satisfy the fractional integral equation

Proof. Consider $u(x, t) \in \, {C_E}(E\, \times \, \, [0, T])$ be a solution of Eq (1.7), from Lemma 2.1, we have that ${D_t}^\alpha u(x, t) \in \, {C_E}(E\, \times \, \, [0, T])$,

Then

Using Lemma 2.2, we get

where,

But

So

From Lemma 2.2 we get

${D_x}^\beta u(x, t) \in \, {C_E}(E\, \times \, \, [0, T])$, so, we can apply ${I^\alpha }$ to both sides of Eq (1.7) and using Eq (3.2) we obtain

Set $y(x, t) = \, {D_x}^\beta u(x, t)$, then $y \in \, {C_E}(E\, \times \, \, [0, T])$ and similar to Eq (3.3) we obtain

From Lemma 2.2, we have

using Lemma 2.1 and Eq (3.4), we get

Now, we will show that ${u^{(m)}}(x, 0)\, = {u_m}$, for $m\, = \, 0, 1, 2, ...n - 1.$ For n = 1, it is easy to see that $u(x, 0)\, = {u_0}.\,$

Now for $n \geqslant 2\, ,$ using Proposition 2.1 and Eq (3.2) we get

Thus, from Lemma 2.1 we have

$u(x, t) \in \, {C_E}(E\, \times \, \, [0, T])$ and u^(m) (x, 0) = u_m, for $m\, = \, 0, 1, 2, ...n - 2.$

At $m\, = \, n - 1$, we have

Using proposition 2.1 we obtain

Then ${u^{(n - 1)}}(x, 0) = \, {u_{n - 1}}.$

Theorem 3.1. Assume that the conditions (C₁–C₄) hold, then Eq (1.7) has a solution

Proof. From Lemma 3.1, it is sufficient to show that Eq (1.7) have a solution $u(x, t) \in \, \, {C_E}(E\, \times \, \, [0, T])$. Using Eq (3.4) to define $(T\, y)(x, t)$ as

Also, set $r\, = \, \alpha - \beta, \, \, s.t\, \, n - 1 \prec \alpha - \beta \prec n.$

Consider $\overline {{{\text{B}}_{\text{r}}}} {\text{ }} = {\text{ }}\left\{ {\, {\text{y }} \in {\text{ }}{{\text{C}}_E}{\text{ (E }} \times {\text{ [0, T]; }}{{\left\| {\text{u}} \right\|}_{{C_E}}}{\text{ }} < {\text{ r}}\, } \right\}.$

Now we need to show that $T\, :\, \overline {{{\text{B}}_{\text{r}}}} {\text{ }} \to {\text{ }}{{\text{C}}_{\text{E}}}$ is a continuous and compact operator.

From the continuity of functions $f(x, t)\, \, and\, g(x, t)$and the operators $\, {I^{\alpha - \beta }}, \, {I^\beta }$ on $C[a, b]$, it is easy to say that $(T\, y)(x, t) \in \, \, {C_E}(E\, \times \, \, [0, T]),$ for $y\, \in \, {\bar B_r}$.

Consider $y, \, z\, \in \, {\bar B_r}$ and using conditions C₃, C₄ we obtain

Set $\mu \, = \, \, \frac{{{N_1}{{(b - a)}^{\alpha - \beta }}}}{{\Gamma \, (\alpha - \beta + 1)}} + \frac{{{N_2}{{(b - a)}^{\alpha + 1}}}}{{\Gamma \, (\alpha + 2)}}$, therefore we find that T is a continuous operator on $\overline {{{\text{B}}_{\text{r}}}}$ when $\mu < \text{1.}$ The set $A\, = \, \{ Tu\, :\, u\, \in \, {\bar B_r}\, \}$ is uniformly bounded where $\left\| {{\text{ }}T{\text{ }}y{\text{ }}} \right\|{\text{ }} \leqslant {\text{ M}}$, such that

Let ${\theta _1}, \, {\theta _2}\, \in \, \, [\, 0, T\, ]$ and $y\, \in \, {\bar B_r}$, we have

From Eq (3.6) we have

Then $T:\, \overline {{{\text{B}}_{\text{r}}}} {\text{ }} \to {\text{ }}{{\text{C}}_{\text{E}}}$ is uniformly bounded and equicontinuous; also, E is a relatively compact subset of C_E (E × [0, T]), and hence we can say that the operator T satisfies Arzela Weierstrass's theory (Theorem 2.1). If we show that $u = \lambda \, Tu$ doesn't have any solution in $\partial {B_r}$ for some $\lambda$, then by (Theorem 2.2), $T$ has a fixed point in $\, \overline {\partial {B_r}}$. Hence, the Eq (1.7) has a solution

Theorem 3.2. Equation (1.7) has a unique solution ${\text{u(x, t) }} \in {{\text{C}}_E}{\text{(E }} \times {\text{ [0, T]}}$.

Proof. We can prove the uniqueness of the solution of Eq (1.7) using Banach contraction principle as follow.

Using the conditions (C₁–C₄), and Eq (3.7) we have

Thus, T is a contraction operator, from the Banach Contraction Principle (Theorem 2.3), we get the existence and uniqueness of the solution of the Eq (1.7).

4.   Bernoulli matrix approach (BMA)

In this Section, we present a computational approach for solving the NFIDEq which is based on the Bernoulli polynomials approximation. (See Tohidi et al. ^[33] and Hassani et al. ^[34]).

We need to approximate the solution of F(x, t) ∈ C_E(E × [0, 1]) by the truncated Bernoulli series F(x, t) = $\sum\limits_{m = 0}^N {\sum\limits_{n = 0}^N {{f_{m, n}}} {B_m}(x){B_n}(t)}$, where the coefficients f_{m, n} extracted a

Lemma 4.1. Let $D_t^\alpha u(x, t)$ be approximated by the Bernoulli polynomials as

then we have

Proof. Applying operator $I_t^\alpha$ on both sides of $D_t^\alpha u(x, t)$ $= {\text{ }}\sum\limits_{m = 0}^N {\sum\limits_{n = 0}^N {{u_{m, n}}} } B_m(x)B_n(t)$, we have

Where $\, \, \, b_{n, r}^\backslash = \, \left({\begin{array}{*{20}{c}} n \\ r \end{array}} \right)\, {B_{n - r}}\frac{{\Gamma (r + 1)}}{{\Gamma (r + 1 + \alpha)}}, \, {\text{ }}$ then we obtain

4.1.   Numerical solution of Eq (1.7) using BMA

The BMA approach turns the NFIDEq to a system of algebraic equations by extending the relevant approximate solutions as the linear combination of the Bernoulli polynomials.

According to Lemma 4.1, we can get an approximation solution of Eq (1.7) as follow

where $y(x, t)\, = \, {\text{D}}_x^\beta {\text{ u(x, t) }} = {\text{ }}\sum\limits_{i = 0}^M {\sum\limits_{j = 0}^M {{u_{i, j}}} } {B_i}(x){B_j}(t).$

Furthermore, we apply Legendre Gauss collocation nodes and also Legendre Gauss quadrature rule for approximating the existing integrals. By collocating the Eq (4.3) at (𝑁+1) points 𝑥_𝑝 s.t $0 = {x_0} < {x_1} < {x_2} < ... < {x_p} < ...{x_N} = 1$, then we have

Where 𝑥_𝑝 $p\, = \, 0, \, 1\, 2\, , \, ...\, N,$ indicates the roots of the shifted Legendre polynomial 𝑃_𝑁+1 (𝑥) in the interval (0, 1). Also to apply the Legendre Gauss quadrature for estimating the following equation involves integrals, we should convert s-interval [0, 1] into 𝜏-interval [−1, 1] by the following change of variable: $\tau = \, 2s - 1$, then Eq (4.4) will be transformed to

Where all of the 𝜏_𝑞's are the 𝑁+1 zeroes of the Legendre polynomial P _𝑁+1 (𝜏) and the 𝑤_𝑞's are the corresponding weights. The solutions of the nonlinear algebraic system (4.5) are the coefficients of the truncated double Bernoulli series, which are defined in the interval [0, 1].

4.2.   Convergence analysis of the nonlinear algebraic system (4.5)

Also, convergence analysis associated to the presented idea is provided as follow

For $u({x_p}, {t_i}) \in {\text{ }}{{\text{c}}_{\text{E}}}{\text{([0, 1]}}\, \times \, {\text{[0, 1]), x}} \in [0, 1]$, define an operator

$A:\, \, \, {c_{\text{E}}}{\text{ ([0, 1]}}\, \times \, {\text{[0, 1]) }} \to {\text{ }}{{\text{c}}_{\text{E}}}{\text{ ([0, 1]}}\, \times \, {\text{[0, 1])}}$, such that

According to conditions (C₁–C₄) which listed above, we get

Consider $u({x_p}, {t_i}), \, \, v({x_p}, {t_i}) \in {\text{ }}{{\text{c}}_{\text{E}}}{\text{([0, 1]}}\, \times \, {\text{[0, 1]), where }}{{\text{x}}_{\text{p}}} \in [0, 1]$.Then we obtain

Thus, A is a contraction operator in the case of ${\eta _{p, i}} \in (0.1)$, and from the Banach contraction. Principle, we get the existence of a unique solution for the Eq (1.7).

5.   Numerical examples

In this Section, we present two numerical examples to make a verification of the theoretical work which presented in Section 4 by using BMA method.

Example 5.1. Consider the following nonlinear fractional integro-differential equation

with initial condition $u(o, x) = u(0, t) = 0$, $x\in \left(\mathrm{0, 1}\right).$

Observe that example (5.1) is a special case of Eq (1.7) with β = 0.4 and functions f and g determined as:${\rm{f}}\left( {{\rm{x}},{\rm{t}},{\rm{y}},{\rm{v}}} \right) = \frac{{{{\rm{x}}^3}}}{{{{\rm{e}}^{\rm{t}}}}}{\rm{y + v,y,v}} \in {\rm{R}}$ and $g\left(x, t, y, v\right) = \frac{x-t}{1+t}tany.$ The exact solution of Eq (5.1) is $u(x, t) = {x}^{2}{e}^{t}$. The results of exact solution, approximate solutions and the absolute error between them are obtained in Table 1.

Example 5.2. Consider the following nonlinear fractional integro-differential equation

with initial condition $u(o, x) = u(0, t) = 0$, $x\in \left(\mathrm{0, 1}\right].$

Also, we observe that example (5.2) is a special case of Eq (1.7) with β = 0.02 and functions f and g determined as: $\mathrm{f}\left(\mathrm{x}, \mathrm{t}, \mathrm{y}, \mathrm{v}\right) = \frac{x}{4}\text{cos y +}\frac{1}{2}\text{v, }$ and $g(x, t, y, v) = {(x-t)}^{2}u,$ $\text{y, v}\text{∈ R.}$ The exact solution of Eq (5.2) is $u\left(x, t\right) = xsint$. The results of exact solution, approximate solutions and the absolute error between them are obtained in Table 2.

6.   Conclusions

The major purpose of this study is to prove the existence and uniqueness of the solution of a nonlinear fractional integro-differential equation in dual Banach space. The desired findings are demonstrated by applying fixed-point theorems after employing fractional calculus. Also, we use the Bernoulli matrix approach method by reducing the NFIDEq to an algebraic system and deriving the numerical solution. We also observed that the matrix approach method is very efficient by verifying the conversion analysis of the numerical solution. Finally, we have chosen two examples as a verification of the theoretical work. The difference between exact solutions and approximate solutions for different levels of α and t are computed as shown in Figures 1–4, and concluded the following:

Throughout Example 5.1, we deduced that

(1) In Table 1, at α = 0.5, the minimum error is 6.93 e−09 at x = 0.8, t = 0.99, and the maximum error is 1.55 e-06 at x = 0.2, t = 0.66.

(2) In Table 1, at α = 0.95, the minimum error is 5.87 e−10 at x = 0.8, t = 0.99, and the maximum error is 3.2 e−07 at x = 0.4, t = 0.33.

Throughout Example 5.2, we deduced that

(3) In Table 2, at α = 0.5, the minimum error 9.99 e−12 at x = 0.2, t = 0.99, and the maximum error is 4.1e−10 at x = 0.4, t = 0.33.

(4) In Table 2, at α = 0.95, the minimum error is 6.94 e−14 at x = 0.8, t = 0.99, and the maximum error is 3.4 e−10 at x = 0.2, t = 0.33.

We can deduce that when the value of α increases to α = 0.8 and the value of t reaches t = 0.99 in each case, the approximation solutions are convergent to exact solutions, hence the error is small. When the value of α decreases to α = 0.2 and the value of t = 0.33, then the approximate solutions are divergent away from exact solutions and the difference between the two solutions is increase.

From Figures 5 and 6, we concluded the following:

(Ⅰ) The BMA method is very powerful in finding precise numerical solutions, which appear clearly in the comparison between the exact solutions and the numerical solutions as shown.

(Ⅱ) The numerical solutions at α = 0.95 are more accurate than the numerical solutions at α = 0.5. The interpretation of it is that the behavior of the function u(x, t) at α = 0.5 represents a potential function that causes a slight perturbation in the values of u. We can declare that we obtain the equilibrium state when $\alpha \to 1$, we have a singular case, which is called Cauchy kernel (see ^[26,37]).

Future work: We would like to expand on this work to investigate the optimality conditions for solving this fractional optimal control problem: $J(x(t), \, u(t), \, \, t)\, = \, \int\limits_a^T {L(x(t), \, u(t), \, \, t)} \, dt\, + \, \phi \, (x(t), \, \, t)$, subject to a dynamical constraint on the form:$D_t^\alpha x(t) = f\left({x(t)\, , \, t, \, D_x^\beta x(t), \, \, \int\limits_{{t_0}}^t {g(x, s, u(s))\, ds} \, } \right)$.

Conflict of interest

The authors declare that there is no conflict of interest.