Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

Abstract

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space $C_E\left(E, J\right)$. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters $\lambda$ and time $t$ as well as how the absolute error in each example is calculated.

1. Introduction

Fractional calculus and fractional order differential equations have been widely applied in many fields of science and engineering. The analysis of time-fractional ordinary and partial differential equations has received considerable attention from numerous researchers, including Petras et al. [1], Patil et al. [2], Taha et al. [3], Baitiche et al. [4], Abdo et al. [5], Y. Zhou et al. [6,7,8], Tarasov et al. [9], Mainardi et al. [10], Atangana et al. [11], Podlubny et al. [12], Miller et al. [13], Baitiche et al. [4], Wahash et al. [14], and many other researchers. Fractional integro-differential equations (FIDEs) are based on derivatives of fractional order, in contrast to conventional partial differential equations, which usually involve derivatives of integer order. This makes FIDEs highly helpful for modeling phenomena because they can represent the long-term consequences of previous events on the present state of a system. Fractional calculus, a discipline of mathematics that deals with the study of fractional derivatives and integrals, is one of the new mathematical ideas and techniques to have resulted from the use of FIDEs. Thus, in this work, we can introduce a hyperbolic nonlinear fractional integro-differential equation. This equation’s prototype is the wave equation, which is used to explain a broad range of events, such as electromagnetic waves, acoustic waves, fluid-structure interaction, and so on.

Over the past few decades, sufficient conditions for the qualitative properties of solutions for nonlinear FDEs have been extensively examined using standard fixed-point theory. Various researchers looked into the existence of solutions to abstract fractional integro-differential equations. Abdo et al. [15] analyzed the existence properties of solutions of the fractional integro-differential equation under nonlocal conditions using Krasnoselskii’s and Banach’s fixed-point theorems. M.Z. Aissaoui et al. [16] studied the existence and uniqueness questions for a nonlinear Volterra–Fredholm integro-differential equation under some suitable conditions. Aisulaiman et al. [17] recently demonstrated the solution of an NFIDE in dual Banach space $C_E(E, \left[0, T\right]).$ Hamdy et al. [18] explored the presence of a mild solution for a nonlinear impulsive delay integro-differential system of fractional order with Sobolev type. Furthermore, Abdou et al. [19] used the semi-group method to explore the validity of the solutions in Banach space for FDEs of the heat type.

Many studies discuss numerical methods for solving differential and integro-differential problems and have captured the interest of many researchers. Recently, Taheri et al. [20] solved a 2D-NFIDE by using the shifted Jacobi polynomials via the collocation method. Babaei et al. [21] considered a sixth-kind Chebyshev collocation method to solve a nonlinear quadratic FIDE of variable order and the cubic B-splines collocation method introduced by Mittal et al. [22]. Additionally, Izadi et al. [23] investigated the local discontinuous Galerkin method for the numerical solution of a fractional logistic differential equation with the aid of shifted Legendre polynomials; the spline collocation method has been presented by Pitolli et al. [24] to establish an approximate solution to the Riesz–Caputo derivative. Moreover, the variational method for deriving an approximate solution to the time fractal heat conduction equation was offered by Shymanskyi et al. [25], among others. Although multiple numerical methods have been suggested for solving many different problems, there have been few research studies that have developed numerical methods based on Lerch polynomials for solving an equation of the hyperbolic type. This is partly due to the mathematical complexity of such equations and the difficulty of developing numerical methods that can handle them effectively. For example, Alhazmi et al. [26] showed a new efficient approach determined by the Lerch polynomial method for solving mixed integral equations, Cayan et al. [27] obtained approximate solutions for the unsteady convection-diffusion equation in 1-D using a hybrid matrix collocation method that is based on Lerch polynomials, and Doaa S. [28] created a powerful approach based on Lerch polynomials that gives an approximate solution for multiple cases of Cauchy-type singular integral equations.

Motivated by the works mentioned above, mainly [17,26,27,28,29], we investigate the qualitative results of the solutions of the following nonlinear hyperbolic fractional integro-differential equation NHFIDEq:

$$\frac{{\partial }^{1+\alpha } u(x,t)}{\partial t^{1+\alpha }}=f\left(x, t, \frac{{\partial }^{1+\beta } u(x,t)}{\partial x^{1+\beta }}, {\displaystyle \underset{t_0}{\overset{t}{\int }}g(x,t,s,u(x,s)) ds} \right),$$

(1)

with initial conditions:

$$u(x,0) = h(x), u_t(x,0) = u_x(0,t) = 0.$$

(2)

where $x\in \left[a,b\right], t \in J = (0, T],$ with Caputo fractional derivatives $\frac{{\partial }^{1+\beta } }{\partial t^{1+\beta }}$ and $\frac{{\partial }^{1+\alpha } }{\partial x^{1+\alpha }}$ orders $\alpha , \beta$, respectively, so that $0<\beta <\alpha <1,$ and $u(x,t)\in C_E(E, J),$ where $C_E(E, J)$ is a dual Banach space. The prototype of the problem (1)–(2) is a wave equation, which is employed to characterize a broad variety of phenomena, including electromagnetic and acoustic waves, fluid–structure interaction, and other domains.

Thus, this paper is regarded as belong to those uncommon papers in mathematical physics that offer a qualitative analysis for solving a NHFIDEq that includes the existence and uniqueness of the solution to the given problem and gives a numerical approach using the Lerch matrix collocation method. LMC applies for the first time to a fractional problem (1)–(2) and supplies a numerical solution in a quickly convergent power series with an attractively computable series of terms. The LMC method has proven to be very effective and results in considerable savings in computation time as well as accuracy (see [26,27,28,29]).

The basic structure of this article has been organized as follows: Fundamental concepts are discussed in Section 2. In Section 3, we provide sufficient conditions to prove the existence and uniqueness of solutions for the problem (1)–(2). In Section 4, the Lerch matrix collocation method LMC is used to obtain the numerical solution of the given problem. To verify the accuracy and efficiency of the LMC method, an alternative convergence criterion along with error analysis depending on the residual error function is enhanced. Afterward, in Section 5, we elucidate numerical examples associated with what we covered in Section 4 to show the accuracy of the method and also determine the absolute errors of the problem. At last, in Section 6, a conclusion is provided.

2. Preliminaries

Definition 1.

([30]). The Riemann–Liouville fractional integral operator of order $\zeta$ > 0 is defined by:

$$I^{\zeta }u(x)= \frac{1}{\Gamma (\zeta )} {\displaystyle \underset{0}{\overset{x}{\int }}{(x - s)}^{\zeta -1} }u(s) \mathit{ds} , u > 0.$$

Definition 2.

([31]). The fractional derivative of $u(x)$ by means of the Caputo sense is defined by:

$$D^{\zeta }u(x)= \frac{1}{\Gamma (n-\zeta )} {\displaystyle \underset{0}{\overset{x}{\int }}{(x - s)}^{n-\zeta -1} }{u}^{(n)}(s) \mathit{ds}.$$

The Caputo derivatives have some features, such as:

• $D^{\zeta }{ I}^{\zeta } u\left(x\right)=u\left(x\right),$
• $I^{\zeta } D^{\zeta } u\left(x\right)=u\left(x\right) - {\displaystyle \sum _{i=0}^{n-1}\frac{u^{(i)} \left(0\right)}{i!}}{ x}^i,$
• $\left(D^{\zeta } I^{\mu } u\right){\left(x\right)=(I}^{\mu -\zeta } u) (x) ,$ for $\mu > \zeta$.

Theorem 1.

([32]). Every contraction mapping on a Banach space admits a unique fixed point.

Theorem 2.

([32]). Let $X$ be a Banach space; a set $F\subset X$ of functions is relatively compact if and only if it is bounded and equicontinuous.

Theorem 3.

([33]). If $T:X\to X$ is a completely continuous operator, then either:

• The operator equation $x=\lambda T x$ has a solution for $\lambda =1$, or
• The set$A = \left\{u \in X : u=\lambda T u\right\}$is unbounded for $\lambda \in \left(0,1\right)$.

Definition 3.

([27]). The Lerch polynomials are distinguished by the generating function formula:

$${\left(\mathit{1} - Z \mathit{log} (1+t)\right)}^{-\lambda }={\displaystyle \sum _{n=0}^{\infty }{L}_n} (\mathit{z}; \lambda {) t}^n , \lambda =\mathit{1}, \mathit{2}, \mathit{3}, \dots$$

Moreover, the representation formula of Lerch polynomials of degree $n$ is:

$$L_n \left(Z ; \lambda \right)={\displaystyle \sum _{i=1}^n\frac{i!}{n!}} s (n,i) \left(\begin{array}{c}i+\lambda -1\\ i\end{array}\right) z{}_i$$

where $s (n,i)$sterling numbers of the first kind satisfy  $s (n+1,0)= 0,$$s (n,n)= 1,$$s (n,n-1)= - {}_{n}C{}_{\mathit{2}}, n\ge 0.$

3. Analysis for Existence and Uniqueness Solutions for the Model (1)–(2)

Our basic objective is to explore the existence and uniqueness theory for the problem (1)–(2). Under some relevant conditions, we set:

$A_1:$ For $p, \delta \in { R}^{+}$ and $\omega , Z \in R, f(x, t)$ is a continuous function, so that  $\left| f(s, t, \omega , z) \right| \le \left(\left|\omega \right|+\left|z\right|\right)+\delta ,$ $S\in \mathit{[}a, b\mathit{]}$ and $t \in J$;

$A_2:$ For $q,\upsilon$ $\in { R}^{+}$, $g(x,t)$ is a continuous function, so that $\left|g{ (s}_1{ , t , s}_2 , \omega )\right| \le g \left|\omega \right|+\upsilon ,$ where; $s_1, s_2\in \mathit{[}a, b\mathit{]}, t \in J$ and $\omega \in R^{+}$;

$A_3:$ For $u , \upsilon , \omega , Z \in R , S \in \mathit{[}a , b\mathit{]}$ and $t \in J,$ $\exists M \in { R}^{+}$, so that $\left|f (s, t , \omega , Z) - f (s , t , u , \upsilon )\right| \le M \left(\left|\omega - u\right|+\left|Z - \upsilon \right|\right)$;

$A_4:$ For $\omega , Z \in R , S \in \mathit{[}a , b\mathit{]}$ and $t \in J,$$\exists M^{+}\in { R}^{+}$, so that $\left|g{ (s}_{\mathit{1}}{ , t , s}_{\mathit{2}} , \omega ) -{ g (s}_{\mathit{1}}{ , t , s}_{\mathit{2}} , Z)\right| \le { M}^{+} \left|\omega - Z\right|.$

Lemma 1.

If $u (x, t) \in { C}_E (E, J),$ then $u (x, t)$can be written in the form:

$$u (x, t)=\frac{1}{\Gamma (1+\beta )} {\displaystyle \underset{0}{\overset{t}{\int }}\frac{y (x , t)}{{(t - S)}^{2-\beta }}} ds+{\displaystyle \sum _{m=0}^{n-1}\frac{u^{(m+1)} (x, 0)}{(m+1) !}{ A}^m},$$

(3)

where $y(x, t)$satisfies the fractional integral equation:

$${y (x, t)=I}^{1+\alpha -\beta } f \left({x, t, y}_x (x, t), {\displaystyle \underset{0}{\overset{t}{\int }}g}\left.\left(x, t, s, {\displaystyle \sum _{m=0}^{n-1}\frac{u^{(m+1)} (s , 0)}{(m+1) !}{ S}^m}+{ I}^{\beta } y\prime (x, S)\right) ds\right). \right.$$

Proof. 

See (Alsulaiman et al. [17]). $\square$

Theorem 4.

If problem (1)–(2) satisfies the conditions (A₁–A₄), then the problem has a solution.

Proof. 

From Lemma 1, we can say that the problem (1)–(2) has a solution.

Construct $\left(\mathit{Fy}\mathit{\right)}(x, t\mathit{)}$ so that:

$${\mathit{\left(}\mathit{Fy}\right)\mathit{(}x, t\mathbf{)}=I}^{1+\alpha -\beta } f \left({x, t, y}_z (x, t), {\displaystyle \underset{0}{\overset{t}{\int }}g}\left.\left(x, t, s, {\displaystyle \sum _{m=0}^{n-1}\frac{u^{(m+1)} (s , 0)}{(m+1) !}{ S}^m}+{ I}^{\beta }{ y}_x (x, S)\right) \mathit{ds}\right). \right.$$

Additionally, set $\mu =\alpha - \beta , 0 < \alpha - \beta < 1.$

Consider ${\tilde{B}}_{\mu }=\left\{y \in { C}_E (E \times J) : {‖u‖}_{C_E} < \mu \right\},$

where:

$$‖ u_{C_{{}_E}}‖=\mathit{sup} \left\{\left| u (x, t) \right| : (x, t) \in \mathit{[}a, b\mathit{]} \mathit{\times } J\right\}.$$

(4)

For $y\in {\tilde{B}}_{\mu }$ and using conditions A₁ and A₂, it is easy to say that the set

$B=\left\{Fy : y \in {\tilde{B}}_{\mu }\right\}$ is uniformly bounded where $‖Fy‖ < \tilde{M}$, so that:

$$\tilde{M}=\sup \left|f (s, t, \omega , z)\right|.$$

(5)

To claim that $F$ is completely continuous, consider $y, \widehat{y}\in {\tilde{B}}_{\mu },$ and using conditions A₁ and A₄, we obtain:

$$\begin{array}{l}\left|(F y)(x, t) - (F \widehat{y} )(x, t)\right| \le \left\{I^{1+\alpha -\beta }\left(M \left|y(x,t) - \widehat{y} (x,s)\right|\right)\right.+{ I}^{\beta } \left.\left(\tilde{M} \left|y(x,s) - \widehat{y} (x, s)\right|\right)\right\},\\ \le {‖y - \widehat{y}‖}_{C_E}\cdot \mathit{sup} \left\{I^{1+\alpha -\beta }{ M (x,t)+I}^{1+\alpha +\beta }{ M}^{*}(x,t)\right\}, \forall (x,t) \in \mathit{[}a, b\mathit{]} \mathit{\times } J.\\ \le \rho {‖y - \widehat{y}‖}_{C_E}.\end{array}$$

where:

$$\rho =\frac{M \Gamma (1+\alpha {) (b-a)}^{1+\alpha +\beta }{+M}^{*} \Gamma (\alpha -\beta {) (b-a)}^{\alpha }}{\Gamma (\alpha -\beta ) \Gamma (1+\alpha )}$$

(6)

Thus, we obtain that $F$ is a continuous operator on ${\tilde{B}}_{\mu }$; now, using Equations (4) and (5), we have:

$$\begin{array}{ll}|(Fy) (x, {\varphi }_1) - (Fy)& (x, {\varphi }_2)| \le \\ & \frac{1}{\Gamma (1+\alpha - \beta )} \left|\frac{Fu(x,s) \left[{({\varphi }_2-s)}^{2-\alpha +\beta } - ({\varphi }_1-s{)}^{2-\alpha +\beta }\right]}{{\left[({\varphi }_1-s)({\varphi }_2-s)\right]}^{2-\alpha +\beta }}\right|\\ & +\frac{1}{\Gamma (1+\alpha - \beta )} \left|{\displaystyle \underset{0}{\overset{t}{\int }}\frac{Fu (x, s)}{{({\varphi }_{{}_2} - {\varphi }_{{}_1})}^{2-\alpha +\beta }}}\right|,\\ & \le \frac{M \tilde{M} \left({\varphi }_1^{1+\alpha -\beta } - {\varphi }_2^{1+\alpha -\beta }+({\varphi }_2 - {\varphi }_1{)}^{1+\alpha -\beta }\right)}{(\alpha - \beta +1) \Gamma (\alpha - \beta +1)}\\ & +\frac{M \tilde{M} ({\varphi }_2 - {\varphi }_1{)}^{1+\alpha -\beta }}{(\alpha - \beta +1) \Gamma (\alpha - \beta +1)}, \end{array}$$

$$\le \frac{2M \tilde{M} ({\varphi }_2 - {\varphi }_1{)}^{1+\alpha -\beta }}{\Gamma (\alpha - \beta +2)}.$$

(7)

Consequently, we obtain ${‖(Fy) (x, {\varphi }_1) - (Fy) (x, {\varphi }_2)‖}_{C_E} \stackrel{}{\to } 0,$ as ${\varphi }_1\stackrel{}{\to }{\varphi }_2$.

Thus, we conclude that $F$ is completely continuous, and hence, from the Arzelà–Ascoli theory, Theorem 2, we obtain that $F$ is relatively compact. From Lemma 1, we can say that $y (x,t) = \left(F y\right)(x,t),$ and hence, from the first part of Schafer’s fixed point theory, Theorem 3, it is confirmed that the problem has a solution. $\square$

Theorem 5.

If the conditions A₁–A₄ are satisfied, then problem (1)–(2) has a unique solution.

Proof. 

From Theorem 4, we obtain:

$$\begin{array}{ll}\left|(Fy) (x,t) - (F \widehat{y} ) (x,t)\right| \le & \left\{I^{1+\alpha -\beta } \left(M\left|y(x,t) - \widehat{y} (x,t)\right|\right)\right.\left.{+I}^{\beta } \left(\tilde{M} \left|y(x,s)- \widehat{y} (x,s)\right|\right)\right\},\\ & \le {‖y - \widehat{y}‖}_{C_E}\mathit{sup} \left\{I^{1+\alpha -\beta }\right. \left(M(x,t) \frac{x-\mu }{\sigma }\right)\left.+I^{1+\alpha +\beta }\left(M^{*} (x,t)\right) \right\},\\ & \le \rho {‖y - \widehat{y}‖}_{C_E}.\end{array}$$

Thus, if $\rho \in (0 ,1),$ then $F$ turns out to be contraction mapping. The Banach fixed-point theory, Theorem 1, immediately established that $F$ possesses a fixed point that concludes the existence and uniqueness of the solution to the problem (1)–(2). $\square$

4. Lerch Matrix Collocation Method

The approximated solution of the problem (1)–(2) has been examined in the truncated Lerch series form as:

$$u\left(x, t\right)\simeq u_N(x,t;\lambda )= {\displaystyle \sum _{m=0}^N}{\displaystyle \sum _{n=0}^N}{a}_{m,n}{ L}_{m,n} (x,t;\lambda ),$$

(8)

where:

$L_{m,n} ( x,t ;\lambda )=L_m(x;\lambda ) L_n(t;\lambda ),$ is the unknown $a_{m,n}$ is parameter and $\lambda$, $N \in z^{+}$ Lerch polynomials coefficients, while $L_m ( x; \lambda )$ and $L_n ( t; \lambda ),$ denote Lerch polynomials with respect to $x$ and $t$, respectively.

From Definition 3, we can construct $N$ of Lerch polynomials on the matrix form as follows:

$$\underset{L ( x ; \lambda {)}^T }{\underbrace{\left[\begin{array}{c}{L}_0 (x ; \lambda )\\ \begin{array}{l}\\ L_1 (x; \lambda )\\ \end{array}\\ L_2 (x ; \lambda )\\ ⋮\\ \begin{array}{l}{L}_n (x ; \lambda )\\ \end{array}\end{array}\right]}}= \underset{C{(\lambda )}^T}{\underbrace{\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ 0& \frac{1!}{1!} S(1, 1) \left(\begin{array}{c}\lambda \\ 1\end{array}\right)& 0& \cdots & 0\\ 0& \frac{1!}{2!} S(2, 1) \left(\begin{array}{c}\lambda \\ 1\end{array}\right)& \frac{2!}{2!} S(2, 2) \left(\begin{array}{c}\lambda +1\\ 2\end{array}\right)& \cdots & 0\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& \frac{1!}{N!} S(N, 1) \left(\begin{array}{c}\lambda \\ 1\end{array}\right)& \frac{2!}{N!} S(N, 1) \left(\begin{array}{c}\lambda +1\\ 2\end{array}\right)& \cdots & \frac{N!}{N!} S(N, N) \left(\begin{array}{c}\lambda +N-1\\ N\end{array}\right)\end{array}\right]}}\times \underset{X{ \left(x\right)}^T }{\underbrace{\left[\begin{array}{c}\begin{array}{l}1\\ \end{array}\\ \begin{array}{l}x\\ \end{array}\\ \begin{array}{l}{x}^2\\ \end{array}\\ ⋮\\ x^N\end{array}\right],}}$$

or simply:

$$L (x ; \lambda {)}^T=C (\lambda {)}^T{ X \left(x\right)}^T=X \left(x\right) C (\lambda ).$$

(9)

4.1. Numerical Solution of Problem (1)–(2) Using LMC Method

According to the matrix relation (9) placed into the truncated Lerch series form (8), the approximate solution to the problem (1)–(2) is obtained as:

$$u_N(x,t;\lambda )=L(x;\lambda )\overline{L}(t;\lambda )A=X(x)C(\lambda )\overline{X}(t)\overline{C}(\lambda )A,$$

where:

$L (x ; \lambda ) = {\left[\begin{array}{ccc}{L}_0 (x ; \lambda )& L_1 (x ; \lambda )& L_2 (x ; \lambda )\cdots L_N (x ; \lambda )\end{array}\right]}^T, \overline{L} (t ; \lambda )= diag \left[L(t ; \lambda ), L(t ; \lambda ), \dots L(t ; \lambda )\right],$$\overline{X}(t) = diag \left[X\left(t\right), X\left(t\right), \dots X\left(t\right)\right], \overline{C}(\lambda ) = diag \left[C(\lambda ),C(\lambda ),\dots C(\lambda )\right] ,$ and $A={\left[\begin{array}{ccc}{a}_{00}& a_{01}\cdots a_{0N}& a_{11}\cdots a_{1N}\cdots a_{N0}\cdots a_{NN}\end{array}\right]}^T.$

Lemma 2.

The matrix relation of $u^{(m+1)}(x,0)$ is $X(x)C(\lambda )\overline{X}(0){\overline{B}}^{m+1}\overline{C} (\lambda )A.$

Proof. 

The connection between the matrix $X(x)$ and its derivative can be represented as:

$$X^{|}(x)=\left(\begin{array}{ccc}0& 1& 2x\cdots N{x}^{N-1}\end{array}\right) = X(x) B,$$

where:

$$B = \left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 2& \begin{array}{cc}& \cdots \end{array}& 0\\ \begin{array}{c}\\ ⋮\end{array}& \begin{array}{c}\\ ⋮\end{array}& \begin{array}{c}\\ ⋮\end{array}& \begin{array}{c}\ddots \\ \cdots \end{array}& \begin{array}{c}⋮\\ N\end{array}\\ 0& 0& 0& \cdots & 0\end{array}\right).$$

Thus:

$X^{\backslash \backslash }(x)= X(x) B^2,$ and hence we obtain:

$$X^{(x)}(x){=X\left(x\right) B}^k.$$

(10)

Thus, the matrix form of $u^{(m+1)}(x,0)$ using Lerch polynomials will be:

$$\begin{gathered} u^{(m+1)}(x,0)=L (x;\lambda {) \overline{L}}^{m+1}(0;\lambda ) A, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=X \left(x\right) C (\lambda {) \overline{X}}^{(m+1)}(0) \overline{C} (\lambda ) A. \end{gathered}$$

Applying Equation (10), we obtain:

$$u^{(m+1)}(x,0)=X \left(x\right) C (\lambda ) \overline{X}(0){ \overline{B}}^{m+1} \overline{C} (\lambda ) A,$$

where:

${ \overline{B}}^{m+1}=diag (B, B, B, \dots {B)}^{m+1}.$ $\square$

Now, the integral part of Equation (3) can be written on the matrix form by using Equations (8) and (9) as follows:

$$\begin{gathered} \frac{1}{\Gamma (1+\beta )} {\displaystyle \underset{0}{\overset{t}{\int }}\frac{y (x , t)}{ (t - S{)}^{2-\beta }}} \mathit{ds}= \frac{1}{\Gamma (1+\beta )} {\displaystyle \underset{0}{\overset{t}{\int }}{(t - S)}^{\beta -2}} X\left(x\right) C(\lambda ) \overline{X}(t) \overline{C}(\lambda ) A \mathit{ds}, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{\Gamma (1+\beta )} X\left(x\right) C(\lambda ) \overline{X} (\lambda {) t}^{\beta -2}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(1 - \frac{s}{t}\right)}^{\beta -2}A} \mathit{ds}. \end{gathered}$$

By applying binomial expansion to ${\left(1 - \frac{s}{t}\right)}^{\beta -2},$ and putting it on the matrix form, we obtain:

$$\frac{1}{\Gamma (1+\beta )} {\displaystyle \underset{0}{\overset{t}{\int }}\frac{y (x , t)}{ (t - S{)}^{2-\beta }}} \mathit{ds} = X\left(x\right) C(\lambda ) \overline{X} (t) \overline{C} (\lambda ) \left\{{\displaystyle \underset{0}{\overset{t}{\int }}B(s,t) \mathit{ds}}\right\} A ,$$

(11)

where:

$$B(s,t)=\frac{t^{\beta -2}}{\Gamma (1+\beta )} \left[\begin{array}{c}\begin{array}{c}1\\ - (\beta - 2)\end{array}\\ \\ \frac{(\beta - 2) (\beta - 3)}{2 !}\\ \begin{array}{c}- \frac{(\beta - 2) (\beta - 3) (\beta - 4)}{3 !}\\ ⋮\end{array}\\ \frac{(\beta - 2) (\beta - 3) \dots (\beta - N - 1)}{N !}\end{array}\right] \left[\begin{array}{c}1\\ \frac{S}{t}\\ \frac{S^2}{t^2}\\ \begin{array}{c}\frac{S^3}{t^3}\\ ⋮\end{array}\\ \frac{S^N}{t^N}\end{array}\right].$$

Using Lemma 2 and the matrix expression (9), we can obtain the principle matrix representation for the problem (1)–(2) as:

$$u_N(x,t;\lambda )={\displaystyle \sum _{M=0}^{n-1}} \frac{t^m}{(m+1) !} X\left(x\right) C(\lambda {) \overline{X}}_{(0)}^{(m+1)}{ \overline{B}}^{m+1} \overline{C} (\lambda ) A +X\left(x\right) C(\lambda ) \overline{X} (t) \overline{C} (\lambda ) F(t) A,$$

(12)

where:

$$F(t) ={\displaystyle \underset{0}{\overset{t}{\int }}B(s,t) \mathit{ds}}.$$

Briefly, Equation (12) can be reduced as:

$$M(x,t) A=0,$$

(13)

where:

$$M(x,t)={\displaystyle \sum _{M=0}^{n-1}} \frac{t^m}{(m+1) !} X\left(x\right) C(\lambda {) \overline{X}}_{(0)}^{(m+1)}{ \overline{B}}^{m+1} \overline{C} (\lambda )+X(x) C(\lambda ) \overline{X} (t) \overline{C} (\lambda ) F(t).$$

Using the collocation nodes $x_{\mu }$ and $t_p$ for $x=[a, b]$ and $t\in J$ into Equation (13), we obtain:

$x_{\mu }$ = a + $\frac{b - a}{N} \mu { , t}_p=\frac{T}{N} p ,$ where $\mu , p=0, 1, 2, \dots , N,$

Thus, the matrix expression will be:

$$M( x_{\mu }{ , t}_p) A=0, \mathrm{or} M_p A = 0,$$

(14)

where:

$$M_p=\left[\begin{array}{c}M{ (x}_0{ , t}_p)\\ M{ (x}_1{ , t}_p)\\ ⋮\\ M{ (x}_N{ , t}_p)\end{array}\right].$$

Similarly, we can use Equations (8) and (9) to obtain the matrix representation form for the initial conditions of problem (1)–(2) as follows:

$$\begin{array}{ll}u(x, 0)=& L (x; \lambda ) \overline{L} (0, \lambda )A,\\ & = X \left(x\right) C(\lambda ) \overline{X} (0) \overline{C} (\lambda )A= U(x, 0) A.\end{array}$$

where:

$$U(x, 0)=X(x) C(\lambda ) \overline{X} (0) \overline{C} (\lambda ).$$

When also applying the collocation nodes $x_{\mu }{ , t}_p$ for $\mu , p=1, 2, \dots , N-1$ into the matrix form $U (x, 0) A=H \left(x\right)$, the system is then transformed into being:

$$U_{\mu }{ A=H}_{\mu }$$

(15)

where $U_{\mu }=\left[\begin{array}{c}U{ (x}_1 , 0)\\ U{ (x}_2 , 0)\\ ⋮\\ U{ (x}_{N-1} , 0)\end{array}\right]$ and $H_{\mu }=\left[\begin{array}{c}h{ (x}_1)\\ h{ (x}_2)\\ ⋮\\ h{ (x}_{N-1})\end{array}\right]$, for $\mu$$= 1, 2, \dots , N-1.$

Additionally, $u_t (x, 0)=L (x; \lambda )$$\overline{L}\prime (0, \lambda ) A$ = $X \left(x\right) C(\lambda ) \overline{X}\prime (0)\overline{C}(\lambda )A,$

= $X \left(x\right) C(\lambda ) \overline{X}(0\left)\overline{B}\overline{C}\right(\lambda )A=0,$

$V(x,0) = X \left(x\right) C(\lambda )\overline{X}(0\left)\overline{B}\overline{C}\right(\lambda ).$ where: $V(x,0) A = 0,$=

Or simply:

$$V_{\mu } A=O,$$

(16)

where: $V_{\mu }=\left[\begin{array}{c}V{ (x}_1 , 0)\\ V{ (x}_2 , 0)\\ ⋮\\ V{ (x}_{N-1} , 0)\end{array}\right]$, $O=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right].$

$$\begin{gathered} u_x(0, t)=L` (0, \lambda ) \overline{L} (t; \lambda ) A=X\prime \left(0\right) C(\lambda ) \overline{X}(t\left)\overline{C}\right(\lambda )A, \\ =X \left(0\right) B C(\lambda \left)\overline{X}\right(t\left)\overline{C}\right(\lambda )A=W(0,t) A = O. \end{gathered}$$

Or simply:

$$W_p A = O,$$

(17)

where: $W_p=\left[\begin{array}{c}W{ (0 , t}_1)\\ W{ (0 , t}_2)\\ ⋮\\ W{ (0 , t}_{N-1})\end{array}\right] , p=1,2,\dots , N-1.$

Now, replacing the rows of matrices in (12) by 3 (N − 1) rows of the matrix in (13), the augmented matrix $\tilde{W}A=\tilde{H}$ is obtained, if $\mathit{Rank} (\tilde{W}){=\mathit{Rank} ( W ˜ : H)=(N+1)}^2$. Then, $A={(\tilde{W})}^{-1} \tilde{G}$ is calculated, and hence A is uniquely determined to obtain a unique solution for $u_{{}_N} (x, t; \lambda ).$ (See Kruchinin et al. [34], and Cayan et al. [29]).

4.2. Error Analysis

In this part of the study, we will introduce the residual error for the presented method. To constitute the residual correction procedure, let us put the approximate solution $u_N(x, t)$ and its derivatives into the problem (1)–(2) and call:

$$R_n\left(x_{\mu },{ t}_{\rho }; \lambda \right)=\left| u_N\right.(x_{\mu },{ t}_{\rho }) - \frac{1}{\Gamma (1+\beta )} {\displaystyle \underset{0}{\overset{t}{\int }}\frac{y(x_{\mu },{ t}_{\rho })}{{{(t}_{\rho } - S)}^{2-\beta }} \mathit{ds}}\left.{\displaystyle \sum _{\mu +p\le 2}{\displaystyle \sum _{m=0}^{n-1}\frac{u_N^{(m+1)}(x_{\mu },0)}{(m+1)!}{ t}_p^m}} \right|\simeq 0.$$

(18)

where $R_N (x\mu ,t_p;\lambda ) \le { 10}^{-k_{\mu p}} ,$$k\in z^{+}.$ If $\max {10}^{-k_{\mu p}}= {\max 10}^{-k}$ are prescribed, then when $N \stackrel{}{\to } \infty$, we have $R_N \stackrel{}{\to } 0$ (See Gokmen [35]).

Subtracting Equation (18) from Equation (19), we have:

$$\begin{gathered} R_N{ (x}_{\mu }{, t}_p; \lambda ) = \left|{ e}_N (x, t; \lambda ) - \frac{1}{\Gamma (1+\beta )}\right. {\displaystyle \underset{0}{\overset{t}{\int }}\frac{e_N^{*} (x , t)}{{(t - S)}^{2-\beta }{ (t}_p - S{)}^{2-\beta }}} \mathit{ds} \\ -\left.{\displaystyle \sum _{m=0}^{n-1}{\displaystyle \sum _{\mu +p\le 2}\frac{u_N^{(m+1)}{ (x}_{\mu }{ , 0) t}_p^m{ - u}^{(m+1)}{ (x, 0) t}^m}{(m+1) !}}}\right| . \end{gathered}$$

(19)

where:

$e_N (x, t; \lambda )=\left| u{ (x, t) - u}_N{ (x}_{\mu }{ , t}_p , \lambda )\right|$, and $e_N^{*} (x, t)=\left|{{(t}_p- s)}^{2-\beta }{ y (x, t) - (t - s)}^{2-\beta }{ y (x}_{\mu }{ , t}_p)\right|.$

Since the exact solution and the Lerch matrix expression satisfy the initial conditions, the conditions for the error will be:

$$e_N (x, 0; \lambda )=0 , \frac{\partial e_N (x, 0)}{\partial t}=\frac{\partial e_N (0, t)}{\partial x}=0.$$

(20)

If we apply the LMC method to the error system (19)–(20) for some $r$, it is not necessary for it to be different from $N$ so it is easy to obtain the solution indicated by ${\widehat{e}}_N^r(x, t; \lambda )$. One of the benefits of the procedure is acquiring a new approximate solution by adding ${\widehat{e}}_N^{}(x, t; \lambda )$ to $u_N(x,t;\lambda );$ the new approximate solution is named the corrected approximate solution $u_N(x,t;\lambda )$ and is represented as:

$$u_N{}_{,r} (x, t; \lambda {)=u}_N (x, t; \lambda )+{\widehat{e}}_N^r (x, t; \lambda ).$$

Through the aid of the residual function $R_N (x, t; \lambda ),$ the accuracy of the solution can be controlled.

If $‖{ e}_N (x, t; \lambda {) - \widehat{e}}_N^r (x, t; \lambda ) ‖ < ‖{ u (x, t) - u}_N^{} (x, t; \lambda ) ‖$, then $u_N{}_{,r} (x, t; \lambda )$ is a better approach than $u_N(x, t; \lambda );$ moreover, we can estimate the error $e_N(x, t; \lambda )$ by ${\widehat{e}}_N^r (x, t; \lambda ),$ whenever $‖{ e}_N (x, t; \lambda {) - \hat{e}}_N^r (x, t; \lambda ) ‖ < \epsilon$.

4.3. Convergence Analysis of the Problem (1)–(2)

Lemma 3.

The residual function sequence  ${\left\{R_N (x, t; \lambda )\right\}}_{N=2}^{\infty }$ is convergent, and the following inequality is satisfied:

$‖{ R}_M (x, t; \lambda ) ‖ \le C ‖{ R}_N (x, t; \lambda ) ‖$, $N < M,$ for $C\in (0, 1).$

Proof. 

We will define $R_N (x, t; \lambda )$ on the interval $[0, b] \times [0, T]$ as:

$R_N (x, t; \lambda ) : \left[a+{\epsilon }_1 , b - {\epsilon }_1\right]\times ({\epsilon }_2 , T - {\epsilon }_2) , {\epsilon }_1 , {\epsilon }_2$ are sufficiently small values.

If we express $R_N (y, t; \lambda )$ in the Taylor expansion form, we then obtain:

$$\begin{array}{ll}‖{ R}_M (x, t; \lambda ) ‖& =\mathit{sup} \left\{\left|{\displaystyle \sum _{\mu =0}^M{\displaystyle \sum _{p=0}^M{\eta }_{\mu .p}{ x}^{\mu }{ t}^p}}\right|\right\} \le \mathit{sup} {\displaystyle \sum _{\mu =0}^M{\displaystyle \sum _{p=0}^M}\left|{\eta }_{\mu .p}{ x}^{\mu }{ t}^p\right| }.\\ & ={ R}_M (b, T, \lambda ).\end{array}$$

(21)

This can be expressed as:

$$‖{ R}_M (b, T; \lambda ) ‖ \le C ‖{ R}_N (b, T; \lambda ) ‖, N < M.$$

(22)

By using inequalities (21) and (22), we have:

$$\begin{gathered} \left|R_M (b, T, \lambda ) -{ R}_N (b, T, \lambda )\right| \le (C-1) \left|R_N (b, T, \lambda )\right|\le { (C-1)}^2\left|R_M (b, T, \lambda )\right| \le \dots \\ \le { (C-1)}^{N-1}\left|R_2 (b, T, \lambda )\right| \end{gathered}$$

Now, consider $Q\in Z^{+}$, so that $N > Q,$

$$\begin{array}{c}\left|R_N (b, T, \lambda ) -{ R}_Q (b, T, \lambda \right)|=\\ |R_N (b, T, \lambda ) -{ R}_{N-1} (b, T, \lambda ) +{ R}_{N-1} (b, T, \lambda )\dots |\left.R_{Q+1} (b, T, \lambda ) -{ R}_Q (b, T, \lambda )\right|,\\ \le |R_N (b, T, \lambda ) -{ R}_{N-1} (b, T, \lambda )|+\left|R_{N-1} (b, T, \lambda {) - R}_{N-2} (b, T, \lambda )\right|+\dots \\ \left|R_{Q+1} (b, T, \lambda ) -{ R}_Q (b, T, \lambda )\right|,\\ \le \left[{(C-1)}^{N-2}+{(C-1)}^{N-3}+\dots {(C-1)}^{Q-1}\right] \left|R_2 (b, T, \lambda )\right|.\end{array}$$

The sum of this finite geometric series will be:

$$\left|R_N (b, T, \lambda ) -{ R}_Q (b, T, \lambda )\right| \le \frac{{(C-1)}^{Q-1} \left[1 - {(C-1)}^{N-1-Q}\right]}{1 - (C-1)} \left|R_2 (b, T, \lambda )\right|.$$

For large $N, Q,$ and $C\in (0 , 1),$ we have:

$$\underset{N,Q\to \infty }{lim}\left|R_N (b, T; \lambda ) -{ R}_Q (b, T; \lambda )\right|=0.$$

This means that ${\left\{R_N (x, t; \lambda )\right\}}_{N=2}^{\infty }$ is a Cauchy sequence, and hence, the residual function $R_N (x, t; \lambda )$ is convergent. $\square$

Lemma 4.

The differences between  $u_M$$(x, t; \lambda )$ and  $u_N$$(x, t; \lambda )$ nearly bound each of the consecutive absolute errors.

Proof. 

Consider $u_{{}_N}(x, t; \lambda )$ and $u_{ N^{\prime }}(x, t; \lambda )$ as two approximate solutions for the system (17).

Suppose that:

$$‖{ u (x, t) - u}_{N\prime } (x, t; \lambda ) ‖ < ‖{ u (x, t) - u}_N (x, t; \lambda ) ‖.$$

Let:

$$‖{ u (x, t) - u}_N (x, t; \lambda ) ‖= \tilde{C}‖{ u (x, t) - u}_{N\prime } (x, t; \lambda ) ‖,$$

Then, using triangle inequality, we obtain:

$‖{ u (x, t) - u}_{N\prime } (x, t; \lambda ) ‖ < \frac{1}{ \tilde{C}-1} ‖{ u}_N{ (x, t) - u}_{N\prime } (x, t; \lambda ) ‖$,

where C > 1.

According to Lemma 2, we can say that $‖{ u}_M (x, t ; \lambda {) - u}_N (x, t; \lambda ) ‖$ bound each of the consecutive absolute errors; thus:

$$\underset{M,N\to \infty }{lim}\left| u_M (x, t; \lambda ) -{ u}_N (x, t; \lambda )\right|=0.$$

□

5. Numerical Examples

At this point, we provide two numerical examples to present a clarification for the theoretical and numerical works covered in Section 3 and Section 4, respectively.

Example 1.

Consider the following NHFIDE:

$$\frac{{\partial }^{1.4} u(x,t)}{\partial { t}^{1.4}} = \frac{x^2}{2} \frac{{\partial }^{1.1} u(x,t)}{\partial { x}^{1.1}} + {\displaystyle \underset{0}{\overset{t}{\int }}\frac{x}{s} e^{u(x,s)} ds},$$

(23)

Under the initial conditions, $u(x, 0)=0{.5, u}_t{(x, 0)=u}_x(0, t)=0, 0 \le x \le 1$ and $t\in ( 0, T).$

$\alpha = 0.4,$$\beta = 0.1$ and $[a,b]= [0, 1]$ with the functions $f$ and $g$ defined by $f(x, t, y, v)= \frac{1}{2}{x}^{2}y+ v, g( x, t, s, y ) = \frac{x}{s}{e}^y,$ where $v, s\in (0, T).$

Since $\left|f( x, t, y_1, v_1 ) - f( x, t, y_2, v_2 )\right| \le \frac{1}{2}{x}^2 \left|y_1 - y_2\right| + \left| v_1 - v_2\right|\le \left(x^2 \left|y_1 - y_2\right| + \left| v_1 - v_2\right|\right), 0 \le x \le 1.$

$A3$ with $M = 1,$ and similarly, one can state in a straightforward manner that the function $g$ satisfies the assumption $A_4$ with $M^{+} = 1.$ Moreover, the value of $\rho$ defined in Equation (6) is calculated so as to obtain $\rho =\frac{M \Gamma (1+\alpha ) ( b-a {)}^{1+\alpha +\beta }{+M}^{*} \Gamma (\alpha -\beta ) ( b-a {)}^{\alpha }}{\Gamma (\alpha -\beta ) \Gamma (1+\alpha )} = \frac{\Gamma (1.4) + 0.341\times \Gamma (0.3)}{\Gamma (0.3)\Gamma (1.4)}=0.831<1.$ According to Theorem 5, we infer that Equation (23) has a unique solution.

The exact solution of Example 1 is $u(x, t)=0{.5 e}^{2\mathrm{xt}}{cos x}^{2t};$ in Figure 1, the exact solution and the approximate solutions of $u_N(x,t; \lambda )$ are presented with dispersing $x, t \in \left[0,1\right]$, as $x=0.2:0.02:0.8,t=0.33:0.033:0.99$, and by taking $N=10, \lambda =1$. Based on the proposed technique for the LMC method, the numerical solutions and the absolute errors of Example 1 are shown in

Table 1. The exact solution and the absolute errors for approximate solutions for Example 1.

x	t	Exact Solution	LMC Method
			N = 10		N = 15		N = 20
			λ = 1	λ = 2	λ = 1	λ = 2	λ = 1	λ = 2
0.2	0.33	0.53415	3.75 × 10−9	1.45 × 10−8	2.42 × 10−8	6.11 × 10−9	4.13 × 10−9	3.32 × 10−10
0.2	0.66	0.77055	2.55 × 10−8	8.89 × 10−10	3.37 × 10−9	2.36 × 10−11	1.42 × 10−9	2.58 × 10−11
0.2	0.99	0.80948	4.54 × 10−11	2.29 × 10−11	7.11 × 10−11	3.64 × 10−11	6.67 × 10−11	6.53 × 10−11
0.4	0.33	0.97055	9.24 × 10−6	7.38 × 10−10	3.85 × 10−9	7.68 × 10−10	8.39 × 10−19	1.89 × 10−10
0.4	0.66	1.04405	6.54 × 10−8	3.17 × 10−11	5.14 × 10−10	1.22 × 10−12	6.83 × 10−10	5.75 × 10−11
0.4	0.99	1.25293	1.45 × 10−10	2.26 × 10−12	6.34 × 10−11	7.17 × 10−12	4.33 × 10−12	2.33e × 10−12
0.6	0.33	1.39947	1.23 × 10−9	8.02 × 10−9	1.29 × 10−10	1.03 × 10−10	2.58 × 10−10	4.62 × 10−11
0.6	0.66	1.44292	3.54 × 10−9	1.26 × 10−10	2.87 × 10−10	3.21 × 10−11	1.72 × 10−11	7.32 × 10−11
0.6	0.99	1.52549	2.45 × 10−11	1.99 × 10−12	3.51 × 10−11	7.29 × 10−11	8.11 × 10−12	8.69 × 10−12
0.8	0.33	1.60109	8.26 × 10−11	3.69 × 10−11	8.09 × 10−10	3.11 × 10−10	4.62 × 10−11	4.03 × 10−11
0.8	0.66	1.64774	1.25 × 10−10	7.14 × 10−11	2.11 × 10−11	1.81 × 10−12	6.39 × 10−11	8.91 × 10−12
0.8	0.99	1.80383	4.24 × 10−12	2.53 × 10−12	2.66 × 10−11	5.07 × 10−12	9.59 × 10−12	3.59 × 10−12

for three values, $N=10, N=15$ and $N= 20$, and for the Lerch parameters $\lambda =1 \mathrm{and} \lambda =2$ in each case of $N.$ The average CPU time in Example 1 is 1.19 s.

Example 2.

Consider the NHFIDE:

$$\frac{{\partial }^{1.8} u(x,t)}{\partial { t}^{1.8}} = \frac{{\partial }^{1.5}}{\partial { x}^{1.5}} \mathit{log} u(x,t) + \frac{4}{5} {\displaystyle \underset{0}{\overset{t}{\int }} e^{2x} u(x,s) ds},$$

(24)

under the initial conditions, $u(x, 0)=0{.5, u}_t{(x, 0)=u}_x(0, t)=0, 0 \le x \le 1$ and $t\in ( 0, T).$ We also notice that Example 2 is an application of $\alpha = 0.8,$$\beta = 0.5$ and $[a,b]= [0, 1]$; also, we have $f$ and $g$ defined as $f(x, t, y, v)= \log y+ v, g( x, t, s, y ) = \frac{4}{5}{e}^{2x} y.$ Similarly to Example 1, it is clear that the $A_3$ and $A_4$ with $M = 1$ and $M^{+} = \frac{4}{5} ,$ respectively. Furthermore, the value of $\rho$ is calculated so to obtain $\rho = 0.791 < 1.$ We use Theorem 5 to confirm that Equation (24) has a unique solution. The exact solution of Equation (24) is $u(x,t)=\sqrt{xt+1} \mathit{Sec} \left(\frac{\pi }{2}{x}^{2}t\right)$, and as shown in Figure 2, the exact solution and the approximate solutions of $u_N(x,t; \lambda )$ are presented with dispersing $x, t \in \left[0,1\right]$, as $x=0.2:0.02:0.8, t=0.33 : 0.033 : 0.99$, and by taking $N=15, \lambda =1$. Based on the proposed technique for the LMC method, the numerical solutions and the absolute errors of Example 2 are shown in

Table 2. The exact solution and the absolute errors for approximate solutions for Example 2.

x	t	Exact Solution	LMC Method
			N = 10		N = 20		N = 30
			λ = 1	λ = 2	λ = 1	λ = 2	λ = 1	λ = 2
0.2	0.33	1.13425	3.32 × 10−11	1.22 × 10−11	9.15 × 10−10	1.35 × 10−11	3.23 × 10−11	2.47 × 10−12
0.2	0.66	1.32644	6.23 × 10−11	8.18 × 10−12	6.01 × 10−11	3.29 × 10−11	1.43 × 10−12	1.24 × 10−12
0.2	0.99	1.32685	2.69 × 10−12	2.04 × 10−12	8.04 × 10−11	2.09 × 10−12	4.04 × 10−12	5.66 × 10−12
0.4	0.33	1.44783	9.17 × 10−10	7.56 × 10−11	6.22 × 10−11	5.21 × 10−11	7.37 × 10−11	7.53 × 10−12
0.4	0.66	1.54373	1.24 × 10−12	6.24 × 10−12	1.38 × 10−12	1.46 × 10−12	4.39 × 10−12	2.58 × 10−13
0.4	0.99	1.60291	8.93 × 10−12	3.21 × 10−13	3.53 × 10−12	1.77 × 10−13	7.28 × 10−12	4.31 × 10−13
0.6	0.33	1.79543	2.09 × 10−9	2.06 × 10−10	2.09 × 10−11	4.03 × 10−11	3.18 × 10−12	7.55 × 10−12
0.6	0.66	1.88256	2.55 × 10−10	7.23 × 10−11	4.26 × 10−10	3.72 × 10−12	9.33 × 10−12	8.83 × 10−13
0.6	0.99	1.92481	7.19 × 10−12	2.82 × 10−12	4.02 × 10−11	2.29 × 10−13	4.95 × 10−13	1.24 × 10−13
0.8	0.33	2.01789	2.02 × 10−11	2.92 × 10−12	5.12 × 10−10	2.26 × 10−12	6.07 × 10−12	7.82 × 10−13
0.8	0.66	2.27456	1.04 × 10−11	5.04 × 10−13	2.28 × 10−12	8.06 × 10−12	7.29 × 10−12	3.31 × 10−14
0.8	0.99	2.65242	2.27 × 10−12	4.25 × 10−12	1.04 × 10−12	7.68 × 10−13	6.13 × 10−13	9.08 × 10−14

for different values of $N=10, N=20$ and $N=30$, and for $\lambda =1 \mathrm{and} \lambda =2$ in each case of $N.$ The average CPU time in Example 2 is 1.37 s. Furthermore, in Figure 3 and Figure 4, we showed a comparison between exact and numerical solutions for different values of the spatial variable x and for time t, respectively of Example 1 For more details, we showed in Figure 5 and Figure 6 a comparison between the exact and numerical solutions for different values of the spatial variable x and for time t, respectively of Example 2.

Comparison of the Findings

• Based on Example 1, the variation among exact solutions and approximate solutions for distinct values of N is computed according to Table 1. From the absolute errors, we noticed the following:
  • For $N=10,$ we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.2	0.66	2.55 × 10−8	0.2	0.33	1.45 × 10−8
Lowest Error	0.8	0.99	4.24 × 10−12	0.8	0.99	2.53 × 10−12

• For $N=15,$ we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.2	0.33	2.24 × 10−8	0.2	0.33	6.11 × 10−8
Lowest Error	0.2	0.99	7.11 × 10−11	0.4	0.99	7.17 × 10−12

• For $N=20,$ we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.2	0.66	1.42 × 10−8	0.4	0.33	1.89 × 10−8
Lowest Error	0.8	0.99	9.95 × 10−12	0.8	0.66	8.91 × 10−12

2.

According to Example 2, for different values of N, the difference between exact solutions and approximate solutions was displayed in Table 2. We reached the following results from the absolute errors:

• For $N=10,$ we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.4	0.99	2.55 × 10−10	0.6	0.33	2.06 × 10−10
Lowest Error	0.4	0.99	8.93 × 10−12	0.8	0.66	5.04 × 10−13

• For $N=20,$, we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.6	0.66	4.26 × 10−10	0.2	0.33	1.35 × 10−8
Lowest Error	0.4	0.99	3.53 × 10−12	0.8	0.99	7.68 × 10−13

• For $N=30,$ we have:

	λ = 1			λ = 2
	x	t	Abs. Error	x	t	Abs. Error
Highest Error	0.2	0.33	3.23 × 10−11	0.2	0.66	1.24 × 10−8
Lowest Error	0.8	0.99	6.13 × 10−13	0.8	0.99	9.08 × 10−14

6. Conclusions

In the context of this study, sufficient conditions for the existence and uniqueness of the NHFIDEq were provided by relying upon the usage of fixed-point theorems and fractional calculus. Additionally, the Lerch matrix collocation method is meant to assist with solving the NHFIDEq by putting the model on the matrix representation form corresponding to a linear system of algebraic equations; moreover, convergence and error criteria are provided to clarify the correctness and validity of the developed method and to indicate that the suggested method is more accurate than other numerical approaches. We provide two examples of numerical computations using [Matlab R2022b] as a final step in validating the theoretical study. These examples show comparisons between exact solutions and numerical solutions for various values of $N$ and for Lerch parameter values set to $\lambda =1$ and $\lambda =2,$ as well as showing the absolute errors in each example, as shown in Table 1 and Table 2. In Figure 3, Figure 4, Figure 5 and Figure 6a–c, we noticed that there is a large convergence between the exact and numerical solutions at time $t= 0.9, \lambda = 2$, especially in Figure 4c and Figure 6c, more than for the solutions at time $t= 0.3, \lambda = 2$ shown in Figure 3c and Figure 5c. Based on the findings, we can conclude that when we take bigger values of N and increase the time to reach to $t= 0.9,$ we obtain a higher accuracy of the numerical solutions, and hence we can conclude that the LMC approach is particularly effective for locating precise numerical solutions and that it significantly reduces calculation time while producing accurate results, which can be observed by comparing the numerical solutions and exact answers in the mentioned examples.

Future Work

In order to solve a fractional optimum control problem with a free end point (see Taha et al. [3]), we would like to continue this work and explore the optimality conditions subject to the dynamical constraint on the hyperbolic form presented in the model (1)–(2).