Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As a matter of fact, the variable-order fractional derivative (VFD) that has been used is in the Caputo sense. The first step of this technique is constructive on the construction of the solution using the shifted Legendre–Laguerre polynomials with unknown coefficients. The second step involves using a combination of the collocation method and the operational matrices (OMs) of the shifted Legendre–Laguerre polynomials, as well as the Newton–Cotes nodal points, to find the unknown coefficients. The final step focuses on solving the resulting algebraic equations by employing Newton’s iterative method. To illustrate and demonstrate the technique’s efficacy and applicability, two examples have been provided.


Introduction
In recent decades, fractional calculus (FC) has played a significant role in science and engineering, and therefore, the scientists focused on its applications to model the real phenomena [2,23,24,29].The fractional derivative and integrals were recognized to be an efficient tool to describe the properties of complex dynamical processes more accurately than the standard integer derivative and integral [14,17,21,27,38].Fractional partial differential equations (FPDEs) are a fascinating subject, because they are frequently used to explain a variety of phenomena in real-world situations, including signal processing control theory, fluid flow, potential theory, information theory, finance, and entropy [7,9,25,32].Samko in 1993 [29] introduces the VOFDEs.These fractional operators can be considered as a generalization of fractional operators of constant orders.Indeed, the variable-order FPDEs extend the fractional fixed-order PDEs and occur in problems in the areas of physics and engineering [11,12,26,32,34].Many models of specific processes or dynamical systems in real-world problems exhibit neutral delay, which is always described using delay differential equations (DDEs) or time-delay systems [6,18,35].Despite the fact that FPDEs have been considered by a few researchers [16,22,37] and the references therein, there has been no work done in the area of VFDPDEs to our knowledge.Therefore, this reason motivates us in this paper to propose a numerical technique to solve a class of VFDDEs using the collocation method and the OMs of the shifted Legendre-Laguerre polynomials.A considerable advantage of the method is that the shifted Legendre-Laguerre polynomial coefficients of the solution are found very easily using computer programs.Also, according to the proposed model, the time of the occurrence of an event does not have fix domain.Therefore, for approximating the time functions in the problem, we apply the Laguerre polynomials, which defined in [0, ∞).Finally, using a few terms of shifted Legendre-Laguerre functions, approximate solution converges to the exact solution.

Pseudo-operational matrix of integer order integral of the shifted Legendre and Laguerre polynomials
The purpose of this section is to find the OMs of the integer order quintessential of SLPs and the (S Ps), respectively, using Taylor polynomials (TPs) [1,28,30], which is described as follows: The SLPs may be expressed by means of the TPs as Then, by integrating G(ξ ), the pseudo-operational matrix of the SLPs is obtained where 1 is the pseudo-operational matrix of the integer order integral of the SLPs and 1 is defined by [10,20,33] where Also, with the aid of integrating (τ ), we gain the operational matrix of integer integration of the (S Ps) as

123
where ϑ 2 = D 2 2 D −1 2 is the pseudo-operational matrix of the integer order integral of the shifted (S Ps) and 2 is given by

Pseudo-operational matrix of the variable-order fractional integral of the (S Ps)
To obtain the operational matrix of the variable-order Riemann-Liouville fractional integration of order δ(ξ, τ ) > 0 of the vector (τ ) defined in Eq. (3.5), we need to calculate first variable-order Riemann-Liouville fractional integral of the TPs which is written as where Also, we need to find where Lemma 5.1 Let (τ ) be the (S Ps) vector defined in (3.5) and q − 1 < δ(ξ, τ) ≤ q ∈ Z + The pseudooperational matrix of variable-order fractional integration of the vector (τ ) can be expressed as where Proof A direct application of the relations (4.2) and (5.1) is given as

The approach
This section is devoted to finding the numerical solution of the following VFDPDEs: So that, ν(ξ, τ ) is an unknown function, the known functions ν 0 (τ ), ν 1 (τ ) , g 0 (ξ ) and g 1 (ξ ) are given continuous functions.Also, q = max (ξ,τ )∈ {δ(ξ, τ )} and q ∈ Z + .For this problem, assume that the easiest order of spinoff with appreciate to ξ and τ is 2. Therefore, we obtain the following approximate functions as: where the unknown matrix U is defined as follows: By integrating of the above equation with (6.4) with respect to τ and using the initial condition (6.3), we have Integrating (6.5) with respect to τ yields where and where and Integrating (6.4) w.r.t.ξ and by the aid of the conditions (6.2) and ( 6.3) yields It is remarkable that ∂ 3 ν(0,τ ) ∂ξ∂τ 2 is unknown function, by integrating (6.11) from 0 to 1 with respect to ξ , we get and + ν0 (τ ). (6.13) By integrating (6.13) for τ , we acquire to (6.14)
Figure 1 represents the AE of Example 7.1 for M=N=8 and distinct values of δ(ξ, τ ).Also, Fig. 2 represents a comparison between the exact solution and the approximate solution using the proposed method.

Conclusion
In this paper, we formulate the collocation method and the OMs of shifted Legendre-Laguerre polynomials to approximate the solutions of VFDDEs.The proposed method transform the VFDDEs to system of algebraic equations using the nodal points of Newton-Cots.By solving the algebraic system using Newton's iterative methods, numerical solutions are obtained.The numerical results approved that the proposed method is accurate and has very high accuracy as M and N increased.
Funding The authors have not disclosed any funding.

Declarations
Conflict of interest The authors have not disclosed any competing interests.

Figure 3 5 − 7 Fig. 1
Figure 3 represents AE of Example 7.2 for M = N = 8 and distinct values of δ(ξ, τ ).A comparison between the exact and the approximate solutions of Example 7.2 is given in Fig. 4.

Fig. 2
Fig. 2 Approximate and exact solution of Example 7.1

Fig. 4
Fig. 4 Approximate and exact solution of Example 7.2