Robust spectral treatment for time-fractional delay partial differential equations

Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.

Over the last four decades, spectral methods have been developed rapidly for several advantages. The uses of spectral methods ensure that all calculations can be performed numerically with an arbitrarily large degree of accuracy. As a result, an accurate solution was obtained with a relatively small number of degrees of freedom and hence at a reasonable computational cost, especially for variable coefficients and nonlinear problems. The choice of trial functions is one of the features which distinguish spectral methods from finite-element and finite-difference methods. The trial functions for spectral methods are differentiable global functions that fit well with the nonlocal definition of fractional derivatives, making them promising candidates for solving different types of fractional differential equations.
Spectral techniques can be separated into three principal types: Galerkin, tau, and collocation. The choice of the appropriate utilized spectral method suggested for solving such differential equations depends certainly on the type of the differential equation and also on the type of the initial or boundary conditions governing it. Also, different trial functions lead to different spectral approximations, for instance, trigonometric polynomials for periodic problems; Chebyshev, Legendre, ultraspherical, and Jacobi polynomials for non-periodic problems; Jacobi rational and Laguerre polynomials for problems on the half line, and Hermite polynomials for problems on the whole line.
In the last few years, Several authors have expressed interest in utilizing various spectral approaches for solving ordinary and partial FDEs. Doha et al. (2013) applied the collocation and tau spectral approaches based on shifted Jacobi polynomials for solving linear and nonlinear FDEs, respectively. Alsuyuti et al. (2019) used a modification of the SG approach for solving a class of ordinary FDEs. Abd-Elhameed et al. (2022) used the third-and fourthkinds Chebyshev polynomials as the basis function of the Petrov-Galerkin method for treating particular types of odd-order boundary value problems.  introduced two efficient SG algorithms for solving multi-dimensional time-fractional advection-diffusionreaction equations based on fractional-order Jacobi functions. Ezz-Eldien et al. (2020) introduced a solution of two-dimensional multi-term time-fractional mixed sub-diffusion and diffusion-wave equation using the time-space spectral collocation (SC) method.
Sometimes, the current state is insufficient to describe the behavior of the process being considered. Still, information on the previous state can benefit the system's behavior. These systems are called time-delay systems. In the last decades, much attention has been paid to time-delay systems because of its many applications in a large number of practical systems, such as economics, transportation, chemical processes, robotics, physics and engineering (Zavarei and Jamshidi (1987); Kuang (1993); Jaradat et al. (2022); Cai and Huang (2002); Bocharov and Rihan (2000); Ji and Leung (2002)). Searching for exact solutions for any system in the presence of delay is a complex process. Therefore, developing analytical and numerical techniques for various types of FDDEs has attracted the attention of many authors. Ezz-Eldien (2018) introduced the spectral tau technique for solving systems of multi-pantograph equations. Cheng et al. (2015) applied the reproducing kernel theory for solving the neutral functional-differential equation with proportional delays. Akkaya et al. (2013) introduced a numerical approach based on first Boubaker polynomials for solve FDDEs. Ahmad and Mukhtar (2015) applied the artificial neural network for the multipantograph differential equation. Ezz-Eldien and Doha (2019) used the SC for pantograph Volterra integro-differential equations. Otherwise, FDDEs have many applications in numerous scientific domains, such as population ecology, control systems, biology and medicine (Magin (2010) (2014)) and Legendre wavelet methods (Yuttanan et al. (2021)) have been applied for solving various types of FDDEs. Recently, spectral methods have been applied to solve different FDDEs. Hafez and Youssri (2022) used the Gegenbauer-Gauss SC method for fractional neutral functionaldifferential equations. Alsuyuti et al. (2021) used the Legendre polynomials as basis function of SG approach for solving a general form of multi-order fractional pantograph equations. Abdelkawy et al. (2020) used the Jacobi SC method for fractional functional differential equations of variable order. Ezz-Eldien et al. (2020) applied Chebyshev spectral tau method for solving multi-order fractional neutral pantograph equations.
The current study aims to develop an accurate numerical solution for the FDDEs and TFDPDEs. The numerical method offered in the spatial and temporal axes is based on the SG technique. Special properties of shifted Jacobi polynomials are exploited to generate new basis functions that meet the problem's initial and boundary conditions. The elements of a system of algebraic equations are determined using advanced techniques and are represented as a matrix.
The following is a breakdown of the article's structure. In Sect. 2, the SG approach is explained, along with a full explanation of how to solve FDDEs using the resulting system of algebraic equations. For TFDPDEs, the SG approach is used in Sect. 3. The convergence analysis and the stability of the proposed numerical scheme is investigated in Sect. 4. Section 5 presents numerical solutions to five test problems compared to those obtained using other algorithms to demonstrate the method's superior performance. Section 6 concludes with some recommendations.

Fractional delay differential equations
In this section, we discuss the numerical approach for solving the FDDEs subject to where 0 ≤ t ≤ τ, k − 1 < ν k ≤ k; (k = 1(1)n) and ν 0 = 0, while η k , α k , β k for k = 0(1)n − 1, are given constants and D ν t Y(t) denotes the Caputo fractional (CF) derivative of order ν, w.r.t. t Podlubny (1999), namely, where (.) and . are the Gamma and Ceilling functions, respectively. As an SG approach, we define the following space and then we choose the basis function as follows: where J (ρ,σ ) j,τ (t) denotes the shifted Jacobi polynomials defined on [0, τ ], have orthogonality relation is given by where ς j,j is the well-known Kronecker delta function, and with and For more details about Jacobi polynomials, one can consult (Ezz-Eldien (2016); Alsuyuti et al. (2022)). Now, we assume that the solution of the FDDEs (1)-(2) is approximated by where Hence, the SG technique is to find Let us denote where 0 ≤ j ≤ N 1 − n. Then, one can deduce that the main problem is equivalent to where C = {c 0 , c 1 , · · · , c N 1 −n }.
Using (9) with the explicit analytic form of the shifted Jacobi polynomial J Applying the CF-derivative, we get Making use of (12)-(13), and after performing some manipulations, we have where In virtue of (10) and (14), we have Finally, we use a suitable solver to find the unknowns vector C.

Time-fractional delay partial differential equations
In this section, we apply the numerical method for the TFDPDEs subject to where 0 ≤ x ≤ , 0 ≤ t ≤ τ, 0 < ν < 1, while μ ι , α ι , β ι for ι = 1(1)6, are given constants, and D ν t Y(x, t) denotes the CF-derivative of order ν, w.r.t. t. Defining the following spaces and As a time-space spectral method, we approximate the solution of the TFDPDE (17)-(18) by a truncated series of shifted Jacobi polynomials as follows where Using (21), we get
Using (22), we can write hence, In the same manner as in the previous section, we get where and while where In virtue of (26) and (32), we have

Theoretical analysis
This section aims to verify how effective the numerical solution of the suggested approach for problems (1) and (17). We start this section with a review of certain auxiliary lemmas that will be important later for the study of convergence and stability analysis for the proposed method.

Theorem 1 If Y(t) = t n f (t) is expanded in infinite series of the basis functions ψ j (t) given as in
Then this series converges uniformly to Y(t), and the expansion coefficients c j satisfy the following inequality with f (nm) (t) ≤ where is a positive constant.
Proof Applying the orthogonality relation (5) to Eq. (34) under the assumption (9), then we can write j,τ and W (ρ,σ ) τ (t) are given as in (6) and (7), respectively. Now, if we assume that Y(t) = t n f (t), and with the aid of the relation (9), then we have Applying the integration by parts nm-times, and in virtue of Lemma 1, we get Hence, Based on Lemmas 2 and 3, and after performing some manipulations, then we get (35), which ends the proof. (22), respectively, i.e.,

Theorem 2 If Y(x, t) = t x ( − x) f (t)f (x) is expanded in infinite series of the basis functions φ i (x) and ψ j (t) given as in
Then this series converges uniformly to Y(x, t), and the expansion coefficients c i j satisfy the following inequality Proof Applying the orthogonality relation (5) to Eq. (36), under the assumptions (22), then we can write where (ρ,σ ) j,τ and W (ρ,σ ) τ (t) are given in (6) and (7), respectively.

Again, with the aid of Eq. (36) and the assumption Y(x, t) = t x ( − x) f (t)f (x), the coefficients c i j can be written as follows
where W (ρ,σ ) (x, t) is given by (23), hence, Applying the integration by parts 2m-times and m-times with respect to x and t, respectively, and in virtue of Lemma 1, with the aid of Theorem 1 and Lemmas 2 and 3, and after performing some manipulations, we get which ends the proof.

Stability analysis
Theorem 3 For the two consecutive approximations Y N 1 (t) and Y N 1 +1 (t), we have the following estimate Proof With the aid of Theorem 1 and Lemma 4, we have Theorem 4 For the two consecutive approximations Y N 2 ,N 1 (x, t) and Y N 2 +1,N 1 +1 (x, t), we have the following estimate Proof Using a similar technique to that introduced for the proof of Theorem 3, we can prove this theorem.

Numerical verification
This section is confined to testing our proposed algorithm. For this purpose, we will present five numerical examples accompanied by presenting comparisons with some other techniques in the literature to demonstrate the efficiency and high accuracy of our proposed numerical algorithm.
Example 3 Consider the following TFDPDE (Hosseinpour et al. 2018) The exact solution is Y(x, t) = x 2 t 5 3 + t 4 3 . To approximate the solution to this problem, Hosseinpour et al. (2018) introduced two new numerical approaches based on the Pade approximation method with Legendre polynomial (PALP) and Muntz-Legendre polynomial (PAMLP). The authors in Hosseinpour et al. (2018) used the Pade approximation and two-sided Laplace transformations with the operational matrix of fractional derivatives to transform the main problem into a system of FPDEs without delay. In Table 2, we compare the absolute errors of Y(x, t) achieved using the proposed approach against the PALP and PAMLP approaches at {N 2 , N 1 } = {5, 5} and ν = 0.9.
and select p(x, t) so that the exact solution is Y(x, t) = t 2 cos(π x). Usman et al. (2020) considered this problem and applied the SC approach for getting the numerical solution. They constructed the operational matrices of fractional-order integration and those of differentiation with the delay operational matrix based on shifted Gegenbauer polynomials to transform them into a system of algebraic equations. The most accurately results obtained by the SC approach (Usman et al. 2020) were around 10 −15 using (N 2 = 20), see Figures 6 and 7 in Usman et al. (2020). In Table 3 To

Concluding remarks
In the current study, we developed a numerical method for solving the FDDEs. The suggested technique is based on the SG method and shifted Jacobi polynomials. A novel approach is used to convert the core problem into another by solving a system of algebraic equations. The SG approach is also applied for TFDPDEs in both temporal and spatial axes. Moreover, the convergence and stability of this scheme are rigorously established. To our knowledge, it is the first attempt to deal with TFDPDEs using the SG algorithm or shifted Jacobi polynomials. By providing five test problems and contrasting the outcomes with those obtained using alternative results and the exact solution, the effectiveness of the suggested approach is validated. The new results ensure that the suggested method is more accurate than the collocation Haar wavelet, Bernoulli wavelet operational matrix, Pade approximation Legendre polynomial, Pade approximation Muntz-Legendre polynomial, collocation Gegenbauer and collocation Genocchi wavelet methods. We also mention that the codes were written and debugged using Mathematica version 12 software using a PC machine, with Intel(R) Core(TM) i5-8500 CPU @ 3.00 GHz, 12.00 GB of RAM.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Conflict of interest This work does not have any conflicts of interest
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