Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed; Hany Mostafa Ahmed; Waleed Mohamed Abd-Elhameed; Hany Mostafa Ahmed

doi:10.3934/math.2024107

AIMS Mathematics

2024, Volume 9, Issue 1: 2137-2166. doi: 10.3934/math.2024107

Previous Article Next Article

Research article Special Issues

Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed ^{1
,
,},
Hany Mostafa Ahmed ²

1.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
2.
Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt; Email: hanyahmed@techedu.helwan.edu.eg

Received: 08 October 2023 Revised: 04 December 2023 Accepted: 13 December 2023 Published: 20 December 2023
MSC : 65XX, 65Mxx, 33C45

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the efficiency and applicability of the proposed algorithms.

Keywords:

Citation: Waleed Mohamed Abd-Elhameed, Hany Mostafa Ahmed. Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials[J]. AIMS Mathematics, 2024, 9(1): 2137-2166. doi: 10.3934/math.2024107

Related Papers:

[1]	Manal Alqhtani, Khaled M. Saad . Numerical solutions of space-fractional diffusion equations via the exponential decay kernel. AIMS Mathematics, 2022, 7(4): 6535-6549. doi: 10.3934/math.2022364
[2]	Shazia Sadiq, Mujeeb ur Rehman . Solution of fractional boundary value problems by $ \psi $-shifted operational matrices. AIMS Mathematics, 2022, 7(4): 6669-6693. doi: 10.3934/math.2022372
[3]	Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830
[4]	Sunyoung Bu . A collocation methods based on the quadratic quadrature technique for fractional differential equations. AIMS Mathematics, 2022, 7(1): 804-820. doi: 10.3934/math.2022048
[5]	Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed . Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations. AIMS Mathematics, 2022, 7(5): 7759-7780. doi: 10.3934/math.2022436
[6]	K. Ali Khalid, Aiman Mukheimer, A. Younis Jihad, Mohamed A. Abd El Salam, Hassen Aydi . Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations. AIMS Mathematics, 2022, 7(5): 8622-8644. doi: 10.3934/math.2022482
[7]	Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi . A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151
[8]	Chang Phang, Abdulnasir Isah, Yoke Teng Toh . Poly-Genocchi polynomials and its applications. AIMS Mathematics, 2021, 6(8): 8221-8238. doi: 10.3934/math.2021476
[9]	Younes Talaei, Sanda Micula, Hasan Hosseinzadeh, Samad Noeiaghdam . A novel algorithm to solve nonlinear fractional quadratic integral equations. AIMS Mathematics, 2022, 7(7): 13237-13257. doi: 10.3934/math.2022730
[10]	Xiaojun Zhou, Yue Dai . A spectral collocation method for the coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2022, 7(4): 5670-5689. doi: 10.3934/math.2022314

Abstract

1. Introduction

Many different areas of mathematics and applied sciences can benefit from Chebyshev polynomials (CPs). For example, the area of approximation theory frequently employs CPs. Moreover, they are very helpful in numerical analysis. Spectral methods can utilize CPs and their various combinations as basis functions to obtain numerical solutions for various differential equations. These methods have the potential to achieve both rapid convergence and highly efficient solutions. For some articles that employ different types of CPs, see, for example, ^[1,2,3,4,5].

Fractional differential equations (FDEs) are crucial in different disciplines of the applied sciences. In fact, they describe many phenomena that cannot be described by ordinary differential equations. This is due to their ability to model complex phenomena involving memory and hereditary properties. For example, they model several biological and physiological processes, such as tumor growth and the behavior of neurons (see ^[6]). These equations are also used to simulate anomalous diffusion, wave propagation in complex media and electromagnetic phenomena (see ^[7]). In addition, the complicated mechanical reaction of viscoelastic materials under stress or strain has been frequently modeled using FDEs (see ^[8]). The use of fractional calculus has been seen in the domain of signal processing, namely in the areas of denoising, filtering and feature extraction; see, for example, ^[9].

Due to the importance of partial FDEs and the non-availability of solving them analytically in most cases, a lot of effort has been put into creating trustworthy numerical and analytical methods for treating these types of equations. Researchers have presented various methods, such as the Adomian decomposition method ^[10], the operational matrix methods ^[11,12] and the splines method ^[13], to solve different partial FDEs.

The various types of time-FDEs have been studied by many researchers. For example, the authors in ^[14] employed a finite difference method for treating the time-fractional diffusion equation. In ^[15], the authors followed an approach for treating the time-fractional Fisher's equations. The authors in ^[16] followed another approach for treating some types of time-fractional PDE. An integral method was followed in ^[17] for treating some time-FDEs. In ^[18], the authors treated other space-time FDEs. The authors in ^[19] combined the dual reciprocity method and the Laplace transformation approach with the singular boundary method to obtain solutions to anomalous heat conduction issues in functionally graded materials. In ^[20], the authors introduced a novel localized collocation method utilizing fundamental solutions to analyze long-term anomalous heat conduction in functionally graded materials. In ^[21], the authors followed a quadratic spline collocation method for the time fractional subdiffusion equation. Another approach is followed in ^[22] to handle the fractional-diffusion equation. In ^[23], the authors followed a certain collocation method for the time tempered fractional diffusion equation.

Among the important time-FDEs is TFHE. Researchers have utilized different numerical algorithms to solve this equation. For instance, the authors in ^[24] utilized an implicit difference scheme to handle the TFHE. The authors of ^[25] followed an approach for treating the TFHE. In ^[26], the authors employed another collocation algorithm to treat the same equation. Because the Caputo derivative is not local, solving TFHE is notoriously hard and takes a long time. For this reason, fast and parallel numerical solutions for these kinds of TFHEs are desirable ^[27,28].

Various types of DEs, including PDEs, can be treated numerically using different versions of spectral methods due to their high accuracy and versatility. When compared to other numerical approaches used to solve PDEs, spectral methods have many benefits. They provide high precision and efficient solutions since the error drops exponentially as you add more terms to the proposed expansion. In these methods, the numerical solution is expressed as combinations of different special functions, which are called basis functions. The choice of suitable basis functions depends on the spectral method that will be applied. For some books regarding the different spectral methods and their applications, one can be referred to ^{[29,30,31,32,33]}. There are three celebrated spectral approaches. The Galerkin method has some restrictions on choosing the basis functions; see, for example, ^{[34,35,36,37,38]}, where such restrictions do not exist when collocation and tau methods are applied; see, for example, ^{[39,40,41,42,43]}.

In this paper, we will introduce a new type of polynomial that generalizes the first and second kinds of CPs. These polynomials are new and differ from the existing generalizing polynomials of CPs, such as Gegenbauer and Jacobi polynomials. This motivates us to study and employ such polynomials. Furthermore, we have two advantages to using these polynomials:

● Several solutions can be obtained if these polynomials are used as basis functions due to the presence of two parameters.

● If these polynomials are used to treat TFHE, it will be shown that the Chebyshev first and second kinds of approximations are not always the best among the other approximations. This demonstrates the benefit of introducing such polynomials.

The article's primary aims can be summed up as follows:

(i) We introduce a new type of polynomials, named unified Chebyshev polynomials (UCPs), that unify CPs of the first and second kinds.

(ii) We implement some new formulas related to the UCPs and the shifted ones that are essential for our suggested algorithms.

(iii) We utilize the introduced polynomials along with the collocation and tau spectral methods to treat the TFHE.

The rest of the paper is organized as follows: The next section gives an overview of CPs and a new set of UCPs, as well as some definitions of fractional calculus. In Section 3, we present new formulas for the UCPs and the shifted ones that are necessary for our proposed algorithms. In Section 4.1, we employ the spectral tau method to treat the TFHE. In Section 4.2, we employ another collocation procedure to solve the same type of equation. In Section 5, we deeply discuss convergence and error analysis. In Section 6, we present several numerical examples that involve tables, figures and comparisons. Finally, Section 7 reports some conclusions.

2. Some fundamentals and preliminaries

In this section, the fractional differential operator in the Caputo sense is presented, and some useful properties are utilized throughout the paper. In addition, some essentials regarding the CPs are presented.

2.1. Some definitions of fractional calculus

Definition 2.1. ^[6,8] The fractional differential operator in Caputo sense is defined as

$\begin{equation} (D^{\nu}f)(t) = \begin{cases} \dfrac{1}{\Gamma(n-\nu)} \int_0^t(t-\tau)^{n-\nu-1} \, f^{(n)}(\tau)\, d\tau, & \nu > 0, \, t > 0, \\ \\ f^{(n)}(t), & \nu = n, \end{cases} \end{equation}$

(2.1)

where $\, n-1\leq\nu\leq n, n\in \mathbb{N}$ .

Here are some properties that are satisfied by $D^{\nu}$ for $n-1\leq \nu \leq n,$

$\begin{align} D^{\nu}(\mu_{1}f(t)+\mu_{2}g(t))& = \mu_{1}\, D^{\nu}f(t)+\mu_{2}\, D^{\nu}g(t), \\ \end{align}$

(2.2)

$\begin{align} D^{\nu}\, t^{k}& = \begin{cases} \frac{\Gamma(k+1)}{\Gamma(k+1-\nu)}\, t^{k-\nu}, & k\in \mathbb{N}, k\geq \lceil\nu\rceil, \\ 0, & k < \lceil\nu\rceil, \end{cases} \end{align}$

(2.3)

where $\lceil\nu\rceil$ denotes the smallest integer greater than or equal to $\nu$ . For more properties of fractional derivatives, see, for example, ^[44].

2.2. An overview on CPs as a type of new UCPs

It is well known that the well-known four kinds of CPs are all particular types of Jacobi polynomials (see, ^[45]). All of these polynomials satisfy the following recurrence relation:

$\begin{equation} \phi_{k}(\theta) = 2\, \theta\, \phi_{k-1}(\theta)-\phi_{k-2}(\theta), \quad k\ge 2, \quad \theta \in [-1, 1], \end{equation}$

(2.4)

but with the following different initial values:

$T_{0}(\theta) = 1, \ T_{1}(\theta) = \theta, \quad U_{0}(\theta) = 1, \ U_{1}(\theta) = 2\theta,$

$V_{0}(\theta) = 1, \ V_{1}(\theta) = 2\theta-1, \quad W_{0}(\theta) = 1, \ W_{1}(\theta) = 2\theta+1,$

where $T_{i}(\theta), U_{i}(\theta), V_{i}(\theta)$ and $W_{i}(\theta)$ denote, respectively, the four kinds of CPs, each of degree $i$ .

Among the important properties of CPs is that $\phi_{-j}(\theta), \, j\ge 0$ can be expressed in terms of $\phi_{j}(\theta)$ . In fact, we have

$T_{-j}(\theta) = T_{j}(\theta), \quad U_{-j}(\theta) = -U_{j-2}(\theta),$

$V_{-j}(\theta) = V_{j-1}(\theta), \quad W_{-j}(\theta) = -W_{j-1}(\theta).$

In this paper, we will introduce a new type of polynomials that unify the first- and second-kinds of CPs. These polynomials will be referred to as "Unified Chebsyhev polynomials (UCPs)". The sequence of UCP, $G^{A, B}_{k}(\theta), \, A, B > 0,$ may be constructed using the recurrence relation:

$\begin{equation} G^{A, B}_{k+1}(\theta) = 2\, \theta\, G^{A, B}_{k-1}(\theta)-G^{A, B}_{k}(\theta), \quad G^{A, B}_{0}(\theta) = A, \ G^{A, B}_{1}(\theta) = B\, \theta, \ k\ge 1. \end{equation}$

(2.5)

The first few UCPs $G^{A, B}_{k}(\theta), \, k = 2, 3, \dots, 7,$ are:

$\begin{align*} G^{A, B}_{2}(\theta)& = 2 B\, \theta^2-A, \\ G^{A, B}_{3}(\theta)& = 4 B\, \theta^3-(2 A+B)\, \theta, \\ G^{A, B}_{4}(\theta)& = 8 B\, \theta^4-4 (A+B)\, \theta^2 +A, \\ G^{A, B}_{5}(\theta)& = 16 B\, \theta^5-4(2 A+3 B)\, \theta^3 +(4 A+B)\, \theta, \\ G^{A, B}_{6}(\theta)& = 32 B\, \theta^6-8 (2 A+4 B)\, \theta^4+3 (4 A+2 B)\, \theta^2-A, \\ G^{A, B}_{7}(\theta)& = 64 B\, \theta^7-16 (2 A+5 B)\, \theta^5+8 (4 A+3 B)\, \theta^3 -(6 A+B)\, \theta. \end{align*}$

Remark 2.1. It is evident that the UCPs are generalizations of both the first- and second-kind CPs. We have

$T_{k}(\theta) = G^{1, 1}_{k}(\theta), \quad U_{k}(\theta) = G^{2, 1}_{k}(\theta).$

3. Some new formulas concerned with UCPs

In this section, we will establish some new formulas concerned with UCPs and their shifted polynomials on $[0, \ell]$ that will be employed in the next section to derive our proposed algorithms.

3.1. Some new results for the UCPs

We are going to state and prove a basic theorem regarding the UCPs. The UCPs can be expressed as a combination of two terms of CPs of the second kind, as we will demonstrate. The following theorem exhibits this important result.

Theorem 3.1. The following expression for the UCPs is valid

$\begin{equation} G^{A, B}_{n}(\theta) = \frac{1}{2} (B-2 A) U_{n-2}(\theta)+\frac{B}{2}\, U_{n}(\theta), \, n\ge 0. \end{equation}$

(3.1)

Proof. Consider the polynomial:

$\xi_{n}(\theta) = \frac{1}{2} (B-2 A) U_{n-2}(\theta)+\frac{B}{2}\, U_{n}(\theta).$

It is easy to see that $\xi_{0}(\theta) = G^{A, B}_{0}(\theta)$ and $\xi_{1}(\theta) = G^{A, B}_{1}(\theta)$ . Now, we are going to prove that $\xi_{n}(\theta) = G^{A, B}_{n}(\theta), \ \forall\ n\ge 2$ . We will prove that they have the same recursive formula. That is, we will prove that

$\begin{split} \xi_{n+2}(\theta)-2\, \theta\, \xi_{n+1}(\theta)+\xi_{n}(\theta) = 0. \end{split}$

We have

$\begin{split} \xi_{n+2}(\theta)-2\, \theta\, \xi_{n+1}(\theta)+\xi_{n}(\theta) = &\frac{1}{2} (B-2 A) U_n(\theta)+\frac{1}{2} B U_{n+2}(\theta)\\ &-2\, \theta \left(\frac{1}{2} (B-2 A)U_{n-1}(\theta)+\frac{1}{2} B U_{n+1}(\theta)\right)\\ &+\frac{1}{2} (B-2 A) U_{n-2}(\theta)+\frac{B}{2}\, U_n(\theta). \end{split}$

Using the well-known formula

$\theta\, U_{n}(\theta) = \frac12\left(U_{n-1}(\theta)+U_{n+1}(\theta)\right),$

we see that

$\xi_{n+2}(\theta)-2\, \theta\, \xi_{n+1}(\theta)+\xi_{n}(\theta) = 0.$

Theorem 3.1 is now proved. □

Theorem 3.2. The analytic form of $G^{A, B}_{n}(\theta)$ is:

$\begin{equation} \begin{split} G^{A, B}_{i}(\theta) = &\sum\limits_{r = 0}^{\left\lfloor \frac{i}{2}\right\rfloor } \frac{(-1)^r 2^{i-2 r-1} (B (i-2 r)+2 A r) (1+i-2 r)_{r-1}}{r!} \theta^{i-2 r}. \end{split} \end{equation}$

(3.2)

Proof. Formula (3.2) can be obtained directly from the expression in (3.1) along with the analytic form of $U_{i}(\theta)$ given by

$\begin{equation} U_{i}(\theta) = \sum\limits_{r = 0}^{\left\lfloor \frac{i}{2}\right\rfloor } \frac{(-1)^r\, 2^{i-2 r}\, (i-r)!}{(i-2 r)!\, r!}\, \theta^{i-2 r}. \end{equation}$

(3.3)

□

The following theorem gives the connection formula between the second-kind CPs $U_{n}(\theta)$ and the UCPs.

Theorem 3.3. For every non-negative integer $j$ , the following expression for $U_{j}(\theta)$ holds

$\begin{equation} U_j(\theta) = 2 \sum\limits_{r = 0}^{\left\lfloor \frac{j}{2}\right\rfloor } \frac{c_{j-2r}\, (2 A-B)^r}{B^{r+1}}\, G_{j-2 r}(\theta), \end{equation}$

(3.4)

where

$\begin{equation} c_{i} = \begin{cases} \dfrac{B}{2\, A}, & i = 0, \\ \\ 1.& i\ge 1. \end{cases} \end{equation}$

(3.5)

Proof. The proof can be easily accomplished by induction on $j$ . □

Now, we will give the inversion formula to $G^{A, B}_{n}(\theta)$ in the following theorem.

Theorem 3.4. For every non-negative integer $m$ , the following formula holds

$\begin{equation} \theta^m = \sum\limits_{i = 0}^{\left\lfloor \frac{m}{2}\right\rfloor} c_{m-2i}\, S_{i, m} \, G^{A, B}_{m-2i} (\theta), \end{equation}$

(3.6)

where

$\begin{equation} S_{i, m} = A^i B^{-i-1} 2^{1-m+i}+\sum\limits_{j = 1}^i \frac{A^{i-j} B^{j-i-1} 2^{1-m-j+i} (m-j-i) (m-i+1)_{j-1}}{j!}. \end{equation}$

(3.7)

Proof. The proof is based on making use of the inversion formula of $U_{j}(\theta)$ given by

$\begin{equation} \theta^j = 2^{-j} \sum\limits_{m = 0}^{\left\lfloor \frac{j}{2}\right\rfloor } \frac{(1+j-2 m)\, j!}{(j-m+1)!\, m!} U_{j-2 m}(\theta), \end{equation}$

(3.8)

along with the connection formula that is given by (3.4). □

3.2. Some formulas of the shifted UCPs

In this paper, it is useful to define the so-called shifted UCPs on $[0, \ell]$ as

$G^{A, B}_{n, \ell}(x) = G^{A, B}_{n}\left(\frac{2\, x}{\ell}-1\right), \quad \, n\ge 1, \, 0\le x\le \ell.$

According to this definition, it is easy to see from (3.1) that $G^{A, B}_{n, \ell}(x)$ can be expressed in terms of the shifted second-kind CPs as in the following corollary:

Corollary 3.1. For every positive integer $n$ , one has

$\begin{equation} G^{A, B}_{n, \ell}(x) = \frac{1}{2} (B-2 A) U_{n-2, \ell}(x)+\frac{B}{2}\, U_{n, \ell}(x), \, n\ge 0, \end{equation}$

(3.9)

where $U_{n, \ell}(x) = U_{n, 1}\left(\frac{2\, x}{\ell}-1\right)$ is the shifted CPs of second kind.

Proof. Formula (3.9) is a direct consequence of (3.1) by replacing $x$ by $\left(\frac{2\, x}{\ell}-1\right)$ . □

The following two corollaries are of interest hereafter. They are regarding the analytic and inversion formulas of the shifted polynomials $G^{A, B}_{n, \ell} (x)$ .

Theorem 3.5. Let $n$ be a positive integer. $G^{A, B}_{n, \ell}(x)$ has the following analytic formula

$\begin{equation} G^{A, B}_{n, \ell}(x) = \sum\limits_{m = 0}^{n} d^{(n)}_{m}\ell^{-m}\, x^{m}, \end{equation}$

(3.10)

where

$\begin{equation} d^{(n)}_{m} = \begin{cases} (-1)^n (A+ (B-A)\, n), & m = 0, \\ \dfrac{4^m (-1)^{n-m} (m+n-1)! \left((B-A) \left(m^2+m+n^2\right)+A (2 m+1) n\right)}{(2 m+1)!\, (n-m)!}, & m = 1, 2, \dots, n. \end{cases} \end{equation}$

(3.11)

Proof. Using the analytical expansion of $U_{n, \ell}(x)$ :

$\begin{equation} U_{n, \ell}(x) = \sum\limits_{i = 0}^{n}(-1)^i \left(\frac{4}{\ell}\right)^{n-i}\frac{(2n-i+1)!}{i!(2n-2i+1)!}\, x^{n-i}, \, n > 0, \end{equation}$

(3.12)

together with (3.9), formula (3.10) can be obtained. □

In the following theorem, we give the inversion formula of $G^{A, B}_{n, \ell} (x)$ which will play a pivotal role in investigating the convergence analysis of the proposed expansion.

Theorem 3.6. For every positive integer $m$ , the following inversion formula holds

$\begin{equation} x^m = \sum\limits_{p = 0}^{m} F_{p, m}\, G^{A, B}_{p, \ell} (x), \end{equation}$

(3.13)

where

$\begin{equation} F_{p, m} = c_{p}(2 m+1)! \ell^m\, \sum\limits_{k = 0}^{\left\lfloor \frac{m-p}{2}\right\rfloor } \frac{4^{1-m} B^{-k-1} (2 A-B)^k (2 k+p+1)}{(m-p-2 k)! (2 k+m+p+2)!}, \end{equation}$

(3.14)

and $c_{p}$ is as defined in (3.5).

Proof. The proof is based on making use of the inversion formula of the shifted second kind of CPs $U_{j, \ell}(x)$ on $[0, \ell]$ given by

$\begin{equation} x^m = (2 m+1)!\ell^m\, 2^{1-2m} \sum\limits_{p = 0}^{m} \frac{(p+1)}{(m-p)! (m+p+2)!} U_{p, \ell}(x), \end{equation}$

(3.15)

along with the connection formula

$\begin{equation} U_{j, \ell}(x) = 2 \sum\limits_{r = 0}^{\left\lfloor \frac{j}{2}\right\rfloor } \frac{c_{j-2r}\, (2 A-B)^r}{B^{r+1}}\, G_{j-2 r, \ell}(x), \end{equation}$

(3.16)

which can be obtained form (3.4) only if $(\frac{2\, x}{\ell}-1)$ is substituted instead of $x$ . □

Remark 3.1. It is worth mentioning here that when $B = \, 2A$ , we get

$G^{A, 2A}_{n}(x) = A\, U_{n}(x), \quad G^{A, 2A}_{n, \ell}(x) = A\, U_{n, \ell}(x),$

while when $B = \, A$ , we get

$G^{A, A}_{n}(x) = A\, T_{n}(x), \quad G^{A, A}_{n, \ell}(x) = A\, T_{n, \ell}(x).$

The following two theorems are pivotal in deriving our proposed numerical algorithms. They consist of the integer and fractional derivatives of $G^{A, B}_{j, \ell}(x)$ .

Theorem 3.7. The $qth$ derivative of $G^{A, B}_{j, \ell}(x)$ can be expressed as:

$\begin{equation} D^q\, G^{A, B}_{n, \ell}(x) = \sum\limits_{m = 0}^{n-q} h^{(q)}_{m, n}\, G^{A, B}_{m, \ell}(x), \, n\ge q \ge 1, \end{equation}$

(3.17)

where

$h^{(q)}_{m, n} = \sum\limits_{r = m}^{n-q}\frac{(r+q)!}{r!\, \ell^{r+q}}\, d^{(n)}_{r+q}\, F_{m, r}.$

Proof. The proof is a direct consequence of Theorems 3.5 and 3.6. □

Theorem 3.8. Let $\alpha$ be a positive real number. We have

$\begin{equation} D^\alpha\, G^{A, B}_{n, \ell}(x) = x^{-\alpha}\, \sum\limits_{m = 0}^{n} \tilde{h}^{(\alpha)}_{m, n}G^{A, B}_{m, \ell}(x), \, n > \alpha > 0, \end{equation}$

(3.18)

where

$\begin{equation} \tilde{h}^{(\alpha)}_{m, n} = \sum\limits_{r = max(m, \, \lceil\alpha\rceil)}^{n}\frac{r!}{\ell^{r}\, \Gamma(r+1-\alpha)}\, d^{(n)}_{r}\, F_{m, r}. \end{equation}$

(3.19)

Proof. The proof of (3.18) can be obtained by applying the fractional derivative $D^{\alpha}$ to (3.10) and using (2.2), relation (3.13) leads to (3.18) and the proof of theorem is complete. □

3.3. Some useful integral formulas involve the UCPs

In this part, we will develop some useful formulas that will be utilized in the derivation of our proposed algorithms.

Lemma 3.1. For every real positive real number $\beta$ , and positive integers $n$ and $m$ , the following integral formula holds:

$\begin{equation} \int_0^\ell \omega_{\ell}(x) x^{\beta } U_{n, \ell}(x) \, dx = I(n, \beta, \ell), \quad n = 0, 1, 2, \dots, \end{equation}$

(3.20)

with

$\begin{equation} I(n, \beta, \ell) = \frac{\sqrt{\pi }\, (n+1)\, \ell^{\beta+2}\, \Gamma \left(\beta +\frac{3}{2}\right)\, \Gamma (\beta +1)}{2\, \Gamma (n+\beta +3)\, \Gamma (\beta +1-n)}, \end{equation}$

(3.21)

and in general

$\begin{equation} \int_0^\ell \omega_{\ell}(x) x^{\beta } \, U_{n, \ell}(x)U_{m, \ell}(x) \, dx = I(n, m, \beta, \ell), \end{equation}$

(3.22)

where

$\omega_{\ell}(x) = \sqrt{x\, (\ell-x)}, \ I(n, m, \beta, \ell) = \sum\limits_{k = 0}^{min(n, m)}I(|n-m|+2k, \beta, \ell).$

Proof. First, we prove (3.20) by induction. It is easy to see that this formula holds for $n = 0$ . Now, suppose that (3.20) holds for all $n < m$ , and we need to show that it holds at $n = m$ . Making use of the recurrence relation

$U_{n+1, \ell}(x) = 2(2\, x/\ell-1)U_{n, \ell}(x)-U_{n-1, \ell}(x),$

we get

$\begin{equation} \int_0^\ell \omega_{\ell}(x) x^{\beta } U_{n, \ell}(x) \, dx = \int_0^\ell \omega_{\ell}(x) x^{\beta } \left(\frac{4}{\ell}x\, U_{n-1, \ell}(x)-2U_{n-1, \ell}(x)-U_{n-2, \ell}(x)\right) \, dx. \end{equation}$

(3.23)

By using the induction hypothesis, we obtain

$\begin{equation} \int_0^\ell \omega_{\ell}(x) x^{\beta } U_{n, \ell}(x) \, dx = \frac{4}{\ell}\, I(n-1, \beta+1, \ell)-2\, I(n-1, \beta, \ell)-I(n-2, \beta, \ell). \end{equation}$

(3.24)

Substitution of (3.21) into (3.24) and performing some calculations lead to (3.20). Now, to prove (3.22), we make use of the well-known linearization formula

$U_{n}(x)U_{m}(x) = \sum\limits_{k = 0}^{n}U_{m-n+2k}(x), \, m\ge n,$

along with (3.20). □

Corollary 3.2. For every real positive real number $\beta$ , and positive integers $n$ and $m$ , the following integral formula holds:

$\begin{equation} \int_0^\ell \omega_{\ell}(x)\, x^{\beta} \, G^{A, B}_{n, \ell}(x)\, U_{m, \ell}(x) \, dx = J(n, m, \beta, \ell), \quad n, m = 0, 1, 2, \dots, \end{equation}$

(3.25)

where

$\begin{equation} J(n, m, \beta, \ell) = \frac{1}{2} (B-2 A) I(n-2, m, \beta, \ell)+\frac{B}{2}\, I(n, m, \beta, \ell). \end{equation}$

(3.26)

Proof. Formula (3.25) is a direct result of Lemma 3.1 along with the expression in (3.9). □

4. Tau and collocation solutions for TFHE

This section is concerned with analyzing two numerical algorithms for handling TFHE, providing a detailed explanation of the proposed algorithms, namely, the unified shifted Chebyshev tau method (USCTM) and the unified shifted Chebyshev collocation method (USCCM) that will be used for the following TFHE (see ^[24,26,46]):

$\begin{equation} \frac{\partial^{\alpha}y(x, t)}{\partial\, t^{\alpha}} = \frac{\partial^{2}y(x, t)}{\partial\, x^{2}}+z(x, t), \, 0 < \alpha\le 1, \end{equation}$

(4.1)

governed by the nonlocal conditions

$\begin{equation} y(x, 0)-y(x, \ell_{2}) = f(x), \quad 0 < x < \ell_{1}, \end{equation}$

(4.2)

and

$\begin{equation} y(0, t) = y(\ell_{1}, t) = 0, \quad 0 < t\le \ell_{2}. \end{equation}$

(4.3)

In this instance, the known functions are $z(x, t)$ and $f(x)$ , whereas the unknown function is given by $y(x, t)$ . Now, consider the space:

$\Omega = \text{span}\{G^{A, B}_{n, \ell_{1}}(x)\, G^{A, B}_{m, \ell_{2}}(t):n, m = 0, 1, \dots, N\},$

and suppose that $y(x, t)\in \Omega$ may be approximately represented as

$\begin{equation} y_{N}(x, t) = \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}\, G^{A, B}_{n, \ell_{1}}(x)\, G^{A, B}_{m, \ell_{2}}(t). \end{equation}$

(4.4)

To be able to apply both tau and collocation methods, we sholud first get the residual of Eq (4.1). This residual may be obtained using the following formula:

$\begin{equation} R_{N}(x, t) = \frac{\partial^{\alpha}y_{N}(x, t)}{\partial\, t^{\alpha}}-\frac{\partial^{2}y_{N}(x, t)}{\partial\, x^{2}}-z(x, t). \end{equation}$

(4.5)

4.1. Tau approach for treating TFHE

This section provides a detailed explanation of the algorithm (USCTM) that will be used to handle TFHE (4.1) using tau spectral method. The tau method's main goal is to identify $y_{N}(x, t)$ such that

$\begin{equation} (t^{\alpha}\, R_{N}(x, t), U_{i, \ell_{1}}(x)\, U_{j, \ell_{2}}(t))_{\tilde{\omega}(x, t)} = 0, \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \end{equation}$

(4.6)

where $\tilde{\omega}(x, t) = \omega_{\ell_{1}}(x)\, \omega_{\ell_{2}}(t)$ .

Now, to be able to compute the integral in the left-hand side of (4.6), use the explicit representation of the fractional derivatives $D^\alpha\, G^{A, B}_{n, \ell}(x)$ in (3.18) to obtain the following two important integrals for a positive integer $q$ and a positive real number $\alpha$

$\begin{equation} \int_0^\ell \omega_{\ell}(t)\, x^{\alpha}D^{\alpha}G^{A, B}_{m, \ell}(t)\, U_{j, \ell}(t) \, dt = a^{(\alpha)}_{m, j}(\ell), \quad m, j = 0, 1, 2, \dots, \end{equation}$

(4.7)

with

$a^{(\alpha)}_{m, j}(\ell) = \sum\limits_{k = 0}^{m} \tilde{h}^{(\alpha)}_{k, m}\, J(k, j, 0, \ell),$

and

$\begin{equation} \int_0^\ell \omega_{\ell}(x)\, x^{\alpha }\, D^{q}G^{A, B}_{n, \ell}(x)U_{i, \ell}(x) \, dt = b^{(q)}_{n, i}(\alpha, \ell), \quad n, i = 0, 1, 2, \dots, \end{equation}$

(4.8)

with

$b^{(q)}_{n, i}(\alpha, \ell) = \sum\limits_{k = 0}^{n-q} h^{(q)}_{k, n}\, J(k, i, \alpha, \ell),$

where $\tilde{h}^{(\alpha)}_{k, m}$ and $J(k, j, 0, \ell)$ can be computed from (3.19) and (3.26), respectively.

The nonlocal conditions (4.2) and (4.3) give

$\begin{equation} y_{N}(x_{i}, 0)-y_{N}(x_{i}, \ell_{2}) = f(x_{i}), \quad 0\le i \le N, \end{equation}$

(4.9)

$\begin{equation} y_{N}(0, t_{j}) = 0, \quad 0 \le j \le N-1, \end{equation}$

(4.10)

and

$\begin{equation} y_{N}(\ell_{1}, t_{j}) = 0, \quad 0 \le j \le N-1, \end{equation}$

(4.11)

where $x_{i}$ and $t_{j}, \, i, j = 0, \dots, M$ , are the zeros of $G^{A, B}_{n, \ell_{1}}(x)$ and $G^{A, B}_{m, \ell_{2}}(t)$ , respectively. The integrals in (4.7) and (4.8) enable one to write (4.6)–(4.11) as follows:

$\begin{equation} \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}H_{n, m, i, j} = z_{i, j}, \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \end{equation}$

(4.12)

and

$\begin{equation} \left.\begin{aligned} \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}K_{n, m, i} = f_{i}, & \quad 0\le i \le N, \\ \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}L_{n, m, j} = 0, & \quad 0\le j \le N-1, \\ \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}M_{n, m, j} = 0, & \quad 0\le j \le N-1, \end{aligned}\right\} \end{equation}$

(4.13)

where

$\begin{equation} H_{n, m, i, j} = J(n, i, 0, \ell_{1})\, a^{(\alpha)}_{m, j}(\ell_{2})-J(m, j, \alpha, \ell_{2})\, b^{(2)}_{n, i}(0, \ell_{1}), \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \end{equation}$

(4.14)

and

$\begin{equation} \left.\begin{aligned} &K_{n, m, i} = G^{A, B}_{n, \ell_{1}}(x_{i})(G^{A, B}_{m, \ell_{2}}(0)-G^{A, B}_{m, \ell_{2}}(\ell_{2})), \quad \quad \quad 0\le i \le N, \\ &L_{n, m, j} = G^{A, B}_{n, \ell_{1}}(0)\, G^{A, B}_{m, \ell_{2}}(t_{j}), \qquad \qquad \qquad \qquad 0\le j \le N-1, \\ &M_{n, m, j} = G^{A, B}_{n, \ell_{1}}(\ell_{1})\, G^{A, B}_{m, \ell_{2}}(t_{j}), \qquad \qquad \quad \qquad \, \, 0\le j \le N-1, \\ &f_{i} = f(x_{i}). \end{aligned}\right\} \end{equation}$

(4.15)

The system (4.12) and (4.13) consists of $(N+1)^{2}$ equations in $(N+1)^{2}$ unknowns, $c_{0, 0}, \, \dots\, , c_{0, N}, \, \dots, \, c_{N, 0}, \, \dots\, , c_{N, N}$ .

4.2. The collocation approach for treating TFHE

This section provides a detailed explanation of the algorithm (USCCM) that will be used to handle TFHE (4.1) using collocation spectral method as follows:

Suppose that $y(x, t)\in \Omega$ may be approximately represented as (4.4). The Collocation method's main goal is to identify $y_{N}(x, t)$ such that

$\begin{equation} R_{N}(x_{i}, t_{j}) = 0, \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \end{equation}$

(4.16)

which gives

$\begin{equation} \sum\limits_{n = 0}^{N} \sum\limits_{m = 0}^{N}c_{n, m}\hat{H}_{n, m, i, j} = z_{i, j}, \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \end{equation}$

(4.17)

where

$\begin{equation} \left.\begin{aligned} &\hat{H}_{n, m, i, j} = G^{A, B}_{n, \ell_{1}}(x_{i})\, D^{\alpha}G^{A, B}_{m, \ell_{2}}(t_{j})-G^{A, B}_{m, \ell_{2}}(t_{j})\, D^{2}G^{A, B}_{n, \ell_{1}}(x_{i}), \quad 0\le i \le N-2, \quad 0 \le j \le N-1, \\ & \quad \quad z_{i, j} = z(x_{i}, t_{j}), \end{aligned}\right\} \end{equation}$

(4.18)

and $x_{i}, \, t_{i}\, (0\le i \le N)$ are choosen to be either the zeros of $G^{A, B}_{N+1, \ell}(x)$ or $x_{i}, \, t_{i} = \dfrac{i+1}{N+2}\, (0\le i \le N)$ , in addition to Eq (4.15) which provide us $(N+1)^{2}$ equations in the $(N+1)^{2}$ unknowns, $c_{0, 0}, \, \dots\, , c_{0, N}, \, \dots, \, c_{N, 0}, \, \dots\, , c_{N, N}$ .

Remark 4.1. Attempting to demonstrate how to use the two algorithms—USCTM and USCCM—presented. While the stages for solving the TFHE using Algorithm 1 are expressed using the USCTM notation, Algorithm 2 uses the USCCM notation. The Mathematica application, version 13.1, is used to do the necessary computations.

Algorithm 1 USCTM Algorithm

Step 1. Given

$A, \, B, \, \ell_{1}, \, \ell_{2}, \, \alpha$ and

$N$
Step 2. Find

$G^{A, B}_{n, \ell_{1}}(x), \, D_{x}^{(2)}G^{A, B}_{n, \ell_{1}}(x)$ and

$D_{t}^{(\alpha)}G^{A, B}_{m, \ell_{2}}(t)$
Step 3. Evaluate

$R_{N}(x, t)$ defined in (4.5)
Step 4. Evaluate the used collocation points

$x_{i}$ and

$t_{i}, \, i = 0, 1, \dots\, , N$
Step 5. Evaluate

$H_{n, m, i, j}, \, K_{n, m, i}, \, L_{n, m, j}, M_{n, m, j}, \, f_{j}$ as defined in (4.14) and (4.15)
Step 6. List the equations system as defined in (4.12) and (4.13)
Step 7. Join [Output 6]
Step 8. Solve [Output 7]

Algorithm 2 USCCM Algorithm

Step 1. Given

$A, \, B, \, \ell_{1}, \, \ell_{2}, \, \alpha$ and

$N$
Step 2. Find

$G^{A, B}_{n, \ell_{1}}(x), \, D_{x}^{(2)}G^{A, B}_{n, \ell_{1}}(x)$ and

$D_{t}^{(\alpha)}G^{A, B}_{m, \ell_{2}}(t, \ell_{2})$
Step 3. Evaluate

$R_{N}(x, t)$ defined in (4.5)
Step 4. Evaluate the used collocation points

$x_{i}$ and

$t_{i}, \, i = 0, 1, \dots\, , N$
Step 5. Evaluate

$\hat{H}_{n, m, i, j}, \, K_{n, m, i}, \, L_{n, m, j}, M_{n, m, j}, \, f_{j}$ as defined in (4.15) and (4.18)
Step 6. List the equations system as defined in (4.13) and (4.17)
Step 7. Join [Output 6]
Step 8. Solve [Output 7]

5. Investigation of convergence and error analysis

In this part, we extensively examine the suggested expansion's convergence and error analysis (4.4). The following lemmas are necessary to continue with our investigation.

Lemma 5.1. The polynomials $G^{A, B}_{n, \ell}(x)$ satisfy the following inequality

$\begin{equation} |G^{A, B}_{n, \ell}(x)| < \rho_{n}, \quad x\in[0, \ell], \end{equation}$

(5.1)

where

$\begin{equation} \rho_{n} = \begin{cases} A, & \quad B = A, \\ A(n+1), & \quad B = 2A, \\ r_{n}, & \quad otherwise, \end{cases} \end{equation}$

(5.2)

such that $r_{n}$ is defined as

$\begin{equation} r_{n} = \begin{cases} A, & \quad n = 0, \\ B, & \quad n = 1, \\ \rho\, n, & \quad n\ge 2, \end{cases} \end{equation}$

(5.3)

where $\rho = max\{|B-2A|, B\}.$

Proof. According to Remark 3.1 and formula (3.1), it is easy to prove (5.1). □

Lemma 5.2. The coefficients $F_{p, m}$ that appear in the inversion formula (3.13) satisfy the following inequality

$\begin{equation} |F_{p, m}| < \frac{\rho\, c_{p}(2m)! \ell^m}{B 2^{2m-1}(m-p)! (m+p)!}. \end{equation}$

(5.4)

Proof. Formula (3.14) leads to the following inequality:

$\begin{equation} |F_{p, m}|\le \rho\, c_{p}B^{-1}(2 m+1)! 4^{1-m} \ell^m\, S_{p, m}, \end{equation}$

(5.5)

where

$S_{p, m} = \sum\limits_{k = 0}^{\left\lfloor \frac{m-p}{2}\right\rfloor} \frac{(2 k+p+1)}{(m-p-2 k)! (2 k+m+p+2)!}.$

By using computer algebra algorithms, especially Zeilberger's algorithm (see, ^[47]), $S_{p, m}$ are able to meet the first-order recurrence relation shown below:

$\begin{equation} (m+p+1) S_{p+1, m}-(m-p) S_{p, m} = 0, \quad S_{0, m} = \frac{1}{2(2 m+1) (m!)^2}. \end{equation}$

(5.6)

The exact solution to this recurrence relation has the form

$\begin{equation} S_{p, m} = \frac{1}{2(2 m+1) (m-p)! (m+p)!}. \end{equation}$

(5.7)

Inserting (5.7) into (5.5) yields (5.4). The lemma's proof is now complete. □

Theorem 5.1. Let $y(x, t)$ be an infinitely differentiable function at the origin with $\left|\frac{\partial^{i+j}y}{\partial\, x^{i}\, \partial\, t^{j}}(0, 0)\right| < M$ . Then it has the following expansion

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^\infty \sum\limits_{j = 0}^{\infty} Y_{i, j}\, G^{A, B}_{i, \ell_{1}}(x)\, G^{A, B}_{j, \ell_{2}}(t), \end{equation}$

(5.8)

where

$Y_{i, j} = \sum\limits_{p = 0}^\infty \sum\limits_{q = 0}^{\infty} y_{i+p, j+q}\, F_{i, p+i}\, F_{j, q+j}, \quad y_{i, j} = \frac{1}{i!j!}\frac{\partial^{i+j}y}{\partial\, x^{i}\, \partial\, t^{j}}(0, 0).$

These expansion coefficients satisfy the following inequalities

$\begin{equation} \left. \begin{split} &|Y_{0, 0}|\le \frac{M \rho^2}{A^2} e^{\ell_{1}}e^{\ell_{2}}, \\ &|Y_{i, 0}| < \frac{8\, M \rho^2\, e^{\ell_{2}}e^{\ell_{1}}}{A\, B}\, \frac{\ell_{1}^{i}}{i!\, 4^i}, \quad i\ge 1, \\ &|Y_{0, j}| < \frac{8\, M \rho^2\, e^{\ell_{1}}e^{\ell_{2}}}{A\, B}\, \frac{\ell_{2}^{j}}{j!\, 4^j}, \quad j\ge 1, \\ & |Y_{i, j}| < \frac{64\, M \rho^2\, e^{\ell_{1}}e^{\ell_{2}}}{B^{2}}\, \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!j!\, 4^i\, 4^j}, \quad i, j\ge 1. \end{split} \right\} \end{equation}$

(5.9)

The inequalities in (5.9) can be combined to give the following expression

$\begin{equation} |Y_{i, j}|\lesssim \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!j!\, 4^i\, 4^j}, \quad \forall i, j\ge 0, \end{equation}$

(5.10)

where $\lesssim$ means that a generic constant $d$ exists such that $|Y_{i, j}|\le \frac{d\, \ell_{1}^{i}\ell_{2}^{j}}{i!j!\, 4^i\, 4^j}$ . The series in (5.8) converges uniformly to $y(x, t)$ .

Proof. First, we expand $y(x, t)$ as

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^{\infty} \sum\limits_{j = 0}^{\infty} y_{i, j}\, x^i \, t^j. \end{equation}$

(5.11)

This expansion can be written in the form:

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^{\infty} b_{i}(t) x^i, \end{equation}$

(5.12)

where

$\begin{equation} b_{i}(t) = \sum\limits_{j = 0}^{\infty} y_{i, j}\, t^j. \end{equation}$

(5.13)

Inserting (3.13) into (5.12) enables one to write

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^{\infty} b_{i}(t) \sum\limits_{p = 0}^{i} F_{p, i}\, G^{A, B}_{p, \ell_{1}}(x). \end{equation}$

(5.14)

Expanding the right hand side of (5.14), and rearranging the similar terms, the following expansion is obtained

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^{\infty} \left(\sum\limits_{p = 0}^{\infty}b_{p+i}(t)\, F_{i, p+i}\right) G^{A, B}_{i, \ell_{1}}(x). \end{equation}$

(5.15)

Now, we have

$\begin{equation} \sum\limits_{p = 0}^{\infty}b_{p+i}(t)\, F_{i, p+i} = \sum\limits_{p = 0}^{\infty}\left( \sum\limits_{j = 0}^{\infty} y_{p+i, j}\, t^j\right)\, F_{i, p+i} = \sum\limits_{j = 0}^{\infty}\left( \sum\limits_{p = 0}^{\infty} y_{p+i, j}\, F_{i, p+i}\right)\, t^j. \end{equation}$

(5.16)

Inserting (3.13) into (5.16) and following the same procedures enables one to write

$\begin{equation} \sum\limits_{p = 0}^{\infty}b_{p+i}(t)\, F_{i, p+i} = \sum\limits_{j = 0}^{\infty}\left( \sum\limits_{p = 0}^{\infty}\, \sum\limits_{q = 0}^{\infty} y_{p+i, j+q}\, F_{i, p+i}\, F_{j, q+j} \right)\, G^{A, B}_{j, \ell_{2}}(t). \end{equation}$

(5.17)

Substituting (5.17) for (5.15) immediately proves (5.8). The first part of Theorem is now proved.

Now, we need to prove (5.9). Using Lemma 5.2, one can write

$\begin{equation} \begin{split} |Y_{i, j}|&\le M \sum\limits_{p = 0}^\infty \sum\limits_{q = 0}^{\infty} \frac{1}{(i+p)!(j+q)!}|F_{i, p+i}||F_{j, q+j}|, \\ &\le \frac{M \rho^2\, c_{i}c_{j}\ell_{1}^{i}\ell_{2}^{j}}{B^{2}\, 2^{2(i+j-1)}}\, \sum\limits_{p = 0}^\infty \sum\limits_{q = 0}^{\infty} \frac{1}{(i+p)!(j+q)!}\frac{(2(p+i))!(2(q+j))! \ell_{1}^{p}\ell_{2}^{q}}{2^{2(p+q)}p!q! (p+2i)!(q+2j)!}, \quad i, j\ge 0. \end{split} \end{equation}$

(5.18)

Using $(2n)! = 2^{2n}n!(1/2)_{n}$ , we obtain

$\begin{equation} |Y_{i, j}|\le \frac{4\, M \rho^2\, c_{i}c_{j}\ell_{1}^{i}\ell_{2}^{j}}{B^{2}}\, \sum\limits_{p = 0}^\infty \sum\limits_{q = 0}^{\infty} \frac{(1/2)_{p+i}(1/2)_{q+j} \ell_{1}^{p}\ell_{2}^{q}}{p!q! (p+2i)!(q+2j)!}, \quad i, j\ge 0. \end{equation}$

(5.19)

Using $\frac{(1/2)_{p}}{p!}\le 1, p \ge 0$ , it is easy to see that

$\begin{equation} |Y_{0, 0}|\le \frac{M \rho^2}{A^2} e^{\ell_{1}}e^{\ell_{2}}, \end{equation}$

(5.20)

$\begin{equation} |Y_{i, 0}|\le \frac{2\, M \rho^2\, \ell_{1}^{i}\, e^{\ell_{2}}}{A\, B}\, \sum\limits_{p = 0}^\infty \frac{(1/2)_{p+i}\, \ell_{1}^{p}}{p! (p+2i)!}, \quad i\ge 1, \end{equation}$

(5.21)

and

$\begin{equation} |Y_{0, j}|\le \frac{2\, M \rho^2\, \ell_{2}^{j}\, e^{\ell_{1}}}{A\, B}\, \sum\limits_{q = 0}^{\infty} \frac{(1/2)_{q+j}\, \ell_{2}^{q}}{q! (q+2j)!}, \quad j\ge 1. \end{equation}$

(5.22)

Using $\frac{\ell^{p}}{p!} < e^{\ell}, \, p\ge 0$ , the three relations (5.19), (5.21) and (5.22) take respectively the forms

$\begin{equation} |Y_{i, j}| < \frac{4\, M \rho^2\, \ell_{1}^{i}\ell_{2}^{j}e^{\ell_{1}}e^{\ell_{2}}}{B^{2}}\, \sum\limits_{p = 0}^\infty \sum\limits_{q = 0}^{\infty} \frac{(1/2)_{p+i}(1/2)_{q+j}}{ (p+2i)!(q+2j)!}, \quad i, j\ge 1, \end{equation}$

(5.23)

$\begin{equation} |Y_{i, 0}| < \frac{2\, M \rho^2\, \ell_{1}^{i}e^{\ell_{2}}e^{\ell_{1}}}{A\, B}\, \sum\limits_{p = 0}^\infty \frac{(1/2)_{p+i}}{(p+2i)!}, \quad i\ge 1, \end{equation}$

(5.24)

and

$\begin{equation} |Y_{0, j}| < \frac{2\, M \rho^2\, \ell_{2}^{j}e^{\ell_{1}}e^{\ell_{2}}}{A\, B}\, \sum\limits_{q = 0}^{\infty} \frac{(1/2)_{q+j}}{(q+2j)!}, \quad j\ge 1. \end{equation}$

(5.25)

Then by using $(a)_{n+k} = (a)_{k}(a+k)_{n}$ and Chu-V Gauss formula ^[48], one can get

$\begin{equation} \sum\limits_{p = 0}^{\infty} \frac{(1/2)_{p+i}}{ (p+2i)!} = \frac{(1/2)_{i}}{(1)_{2i}} \sum\limits_{p = 0}^{\infty} \frac{(1/2+i)_{p}(1)_{p}}{p!(1+2i)_{p}} = \frac{(1/2)_{i}}{(1)_{2i}}\, \,_{2}F_{1}\left( \begin{array}{c} 1/2+i, \, 1 \\ 1+2i \end{array} \Big|1\right) = \frac{\Gamma \left(\frac{1}{2} (2 i-1)\right)}{\sqrt{\pi}\, \Gamma (2i)}. \end{equation}$

(5.26)

Again, using $(2n)! = 2^{2n}n!(1/2)_{n}$ , leads to

$\begin{equation} \sum\limits_{p = 0}^{\infty} \frac{(1/2)_{p+i}}{ (p+2i)!} = \frac{2i\, \Gamma \left(\frac{1}{2} (2 i-1)\right)}{\sqrt{\pi}\, (2i)!} = \frac{1}{(i-1)!(2i-1)4^{i-1}}\le \frac{1}{i!4^{i-1}}, \quad i\ge 1. \end{equation}$

(5.27)

Hence the three relations (5.23)–(5.25) take respectively the forms

$\begin{equation} |Y_{i, j}| < \frac{64\, M \rho^2\, \ell_{1}^{i}\ell_{2}^{j}e^{\ell_{1}}e^{\ell_{2}}}{B^{2}}\, \frac{1}{4^{i}\, i!}\frac{1}{4^{j}\, j!}, \quad i, j\ge 1, \end{equation}$

(5.28)

$\begin{equation} |Y_{i, 0}| < \frac{8\, M \rho^2\, \ell_{1}^{i}e^{\ell_{2}}e^{\ell_{1}}}{A\, B}\, \frac{1}{4^{i}\, i!}, \quad i\ge 1, \end{equation}$

(5.29)

and

$\begin{equation} |Y_{0, j}| < \frac{8\, M \rho^2\, \ell_{2}^{j}e^{\ell_{1}}e^{\ell_{2}}}{A\, B}\, \frac{1}{4^{j}\, j!}, \quad j\ge 1. \end{equation}$

(5.30)

At this point, the second part of the theorem is now proved.

Now, in view of Lemma 5.1, we can see that

$\begin{equation} \sum\limits_{i = 0}^{\infty} \sum\limits_{j = 0}^{\infty} |Y_{i, j}\, G^{A, B}_{i, \ell_{1}}(x)\, G^{A, B}_{j, \ell_{2}}(t)|\le \, d\, \sum\limits_{i = 0}^{\infty} \sum\limits_{j = 0}^{\infty} \frac{\ell_{1}^{i}}{4^{i}\, i!}\frac{\ell_{2}^{j}}{4^{j}\, j!}\, \rho_{i}\, \rho_{j}\le C\, e^{\ell_{1}/4}e^{\ell_{2}/4}, \end{equation}$

(5.31)

where $C$ is a constant depending on the two constants A and B. This shows that the series in (5.8) converges uniformly to $y(x, t)$ . The theorem's proof is complete. □

Theorem 5.2. If $y(x, t)$ satisfies the hypothesis of Theorem 5.1, and if

$\begin{equation} y(x, t) = \sum\limits_{i = 0}^{N} \sum\limits_{j = 0}^{N} Y_{i, j}\, G^{A, B}_{i, \ell_{1}}(x)\, G^{A, B}_{j, \ell_{2}}(t), \end{equation}$

(5.32)

then the following error estimate is satisfied:

$\begin{equation} |y-y_{N}|\lesssim \frac{1}{4^{N+1}}. \end{equation}$

(5.33)

Proof. The truncation error may be written as:

$\begin{equation} \begin{split} \left|y-y_{N}\right| = &\left| \sum\limits_{i = 0}^{\infty} \sum\limits_{j = 0}^{\infty} Y_{i, j}\, G^{A, B}_{i, \ell_{1}}(x)\, G^{A, B}_{j, \ell_{2}}(t)- \sum\limits_{i = 0}^{N} \sum\limits_{j = 0}^{N} Y_{i, j}\, G^{A, B}_{i, \ell_{1}}(x)\, G^{A, B}_{j, \ell_{2}}(t)\right|\\ &\le \sum\limits_{i = 0}^{N} \sum\limits_{j = N+1}^{\infty} |Y_{i, j}|\, |G^{A, B}_{i, \ell_{1}}(x)|\, |G^{A, B}_{j, \ell_{2}}(t)|+ \sum\limits_{i = N+1}^{\infty} \sum\limits_{j = 0}^{\infty} |Y_{i, j}|\, |G^{A, B}_{i, \ell_{1}}(x)|\, |G^{A, B}_{j, \ell_{2}}(t)|\\ &\le \, d\, \sum\limits_{i = 0}^{N} \sum\limits_{j = N+1}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{\, i!j!\, 4^{j}\, 4^{i}}\, \rho_{i}\, \rho_{j}+d\, \sum\limits_{i = N+1}^{\infty} \sum\limits_{j = 0}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{i}\, 4^{j}}\, \rho_{i}\, \rho_{j}, \\ &\le \, d\, \frac{1}{4^{N+1}} \sum\limits_{i = 0}^{N} \sum\limits_{j = N+1}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{i}}\, \rho_{i}\, \rho_{j}+d\, \frac{1}{4^{N+1}} \sum\limits_{i = N+1}^{\infty} \sum\limits_{j = 0}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{j}}\, \rho_{i}\, \rho_{j}, \end{split} \end{equation}$

(5.34)

we have $\rho_{i}$ has the forms $\lambda$ , $\lambda\, i$ or $\lambda (i+1)$ , where $\lambda$ is a constant. So, we have three cases:

Case 1: $\rho_{i} = \lambda$

$\begin{equation} \begin{split} |y-y_{N}|&\le \, d\, \frac{\lambda}{4^{N+1}} \sum\limits_{i = 0}^{N} \sum\limits_{j = N+1}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{i}}+d\, \frac{\lambda}{4^{N+1}} \sum\limits_{i = N+1}^{\infty} \sum\limits_{j = 0}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{j}}\\ &\le d\, \frac{\lambda}{4^{N+1}}\, e^{\ell_{1}/4}e^{\ell_{2}}+d\, \frac{\lambda}{4^{N+1}}\, e^{\ell_{1}}e^{\ell_{2}/4}\lesssim \frac{1}{4^{N+1}}. \end{split} \end{equation}$

(5.35)

Case 2: $\rho_{i} = \lambda\, i$

$\begin{equation} \begin{split} |y-y_{N}|&\le \, d\, \frac{\lambda \, \ell_{1}}{4^{N+2}} \sum\limits_{i = 0}^{N-1} \sum\limits_{j = N+1}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{i}}+d\, \frac{\lambda\, \ell_{2}}{4^{N+2}} \sum\limits_{i = N+1}^{\infty} \sum\limits_{j = 0}^{\infty} \frac{\ell_{1}^{i}\ell_{2}^{j}}{i!\, j!\, 4^{j}}\\ &\le d\, \frac{\lambda\, \ell_{1}}{4^{N+2}}\, e^{\ell_{1}/4}e^{\ell_{2}}+d\, \frac{\lambda\, \ell_{2}}{4^{N+2}}\, e^{\ell_{1}}e^{\ell_{2}/4}\lesssim \frac{1}{4^{N+1}}. \end{split} \end{equation}$

(5.36)

Case 3: $\rho_{i} = \lambda\, (i+1)$ By the same way, it can be proven that

$\begin{equation} |y-y_{N}|\lesssim \frac{1}{4^{N+1}}. \end{equation}$

(5.37)

□

Error stability is further emphasized in the following theorem through an estimation of error propagation.

Theorem 5.3. For any two successive approximations of $y(x, t)$ , we get:

$\begin{equation} |y_{N+1}-y_{N}|\lesssim \frac{1}{4^{N+1}}. \end{equation}$

(5.38)

Proof. In view of (5.33), it is not difficult to obtain (5.38). □

6. Numerical examples

To demonstrate the effectiveness, high accuracy and application of the two suggested algorithms, this part focuses on the presentation of some numerical results followed by comparisons with certain numerical findings from the literature. The error is measured using the maximum absolute error (MAE) in the tests that follow, namely:

$\begin{equation} E_{N} = \max\limits_{(x, t)\in I}E_{N}(x, t), \ I = [0, 1]\times [0, 1], \end{equation}$

(6.1)

where

$\begin{equation} E_{N}(x, t) = |y(x, t)-y_N(x, t)|, \ (x, t)\in I. \end{equation}$

(6.2)

Example 6.1. Consider the following fractional initial value problem:

$\begin{equation} \begin{cases} \dfrac{\partial^{0.5}y(x, t)}{\partial\, t^{0.5}} = \dfrac{\partial^{2}y(x, t)}{\partial\, x^{2}}+2 t-\dfrac{2 \sqrt{t} (x-1) x}{\sqrt{\pi }}, & \quad 0 < x, t < 1, \\ y(x, 0)-y(x, 1) = -x\, (1-x), & \quad 0 < x < 1, \\ y(0, t) = y(1, t) = 0, & \quad 0 < t\le 1. \end{cases} \end{equation}$

(6.3)

If USCTM or USCCM are applied with $N = 2$ , then the following nine coefficients are obtained:

$\begin{equation} c_{0, 0} = c_{0, 1} = -\frac{A-2 B}{16 A^2 B}, \quad c_{2, 0} = -\frac{1}{16 A B}, \quad c_{2, 1} = -\frac{1}{16 B^2}, \quad c_{0, 2} = c_{1, 0} = c_{1, 1} = c_{1, 2} = c_{2, 2} = 0, \\ \end{equation}$

(6.4)

and consequently $y_2(x, t) = x(1-x)\, t,$ which is the exact solution.

If USCTM and USCCM are applied to the following three TFHEs (6.5)–(6.7) using some different values of $N$ , then the obtained numerical results are presented in Tables 1–12 and they affirm that compared to other approaches, the suggested methods provide more accurate findings. Additionally, – show that the exact and approximate solutions to the given problems agree very well. They also show how error depends on $N$ and how the solutions to Examples 6.2–6.5 converge when USCTM and USCCM are used. Furthermore, the stability of solutions is demonstrated.

Table 1. Maximum absolute error

$E_{N}$ for Example 6.2 (

$\alpha = 0.5$ ).

$A$	$B$	Method	$N=4$	$N=8$	$N=12$	$N=16$	$N=19$	$N=20$
$1$	$1$	USCTM	1.4 $.\, 10^{-2}$	1.5 $.\, 10^{-5}$	3.1 $.\, 10^{-9}$	2.7 $.\, 10^{-13}$	2.3 $.\, 10^{-15}$	1.1 $.\, 10^{-15}$
		USCCM	2.8 $.\, 10^{-2}$	1.3 $.\, 10^{-4}$	1.2 $.\, 10^{-7}$	1.3 $.\, 10^{-10}$	2.9 $.\, 10^{-13}$	6.9 $.\, 10^{-13}$
$0.9$	$0.9$	USCTM	1.5 $.\, 10^{-2}$	1.7 $.\, 10^{-5}$	3.5 $.\, 10^{-9}$	2.4 $.\, 10^{-12}$	4.1 $.\, 10^{-14}$	1.2 $.\, 10^{-14}$
		USCCM	2.3 $.\, 10^{-2}$	1.4 $.\, 10^{-4}$	1.3 $.\, 10^{-7}$	2.3 $.\, 10^{-10}$	6.9 $.\, 10^{-13}$	5.1 $.\, 10^{-14}$
$0.9$	$1.1$	USCTM	2.4 $.\, 10^{-2}$	3.3 $.\, 10^{-5}$	8.0 $.\, 10^{-9}$	6.0 $.\, 10^{-13}$	4.0 $.\, 10^{-15}$	2.9 $.\, 10^{-15}$
		USCCM	2.3 $.\, 10^{-2}$	1.5 $.\, 10^{-4}$	1.4 $.\, 10^{-7}$	4.6 $.\, 10^{-10}$	4.9 $.\, 10^{-13}$	2.1 $.\, 10^{-13}$
$0.6$	$0.8$	USCTM	1.8 $.\, 10^{-2}$	3.3 $.\, 10^{-5}$	5.1 $.\, 10^{-9}$	8.0 $.\, 10^{-13}$	4.1 $.\, 10^{-15}$	3.8 $.\, 10^{-15}$
		USCCM	2.4 $.\, 10^{-2}$	1.1 $.\, 10^{-4}$	1.5 $.\, 10^{-7}$	2.3 $.\, 10^{-10}$	6.9 $.\, 10^{-13}$	1.5 $.\, 10^{-14}$

| Show Table

DownLoad: CSV

Table 2. Comparison between different methods of Example 6.2 (

$A = 0.6, \, B = 0.8, \, \alpha = 0.5$ ).

USCTM		USCCM		^[46]		^[24] $(N=M)$		^[26]		^[25]
$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE
4	1.8 $.\, 10^{-2}$	4	2.4 $.\, 10^{-2}$	6	2.5 $.\, 10^{-2}$	4	2.3 $.\, 10^{-1}$	4	3.6 $.\, 10^{-1}$	4	3.0 $.\, 10^{-2}$
8	3.3 $.\, 10^{-5}$	8	1.1 $.\, 10^{-4}$	8	8.8 $.\, 10^{-4}$	8	5.4 $.\, 10^{-2}$	6	5.9 $.\, 10^{-1}$	6	2.9 $.\, 10^{-3}$
12	5.1 $.\, 10^{-9}$	12	1.5 $.\, 10^{-7}$	10	2.1 $.\, 10^{-5}$	16	1.4 $.\, 10^{-2}$	8	5.0 $.\, 10^{-3}$	8	1.6 $.\, 10^{-4}$
16	8.0 $.\, 10^{-13}$	16	2.3 $.\, 10^{-10}$	12	1.8 $.\, 10^{-6}$	32	3.8 $.\, 10^{-3}$	10	2.7 $.\, 10^{-4}$	10	6.4 $.\, 10^{-6}$
19	4.1 $.\, 10^{-14}$	19	6.9 $.\, 10^{-13}$	16	1.0 $.\, 10^{-7}$	64	1.2 $.\, 10^{-3}$	12	9.6 $.\, 10^{-6}$	12	1.7 $.\, 10^{-7}$
20	3.8 $.\, 10^{-15}$	20	1.5 $.\, 10^{-14}$	18	4.4 $.\, 10^{-7}$	256	1.4 $.\, 10^{-4}$			14	3.6 $.\, 10^{-9}$

| Show Table

DownLoad: CSV

Table 3. Maximum absolute error

$E_{N}$ for Example 6.3 (

$\beta = 2, \, A = 1, \, B = 1$ ).

$\alpha=0.1$				$\alpha=0.5$				$\alpha=0.95$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	5.8. $10^{-04}$	4	5.1. $10^{-04}$	4	1.5. $10^{-04}$	5	5.1. $10^{-04}$	4	1.1. $10^{-04}$	5	2.0. $10^{-04}$
$8$	1.1. $10^{-09}$	8	1.5. $10^{-08}$	8	3.1. $10^{-10}$	8	1.2. $10^{-08}$	8	2.2. $10^{-10}$	10	1.5. $10^{-11}$
$11$	2.2. $10^{-14}$	12	1.1. $10^{-14}$	11	6.2. $10^{-15}$	13	1.1. $10^{-15}$	11	3.1. $10^{-15}$	13	1.2. $10^{-15}$
$15$	6.1. $10^{-15}$	13	8.1. $10^{-16}$	15	5.2. $10^{-16}$	14	7.2. $10^{-16}$	12	1.9. $10^{-16}$	15	8.1. $10^{-16}$

| Show Table

DownLoad: CSV

Table 4. Maximum absolute error

$E_{N}$ for Example 6.3 (

$\beta = 2, \, A = 1, \, B = 2$ ).

$\alpha=0.1$				$\alpha=0.5$				$\alpha=0.95$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	1.0. $10^{-3}$	4	6.1. $10^{-04}$	4	2.0. $10^{-04}$	4	6.0. $10^{-04}$	4	1.5. $10^{-04}$	4	6.0. $10^{-03}$
$8$	3.0. $10^{-9}$	8	1.6. $10^{-08}$	8	8.0. $10^{-10}$	8	1.5. $10^{-08}$	8	4.0. $10^{-10}$	8	1.5. $10^{-08}$
$11$	1.5. $10^{-14}$	12	2.2. $10^{-14}$	11	1.5. $10^{-14}$	12	2.0. $10^{-14}$	11	1.5. $10^{-14}$	11	1.4. $10^{-12}$
$13$	1.0. $10^{-14}$	13	1.3. $10^{-15}$	12	2.0. $10^{-15}$	13	2.0. $10^{-15}$	12	8.0. $10^{-16}$	12	2.0. $10^{-14}$
$15$	6.0. $10^{-15}$	14	1.4. $10^{-15}$	15	6.0. $10^{-16}$	14	1.5. $10^{-15}$	15	8.3. $10^{-16}$	13	1.0. $10^{-15}$

| Show Table

DownLoad: CSV

Table 5. Maximum absolute error

$E_{N}$ for Example 6.3 (

$\beta = 2, \, A = 0.5, \, B = 0.6$ ).

$\alpha=0.1$				$\alpha=0.45$				$\alpha=0.9$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	1.3. $10^{-4}$	4	2.3. $10^{-04}$	4	2.1. $10^{-04}$	4	5.1. $10^{-04}$	4	1.6. $10^{-04}$	5	1.8. $10^{-04}$
$8$	3.2. $10^{-9}$	9	1.2. $10^{-09}$	8	8.1. $10^{-10}$	9	1.0. $10^{-09}$	8	4.0. $10^{-09}$	9	1.1. $10^{-09}$
$12$	3.1. $10^{-14}$	12	1.5. $10^{-14}$	12	2.2. $10^{-14}$	12	1.4. $10^{-14}$	12	2.4. $10^{-14}$	12	1.5. $10^{-14}$
$13$	4.0. $10^{-15}$	13	5.5. $10^{-16}$	13	7.1. $10^{-16}$	13	1.1. $10^{-16}$	13	3.1. $10^{-16}$	13	1.4. $10^{-16}$
$15$	3.1. $10^{-16}$	14	1.2. $10^{-16}$	14	5.8. $10^{-16}$	14	6.3. $10^{-16}$	14	2.3. $10^{-16}$	14	4.1. $10^{-16}$

| Show Table

DownLoad: CSV

Table 6. Maximum absolute error

$E_{N}$ for Example 6.3 (

$\alpha = 0.5, \, A = 0.9, \, B = 1.9$ ).

$\beta=0.1$				$\beta=0.5$				$\beta=0.95$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	1.2. $10^{-2}$	4	4.3. $10^{-03}$	4	1.1. $10^{-04}$	4	2.0. $10^{-04}$	4	1.4. $10^{-04}$	4	1.8. $10^{-04}$
$8$	2.2. $10^{-8}$	9	2.5. $10^{-08}$	8	2.0. $10^{-10}$	9	1.7. $10^{-09}$	6	2.2. $10^{-07}$	6	1.2. $10^{-08}$
$12$	7.2. $10^{-13}$	12	2.2. $10^{-13}$	12	3.0. $10^{-14}$	12	1.5. $10^{-15}$	8	5.0. $10^{-10}$	8	3.3. $10^{-10}$
$13$	5.0. $10^{-14}$	13	4.4. $10^{-14}$	13	5.0. $10^{-15}$	13	2.2. $10^{-15}$	12	2.0. $10^{-15}$	12	1.2. $10^{-15}$
$14$	3.1. $10^{-14}$	14	3.2. $10^{-15}$	15	4.0. $10^{-16}$	14	2.4. $10^{-16}$	14	1.7. $10^{-16}$	14	2.5. $10^{-16}$

| Show Table

DownLoad: CSV

Table 7. Comparison between different methods of Example 6.3 (

$\beta = 2, \, A = 0.7, \, B = 1.5, \, \alpha = 0.95$ ).

USCTM		USCCM		^[26]		^[25]
$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE
6	1.3 $.\, 10^{-07}$	7	7.4 $.\, 10^{-07}$	4	1.7 $.\, 10^{-4}$	4	8.5 $.\, 10^{-7}$
8	2.0 $.\, 10^{-10}$	10	2.1 $.\, 10^{-11}$	6	9.9 $.\, 10^{-6}$	6	1.1 $.\, 10^{-9}$
10	1.1 $.\, 10^{-13}$	12	2.0 $.\, 10^{-14}$	8	7.7 $.\, 10^{-6}$	8	1.8 $.\, 10^{-12}$
11	3.0 $.\, 10^{-15}$	13	3.1 $.\, 10^{-15}$	10	6.0 $.\, 10^{-6}$	10	1.4 $.\, 10^{-13}$
12	1.7 $.\, 10^{-16}$	14	2.9 $.\, 10^{-15}$	12	4.4 $.\, 10^{-6}$	12	2.3 $.\, 10^{-12}$

| Show Table

DownLoad: CSV

Table 8. Maximum absolute error

$E_{N}$ for Example 6.4 (

$A = 0.8, \, B = 1.2$ ).

$\alpha=0.1$				$\alpha=0.5$				$\alpha=0.9$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	4.0. $10^{-03}$	4	2.1. $10^{-3}$	4	1.0. $10^{-03}$	4	6.0. $10^{-3}$	4	8.0. $10^{-04}$	4	1.3. $10^{-3}$
$8$	1.0. $10^{-05}$	9	1.0. $10^{-5}$	8	3.0. $10^{-06}$	9	2.6. $10^{-5}$	8	2.0. $10^{-06}$	9	2.0. $10^{-5}$
$11$	4.0. $10^{-08}$	12	3.1. $10^{-7}$	14	4.0. $10^{-10}$	12	5.1. $10^{-7}$	12	6.0. $10^{-09}$	12	4.1. $10^{-7}$
$15$	1.5. $10^{-10}$	15	4.3. $10^{-9}$	18	4.2. $10^{-12}$	16	4.3. $10^{-9}$	14	3.0. $10^{-10}$	16	5.3. $10^{-9}$
$18$	6.0. $10^{-13}$	20	1.3. $10^{-11}$	20	2.1. $10^{-14}$	20	2.4. $10^{-11}$	18	3.0. $10^{-13}$	20	1.1. $10^{-11}$
$22$	6.0. $10^{-15}$	22	4.3. $10^{-13}$	22	1.5. $10^{-15}$	22	4.3. $10^{-13}$	22	3.0. $10^{-15}$	22	5.3. $10^{-13}$

| Show Table

DownLoad: CSV

Table 9. Maximum absolute error

$E_{N}$ for Example 6.4 (

$A = 1.6, \, B = 2.2$ ).

$\alpha=0.1$				$\alpha=0.5$				$\alpha=0.9$
$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM	$N$	USCTM	$N$	USCCM
$4$	4.0. $10^{-03}$	4	3.0. $10^{-3}$	4	1.0. $10^{-03}$	4	1.3. $10^{-3}$	4	8.0. $10^{-04}$	4	1.5. $10^{-3}$
$8$	7.0. $10^{-06}$	9	1.1. $10^{-6}$	8	2.0. $10^{-06}$	9	2.5. $10^{-5}$	8	2.0. $10^{-06}$	9	4.0. $10^{-5}$
$11$	2.0. $10^{-08}$	12	3.1. $10^{-7}$	12	6.0. $10^{-09}$	12	5.4. $10^{-7}$	12	4.0. $10^{-09}$	12	6.0. $10^{-7}$
$15$	1.0. $10^{-10}$	16	2.5. $10^{-9}$	16	2.0. $10^{-11}$	16	4.3. $10^{-9}$	14	3.0. $10^{-10}$	16	4.0. $10^{-9}$
$19$	4.0. $10^{-13}$	20	1.5. $10^{-11}$	19	4.0. $10^{-13}$	20	4.3. $10^{-11}$	18	6.0. $10^{-13}$	19	5.3. $10^{-11}$
$21$	2.0. $10^{-14}$	24	1.5. $10^{-13}$	20	4.0. $10^{-14}$	24	2.3. $10^{-13}$	20	3.0. $10^{-14}$	24	3.5. $10^{-13}$

| Show Table

DownLoad: CSV

Table 10. Comparison between different methods of Example 6.4 (

$A = 0.4, \, B = 0.6, \, \alpha = 0.95$ ).

USCTM		USCCM		^[46]		^[24](M=16)		^[26]		^[25]
$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE	$N$	MAE
4	8.0 $.\, 10^{-4}$	5	6.0 $.\, 10^{-03}$	4	1.1 $.\, 10^{-3}$	4	7.5 $.\, 10^{-3}$	4	1.3 $.\, 10^{-03}$	4	6.8 $.\, 10^{-04}$
8	2.0 $.\, 10^{-6}$	9	5.1 $.\, 10^{-05}$	8	5.9 $.\, 10^{-5}$	8	2.3 $.\, 10^{-3}$	6	1.1 $.\, 10^{-04}$	6	1.8 $.\, 10^{-05}$
12	6.0 $.\, 10^{-9}$	16	1.2 $.\, 10^{-09}$	12	5.1 $.\, 10^{-7}$	16	1.4 $.\, 10^{-3}$	8	1.0 $.\, 10^{-05}$	8	1.3 $.\, 10^{-06}$
16	1.5 $.\, 10^{-11}$	19	1.5 $.\, 10^{-10}$	14	2.8 $.\, 10^{-7}$	32	9.6 $.\, 10^{-4}$	10	1.3 $.\, 10^{-05}$	10	1.2 $.\, 10^{-07}$
20	4.0 $.\, 10^{-14}$	20	2.0 $.\, 10^{-11}$	16	1.7 $.\, 10^{-7}$	64	7.3 $.\, 10^{-4}$	12	1.1 $.\, 10^{-05}$	12	1.2 $.\, 10^{-08}$
22	1.7 $.\, 10^{-15}$	22	1.5 $.\, 10^{-13}$	18	1.7 $.\, 10^{-7}$	256	5.6 $.\, 10^{-4}$			16	4.1 $.\, 10^{-09}$

| Show Table

DownLoad: CSV

Table 11. Comparison between different methods of Example 6.4 (

$A = 1.4, \, B = 1.6, \, \alpha = 0.45$ ).

USCTM		USCCM		^[46]		^[24]		^[26]
$N$	MAE	$N$	MAE	$N$	MAE	$M(N=16)$	MAE	$N$	MAE
4	1.0 $.\, 10^{-3}$	4	8.0 $.\, 10^{-03}$	4	1.1 $.\, 10^{-3}$	4	9.5 $.\, 10^{-3}$	4	1.4 $.\, 10^{-03}$
8	2.0 $.\, 10^{-6}$	7	7.1 $.\, 10^{-04}$	8	6.3 $.\, 10^{-6}$	8	3.7 $.\, 10^{-3}$	6	1.2 $.\, 10^{-04}$
12	4.0 $.\, 10^{-9}$	10	5.2 $.\, 10^{-06}$	12	9.0 $.\, 10^{-7}$	16	2.3 $.\, 10^{-3}$	8	1.1 $.\, 10^{-05}$
15	8.0 $.\, 10^{-11}$	12	6.1 $.\, 10^{-07}$	14	5.1 $.\, 10^{-7}$	32	1.9 $.\, 10^{-3}$	10	4.1 $.\, 10^{-06}$
18	6.1 $.\, 10^{-13}$	16	3.2 $.\, 10^{-09}$	16	3.1 $.\, 10^{-7}$	64	1.8 $.\, 10^{-3}$	12	3.3 $.\, 10^{-06}$
20	2.9 $.\, 10^{-14}$	20	4.0 $.\, 10^{-12}$	18	1.9 $.\, 10^{-7}$	256	1.8 $.\, 10^{-3}$
23	1.9 $.\, 10^{-15}$	24	3.1 $.\, 10^{-13}$

| Show Table

DownLoad: CSV

Table 12. Maximum absolute error

$E_{N}$ for Example 6.5.

$A$	$B$	Method	$N=2$	$N=4$	$N=6$	$N=10$	$N=15$	$N=20$
$1$	$1$	USCTM	2.2 $.\, 10^{-2}$	1.8 $.\, 10^{-3}$	3.4 $.\, 10^{-4}$	3.2 $.\, 10^{-5}$	4.4 $.\, 10^{-6}$	4.0 $.\, 10^{-8}$
		USCCM	3.1 $.\, 10^{-2}$	2.3 $.\, 10^{-3}$	4.2 $.\, 10^{-4}$	1.5 $.\, 10^{-5}$	2.7 $.\, 10^{-6}$	4.9 $.\, 10^{-7}$
$1$	$2$	USCTM	1.5 $.\, 10^{-2}$	1.7 $.\, 10^{-3}$	3.5 $.\, 10^{-4}$	2.4 $.\, 10^{-5}$	4.1 $.\, 10^{-6}$	1.2 $.\, 10^{-8}$
		USCCM	3.3 $.\, 10^{-2}$	2.1 $.\, 10^{-3}$	3.3 $.\, 10^{-4}$	2.8 $.\, 10^{-5}$	4.2 $.\, 10^{-6}$	5.5 $.\, 10^{-8}$
$0.8$	$1.8$	USCTM	1.4 $.\, 10^{-2}$	2.1 $.\, 10^{-3}$	5.0 $.\, 10^{-4}$	2.2 $.\, 10^{-5}$	3.1 $.\, 10^{-7}$	2.9 $.\, 10^{-8}$
		USCCM	2.2 $.\, 10^{-2}$	2.5 $.\, 10^{-3}$	1.7 $.\, 10^{-4}$	1.5 $.\, 10^{-5}$	1.3 $.\, 10^{-7}$	1.9 $.\, 10^{-8}$
$0.5$	$0.6$	USCTM	2.2 $.\, 10^{-2}$	3.5 $.\, 10^{-3}$	5.5 $.\, 10^{-4}$	4.2 $.\, 10^{-5}$	2.2 $.\, 10^{-6}$	3.7 $.\, 10^{-8}$
		USCCM	1.2 $.\, 10^{-2}$	1.3 $.\, 10^{-3}$	1.7 $.\, 10^{-4}$	2.0 $.\, 10^{-5}$	5.8 $.\, 10^{-7}$	2.5 $.\, 10^{-7}$
$1.5$	$1.5$	USCTM	3.2 $.\, 10^{-2}$	4.6 $.\, 10^{-3}$	2.5 $.\, 10^{-4}$	1.2 $.\, 10^{-5}$	2.7 $.\, 10^{-7}$	4.5 $.\, 10^{-8}$
		USCCM	3.2 $.\, 10^{-2}$	1.1 $.\, 10^{-3}$	1.4 $.\, 10^{-4}$	1.3 $.\, 10^{-5}$	5.1 $.\, 10^{-7}$	2.6 $.\, 10^{-7}$

| Show Table

DownLoad: CSV

Figure 1. Figures of exact and approximate solutions for Example 6.2.

DownLoad: Full-Size Img PowerPoint

Figure 2. Obtained Errors for Example 6.2 at

$\alpha = 0.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Errors results using USCTM and USCCM (

$A = 0.6$ and

$B = 0.8$ ) for Example 6.2.

DownLoad: Full-Size Img PowerPoint

Figure 4. Figures of exact and approximate solutions for Example 6.3 at

$\alpha = 0.95$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Obtained errors for Example 6.3 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Figure 6. The behavior of exact solution and approximate solution for Example 6.3 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Figure 7. Obtained errors for Example 6.4 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Figure 8. Obtained errors for Example 6.4 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Figure 9. Obtained errors for Example 6.5 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Figure 10. Obtained errors for Example 6.5 using USCTM and USCCM.

DownLoad: Full-Size Img PowerPoint

Example 6.2. Consider the following TFHE:

$\begin{equation} \begin{cases} \dfrac{\partial^{\alpha}y(x, t)}{\partial\, t^{\alpha}} = \dfrac{\partial^{2}y(x, t)}{\partial\, x^{2}}+2\, t^2\left(\dfrac{t^{-\alpha}}{\Gamma (3-\alpha)}+2 \pi ^2\right)\sin (2 \pi x), & \quad 0 < x < 1, \, 0 < \, t\le 1, \\ y(x, 0)-y(x, 1) = -\sin(2\, \pi\, x), & \quad 0 < x < 1, \\ y(0, t) = y(1, t) = 0, & \quad 0 < t\le 1, \end{cases} \end{equation}$

(6.5)

where the exact solution is $y(x, t) = t^2\, \sin(2\, \pi\, x).$ presents MAE for Example 6.2 for different $N, A, B$ and $\alpha = 0.5$ using the two proposed numerical methods, while Table 2 presents a comparison with some other methods. The results of this table show that our methods are more accurate.

Example 6.3. Consider the following TFHE:

$\begin{equation} \begin{cases} \dfrac{\partial^{\alpha}y(x, t)}{\partial\, t^{\alpha}} = \dfrac{\partial^{2}y(x, t)}{\partial\, x^{2}}+\dfrac{\Gamma (\beta+1)}{\Gamma (\beta-\alpha +1)} t^{\beta-\alpha }(1-x) \sin (x)+ t^{\beta}(2\cos(x)+(1-x)\sin (x)), & \quad 0 < x < 1, \, 0 < \, t\le 1, \\ y(x, 0)-y(x, 1) = (x-1)\sin(x), & \quad 0 < x < 1, \\ y(0, t) = y(1, t) = 0, & \quad 0 < t\le 1, \end{cases} \end{equation}$

(6.6)

where the exact solution is $y(x, t) = t^{\beta}\, (1-x)\sin\, x.$

Remark 6.1. It is known that the exact solution of TFHE has a weak singularity near the initial time point, i.e., the exact solution is nonsmooth near the initial time $t = 0$ (see ^[49]). Table 6 presents the numerical solutions obtained for the TFHE Eq (6.6) (for $\alpha = 0.5$ ), whose exact solution is a nonsmooth solution for values of $\beta, \, 0 < \beta < 1$ . These results show that our algorithm still provides accurate solutions.

Example 6.4. Consider the following TFHE:

$\begin{equation} \begin{cases} \dfrac{\partial^{\alpha}y(x, t)}{\partial\, t^{\alpha}} = \dfrac{\partial^{2}y(x, t)}{\partial\, x^{2}}+\dfrac{2}{\Gamma (3-\alpha)}t^{2-\alpha}\, \ln(1+x-x^2)+t^2\, \dfrac{2\, x^2-2\, x+3}{(x^2-x-1)^2}, & \quad 0 < x < 1, \, 0 < \, t\le 1, \\ y(x, 0)-y(x, 1) = -\ln(1+x-x^2), & \quad 0 < x < 1, \\ y(0, t) = y(1, t) = 0, & \quad 0 < t\le 1, \end{cases} \end{equation}$

(6.7)

where the exact solution is $y(x, t) = t^2\, \ln(1+x-x^2).$

Example 6.5. Consider the following TFHE:

$\begin{equation} \begin{cases} \dfrac{\partial^{0.5}y(x, t)}{\partial\, t^{0.5}} = \dfrac{\partial^{2}y(x, t)}{\partial\, x^{2}}+z(x, t), & \quad 0 < x < 1, \, 0 < \, t\le 1, \\ y(x, 0)-y(x, 1) = -e^{x^2} (1-erf(x))+(e\, (1-erf(1))-1) x+1, & \quad 0 < x < 1, \\ y(0, t) = y(1, t) = 0, & \quad 0 < t\le 1, \end{cases} \end{equation}$

(6.8)

where the function $z(x, t)$ is chosen such that the exact solution is

$\begin{equation} y(x, t) = E_{\frac{1}{2}, 1}\left(-x \sqrt{t}\right)+x \left(-E_{\frac{1}{2}, 1}\left(-\sqrt{t}\right)\right)+x-1, \end{equation}$

(6.9)

where the functions

$\begin{equation} erf(x) = \frac{2}{\sqrt{\pi }}\int_0^x e^{-t^2}\, dt, \quad \text{and} \quad E_{\alpha, \beta }(x) = \sum\limits_{k = 0}^{\infty } \frac{x^k}{\Gamma (k \alpha +\beta )}, \end{equation}$

(6.10)

are the Gaussian error function and the generalized Mitta-Leffler function, respectively.

Remark 6.2. The results of Table 1 show that the first and second kinds of Chebyshev approximations are not always the best, along with other approximations for the UCPs. This, of course, clarifies the importance of our generalization to the CPs in this paper.

7. Conclusions

In order to solve TFHE in non-local conditions, this work developed spectral tau and collocation methods. To choose appropriate sets of basis functions, UCPs, and their shifted polynomials were employed. An approximate solution can be obtained by solving the given system of algebraic equations using an appropriate solver. We emphasize the benefit of using the properties of second-kind CPs, which help us calculate some of the computational formulas. In Section 6, we illustrated the accuracy and usefulness of our methods by comparing them to other methodologies in the literature. To the best of our knowledge, this is the first time that this type of polynomial has been utilized in numerical analysis. It is shown that the first- and second-kinds are not always the best among other Chebyshev approximations. In addition, in future work, we aim to employ these polynomials to treat other types of differential equations.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgment

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant No. (UJ-23-DR-202). Therefore, the authors thank the University of Jeddah for its technical and financial support.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	Y. H. Youssri, W. M. Abd-Elhameed, H. M. Ahmed, New fractional derivative expression of the shifted third-kind Chebyshev polynomials: Application to a type of nonlinear fractional pantograph differential equations, J. Funct. Space., 2022 (2022). https://doi.org/10.1155/2022/3966135
[2]	H. M. Ahmed, Numerical solutions for singular Lane-Emden equations using shifted Chebyshev polynomials of the first kind, Contemp. Math., 4 (2023), 132–149. https://doi.org/10.37256/cm.4120232254 doi: 10.37256/cm.4120232254
[3]	E. H Doha, W. M Abd-Elhameed, M. A. Bassuony, On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds, Acta Math. Sci., 35 (2015), 326–338. https://doi.org/10.1016/s0252-9602(15)60004-2 doi: 10.1016/s0252-9602(15)60004-2
[4]	W. M Abd-Elhameed, H. M. Ahmed, Tau and Galerkin operational matrices of derivatives for treating singular and Emden-Fowler third-order-type equations, Internat. J. Modern Phys. C, 33 (2022), 2250061. https://doi.org/10.1142/s0129183122500619 doi: 10.1142/s0129183122500619
[5]	A. T. Dincel, S. N. T. Polat, Fourth kind Chebyshev wavelet method for the solution of multi-term variable order fractional differential equations, Eng. Comput., 39 (2022), 1274–1287. https://doi.org/10.1108/ec-04-2021-0211 doi: 10.1108/ec-04-2021-0211
[6]	R. Magin, Fractional calculus in bioengineering, part 1. Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/critrevbiomedeng.v32.10
[7]	V. E. Tarasov, Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media, Springer Science & Business Media, 2011.
[8]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, World Scientific, 2022.
[9]	S. Das, I. Pan, Fractional order signal processing: Introductory concepts and applications, Springer Science & Business Media, 2011.
[10]	S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488–494. https://doi.org/10.1016/j.amc.2005.11.025 doi: 10.1016/j.amc.2005.11.025
[11]	S. Abbasbandy, S. Kazem, M. S. Alhuthali, H. H. Alsulami, Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection-diffusion equation, Appl. Math. Comput., 266 (2015), 31–40. https://doi.org/10.1016/j.amc.2015.05.003 doi: 10.1016/j.amc.2015.05.003
[12]	H. Dehestani, Y. Ordokhani, M. Razzaghi, Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations, Math. Method. Appl. Sci., 42 (2019), 7296–7313. https://doi.org/10.1002/mma.5840 doi: 10.1002/mma.5840
[13]	T. Akram, M. Abbas, M. B. Riaz, A. I. Ismail, N. M. Ali, An efficient numerical technique for solving time fractional Burgers equation, Alex. Eng. J., 59 (2020), 2201–2220. https://doi.org/10.1016/j.aej.2020.01.048 doi: 10.1016/j.aej.2020.01.048
[14]	Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[15]	F. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y. M. Chu, Numerical solution of traveling waves in chemical kinetics: Time-fractional fishers equations, Fractals, 30 (2022), 2240051. https://doi.org/10.1142/s0218348x22400515 doi: 10.1142/s0218348x22400515
[16]	M. Shakeel, I. Hussain, H. Ahmad, I. Ahmad, P. Thounthong, Y. F. Zhang, Meshless technique for the solution of time-fractional partial differential equations having real-world applications, J. Funct. Space., 2020 (2020). https://doi.org/10.1155/2020/8898309
[17]	B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684–693. https://doi.org/10.1016/j.jmaa.2012.05.066 doi: 10.1016/j.jmaa.2012.05.066
[18]	K. S. Al-Ghafri, H. Rezazadeh, Solitons and other solutions of (3+ 1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation, Appl. Math. Nonlinear Sci., 4 (2019), 289–304. https://doi.org/10.2478/amns.2019.2.00026 doi: 10.2478/amns.2019.2.00026
[19]	Z. J. Fu, L. W. Yang, Q. Xi, C. S. Liu, A boundary collocation method for anomalous heat conduction analysis in functionally graded materials, Comput. Math. Appl., 88 (2021), 91109. https://doi.org/10.1016/j.camwa.2020.02.023 doi: 10.1016/j.camwa.2020.02.023
[20]	Q. Xi, Z. Fu, T. Rabczuk, D. Yin, A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials, Int. J. Heat Mass Transf., 180 (2021), 121778. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121778 doi: 10.1016/j.ijheatmasstransfer.2021.121778
[21]	W. H. Luo, T. Z. Huang, G. C. Wu, X. M. Gu, Quadratic spline collocation method for the time fractional subdiffusion equation, Appl. Math. Comput., 276 (2016), 252–265. https://doi.org/10.1016/j.amc.2015.12.020 doi: 10.1016/j.amc.2015.12.020
[22]	W. H. Luo, C. Li, T. Z. Huang, X. M. Gu, G. C. Wu, A high-order accurate numerical scheme for the Caputo derivative with applications to fractional diffusion problems, Numer. Funct. Anal. Optim., 39 (2018), 600–622. https://doi.org/10.1080/01630563.2017.1402346 doi: 10.1080/01630563.2017.1402346
[23]	W. H. Luo, X. M. Guo, L. Yang, J. Meng, A Lagrange-quadratic spline optimal collocation method for the time tempered fractional diffusion equation, Math. Comput. Simulat., 182 (2021), 1–24. https://doi.org/10.1016/j.matcom.2020.10.016 doi: 10.1016/j.matcom.2020.10.016
[24]	I. Karatay, S. R. Bayramoğlu, A. Şahin, Implicit difference approximation for the time fractional heat equation with the nonlocal condition, Appl. Numer. Math., 61 (2011), 1281–1288. https://doi.org/10.1016/j.apnum.2011.08.007 doi: 10.1016/j.apnum.2011.08.007
[25]	E. M. Abdelghany, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, A. G. Atta, A tau approach for solving time-fractional heat equation based on the shifted sixth-kind Chebyshev polynomials, Symmetry, 15 (2023), 594. https://doi.org/10.3390/sym15030594 doi: 10.3390/sym15030594
[26]	M. El-Gamel, M. El-Hady, A fast collocation algorithm for solving the time fractional heat equation, SeMA J., 78 (2021), 501–513. https://doi.org/10.1007/s40324-021-00245-2 doi: 10.1007/s40324-021-00245-2
[27]	X. M. Gu, S. L. Wu, A parallel-in-time iterative algorithm for volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576. https://doi.org/10.1016/j.jcp.2020.109576 doi: 10.1016/j.jcp.2020.109576
[28]	S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.oa-2016-0136 doi: 10.4208/cicp.oa-2016-0136
[29]	J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral methods for time-dependent problems, volume 21, Cambridge University Press, 2007.
[30]	J. Shen, T. Tang, L. L. Wang, Spectral methods: Algorithms, analysis and applications, volume 41, Springer Science & Business Media, 2011.
[31]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, 1988.
[32]	J. P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.
[33]	L. N. Trefethen, Spectral methods in MATLAB, volume 10, SIAM, 2000.
[34]	W. M. Abd-Elhameed, M. M. Alsuyuti, Numerical treatment of multi-term fractional differential equations via new kind of generalized Chebyshev polynomials, Fractal Fract., 7 (2023), 74. https://doi.org/10.3390/fractalfract7010074 doi: 10.3390/fractalfract7010074
[35]	W. M. Abd-Elhameed, A. M. Alkenedri, Spectral solutions of linear and nonlinear BVPs using certain Jacobi polynomials generalizing third-and fourth-kinds of Chebyshev polynomials, CMES Comput. Model. Eng. Sci., 126 (2021), 955–989. https://doi.org/10.32604/cmes.2021.013603 doi: 10.32604/cmes.2021.013603
[36]	Q. M. Al-Mdallal, On fractional-Legendre spectral Galerkin method for fractional Sturm-Liouville problems, Chaos Soliton. Fract., 116 (2018), 261–267. https://doi.org/10.1016/j.chaos.2018.09.032 doi: 10.1016/j.chaos.2018.09.032
[37]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, B. I. Bayoumi, D. Baleanu, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Method. Appl. Sci., 42 (2019), 1389–1412. https://doi.org/10.1002/mma.5431 doi: 10.1002/mma.5431
[38]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, Galerkin operational approach for multi-dimensions fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 114 (2022), 106608. https://doi.org/10.1016/j.cnsns.2022.106608 doi: 10.1016/j.cnsns.2022.106608
[39]	F. Ghoreishi, S. Yazdani, An extension of the spectral tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. Math. Appl., 61 (2011), 30–43. https://doi.org/10.1016/j.camwa.2010.10.027 doi: 10.1016/j.camwa.2010.10.027
[40]	P. Mokhtary, F. Ghoreishi, H. M. Srivastava, The Müntz-Legendre Tau method for fractional differential equations, Appl. Math. Model., 40 (2016), 671–684. https://doi.org/10.1016/j.apm.2015.06.014 doi: 10.1016/j.apm.2015.06.014
[41]	W. M. Abd-Elhameed, Novel expressions for the derivatives of sixth kind Chebyshev polynomials: Spectral solution of the non-linear one-dimensional Burgers' equation, Fractal Fract., 5 (2021), 53. https://doi.org/10.3390/fractalfract5020053 doi: 10.3390/fractalfract5020053
[42]	M. A. Abdelkawy, A. Z. M. Amin, A. M. Lopes, Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations, Comput. Appl. Math., 41 (2022), 1–21. https://doi.org/10.1007/s40314-021-01702-4 doi: 10.1007/s40314-021-01702-4
[43]	C. Liu, Z. Yu, X. Zhang, B. Wu, An implicit wavelet collocation method for variable coefficients space fractional advection-diffusion equations, Appl. Numer. Math., 177 (2022), 93–110. https://doi.org/10.1016/j.apnum.2022.03.007 doi: 10.1016/j.apnum.2022.03.007
[44]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[45]	J. C. Mason, D. C. Handscomb, Chebyshev polynomials, CRC Press, 2002.
[46]	A. H. Bhrawy, M. A. Alghamdi, A Legendre Tau-spectral method for solving time-fractional heat equation with nonlocal conditions, Sci. World J., 2014 (2014), 706296. https://doi.org/10.1155/2014/706296 doi: 10.1155/2014/706296
[47]	W. Koepf, Hypergeometric summation, 2 Eds., Springer Universitext Series, 2014, Available from: http://www.hypergeometric-summation.org.
[48]	G. E. Andrews, R. Askey, R. Roy, Special functions, volume 71, Cambridge University Press, 1999.
[49]	Y. L. Zhao, X. M. Gu, A. Ostermann, A preconditioning technique for an all-at-once system from volterra subdiffusion equations with graded time steps, J. Sci. Comput., 88 (2021), 11. https://doi.org/10.1007/s10915-021-01527-7 doi: 10.1007/s10915-021-01527-7

This article has been cited by:

1.	Chaeyoung Lee, Yunjae Nam, Minjoon Bang, Seokjun Ham, Junseok Kim, Numerical investigation of the dynamics for a normalized time-fractional diffusion equation, 2024, 9, 2473-6988, 26671, 10.3934/math.20241297
2.	M. A. Abdelkawy, H. Almadi, E. M. Solouma, M. M. Babatin, Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models, 2024, 35, 0129-1831, 10.1142/S0129183124501493
3.	H M Ahmed, R M Hafez, W M Abd-Elhameed, A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices, 2024, 99, 0031-8949, 045250, 10.1088/1402-4896/ad3482
4.	A. N. Nirmala, S. Kumbinarasaiah, A Robust numerical technique based on the chromatic polynomials for the European options regulated by the time-fractional Black–Scholes equation, 2024, 2731-6734, 10.1007/s43994-024-00193-3
5.	H. M. Ahmed, New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations, 2024, 2024, 1687-2770, 10.1186/s13661-024-01948-x
6.	H. M. Ahmed, Enhanced shifted Jacobi operational matrices of integrals: spectral algorithm for solving some types of ordinary and fractional differential equations, 2024, 2024, 1687-2770, 10.1186/s13661-024-01880-0
7.	Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori, Abdulrahman Khalid Al-Harbi, Mohammed H. Alharbi, Ahmed Gamal Atta, Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem, 2024, 32, 2688-1594, 6200, 10.3934/era.2024288
8.	M.H. Heydari, F. Heydari, O. Bavi, M. Bayram, Logarithmic Bernstein functions for fractional Rosenau–Hyman equation with the Caputo–Hadamard derivative, 2024, 67, 22113797, 108055, 10.1016/j.rinp.2024.108055
9.	N M Yassin, Emad H Aly, A G Atta, Novel approach by shifted Schröder polynomials for solving the fractional Bagley-Torvik equation, 2025, 100, 0031-8949, 015242, 10.1088/1402-4896/ad9963
10.	Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta, Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation, 2024, 9, 2473-6988, 34567, 10.3934/math.20241646
11.	M.H. Heydari, M. Razzaghi, M. Bayram, A numerical approach for multi-dimensional ψ-Hilfer fractional nonlinear Galilei invariant advection–diffusion equations, 2025, 68, 22113797, 108067, 10.1016/j.rinp.2024.108067
12.	Minilik Ayalew, Mekash Ayalew, Mulualem Aychluh, Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method, 2025, 2661-3352, 10.1142/S2661335224500291
13.	Zehui Ma, Rahmatjan Imin, A High‐Precision Meshless Method for Time‐Fractional Mixed Diffusion and Wave Equations, 2025, 126, 0029-5981, 10.1002/nme.70020
14.	Atallah El-Shenawy, Mohammad Izadi, Mahmoud Abd El-Hady, An eye surgery corneal nonlinear model: mathematical analysis and simulation via Dickson polynomials series, 2025, 14, 2314-8543, 10.1186/s43088-025-00616-y
15.	S. Akansha, Aditya Subramaniam, Exploring Chebyshev polynomial approximations: Error estimates for functions of bounded variation, 2025, 10, 2473-6988, 8688, 10.3934/math.2025398
16.	N. M. Yassin, A. G. Atta, Emad H. Aly, Numerical solutions for nonlinear ordinary and fractional Newell–Whitehead–Segel equation using shifted Schröder polynomials, 2025, 2025, 1687-2770, 10.1186/s13661-025-02041-7
17.	Hais Azin, Omid Baghani, Ali Habibirad, A numerical scheme to simulate the distributed-order time 2D Benjamin Bona Mahony Burgers equation with fractional-order space, 2025, 30, 1648-3510, 277, 10.3846/mma.2025.20964

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(1656) PDF downloads(67) Cited by(17)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(10) / Tables(12)

AIMS Mathematics

Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Related Papers:

Abstract

1. Introduction

2. Some fundamentals and preliminaries

2.1. Some definitions of fractional calculus

2.2. An overview on CPs as a type of new UCPs

3. Some new formulas concerned with UCPs

3.1. Some new results for the UCPs

3.2. Some formulas of the shifted UCPs

3.3. Some useful integral formulas involve the UCPs

4. Tau and collocation solutions for TFHE

4.1. Tau approach for treating TFHE

4.2. The collocation approach for treating TFHE

5. Investigation of convergence and error analysis

6. Numerical examples

7. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials

Related Papers:

Abstract

1. Introduction

2. Some fundamentals and preliminaries

2.1. Some definitions of fractional calculus

2.2. An overview on CPs as a type of new UCPs

3. Some new formulas concerned with UCPs

3.1. Some new results for the UCPs

3.2. Some formulas of the shifted UCPs

3.3. Some useful integral formulas involve the UCPs

4. Tau and collocation solutions for TFHE

4.1. Tau approach for treating TFHE

4.2. The collocation approach for treating TFHE

5. Investigation of convergence and error analysis

6. Numerical examples

7. Conclusions

Use of AI tools declaration

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog