# Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

## Abstract

This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.

## 1 Introduction

Because of their versatility in describing a broad spectrum of physical, chemical, biological, and other systems, fractional differential equations (DEs) have gained much attention in recent decades [1, 2]. These equations have become indispensable for modeling complicated phenomena with long-range interactions and memory effects [3, 4]. Recent articles have demonstrated an interest in solving fractional differential models encountered in various applied sciences. For instance, some fractional order epidemic models were examined in [5, 6]. InÂ [7], the authors developed some solutions for a specific fractional financial model. An additional fractional differential equation that arises in fluid mechanics was examined in [8].

The novel results for approximating solutions are significant since analytic solutions for fractional DEs are typically not explicitly found. As a result, numerous numerical approaches for solving fractional DEs have been of great interest. Researchers used a variety of numerical algorithms to solve various fractional DEs. For example, two finite difference methods are followed in [9] for treating the time-fractional Fokker-Planck equation. The Galerkin approach is followed in [10] to solve a linear hyperbolic first-order partial DEs. The collocation method is applied in [11] to solve a specific fractional model. The homotopy analysis method treats partial fractional DEs in [12].

The time-fractional diffusion equation (TFDE), a basic modification of the classical diffusion equation that considers fractional derivatives in time, is at the core of this mathematical framework [13]. The importance of the TFDE extends beyond academic fields, providing essential insights into anomalous diffusion processes found in biological, physical, and engineering systems, including viscoelastic materials, subdiffusion phenomena, and porous media flow [14â€“16].

One class of older polynomials defined by Pafnuty Chebyshev is the set of orthogonal polynomials known as Chebyshev polynomials (CPs), see [17]. Most CPs utilized in various applied science domains are of the first and second kinds. These polynomials are helpful in many fields. Their significance in electrohydrodynamics, optimum control, and signal processing has been noted [18â€“20]. Many authors used the first- and second-kinds of CPs. For example, the authors of [21] handled some real-life problems via first-derivative CPs of the first kind. Laneâ€“Emden equations are solved via CPs of the first kind in [22]. The authors in [23] used the second-kind CPs for higher-order DEs. The kinds of CPs are not restricted to the first- and second-kinds. Other contributions were presented, but other types of CPs were used. The third and fourth kinds of CPs are special non-symmetric Jacobi polynomials. The authors of [24] solved some variable-order fractional DEs by using the shifted CPs of the third kind; the authors in [25] treated some fractional DEs. Some singular Lane-Emden equations were treated using the third- and fourth-kinds in [26]. Several applications have recently been utilized by the other four types of CPs, classified as special ones of the generalized Gegenbauer polynomials ([27, 28]). For example, the authors in [29] solved some mixed-type fractional DEs using the fifth-kind CPs. The authors of [30] employed the sixth-kind CPs to treat some telegraph type-equations. More recently, the authors of [31] introduced and utilized the seventh-kind CPs to solve some fractional delay DEs. Furthermore, the polynomials, namely, the eighth-kind CPs, solve the generalized Kawahara DEs using two different approaches in [32, 33].

Due to their incredible accuracy and computing efficiency, spectral methods have become famous for solving DEs [34]. Many contributions employ the different versions of spectral methods; see, for example, [35â€“40]. Among these techniques for fractional DEs is the spectral tau approach, which converts the original problems into algebraic systems that can be solved computationally effectively [41, 42]. In this study, we suggest a new use of the spectral tau approach for solving the time-fractional DEs, with basis functions being seventh-kind CPs.

Since CPs are a desirable option for spectral approximations because of their rapid convergence, numerical stability, and orthogonality, we create a strong basis expansion designed explicitly for solving TFDE. Our main goal is to provide accurate numerical estimates for TFDE using the spectral tau approach with seventh-kind CPs. We will investigate the theoretical and numerical aspects of our method. The suggested methodâ€™s accuracy and convergence behavior will be carefully studied. Moreover, we evaluate our methodologyâ€™s effectiveness and computational power using a set of numerical examples that demonstrate its faithful representation of the complex dynamics of TFDE solutions.

This paper is organized into the following sections: Sect.Â 2 presents some fundamentals and useful preliminaries. A tau approach for solving the TFDE is followed in Sect.Â 3. AÂ thorough error analysis of our methodology is provided in Sect.Â 4. Three examples are provided in Sect.Â 5. Lastly, Sect.Â 6 reports a few conclusions.

## 2 Some key relations and formulas

This section is dedicated to reviewing the fundamentals of fractional calculus. Moreover, some relations and properties of the seventh-kind CPs and their corresponding shifted ones are exhibited.

### Definition 1

In Caputoâ€™s sense, the fractional-order derivative is [3]:

\begin{aligned}& D^{\nu}g(\xi )=\frac{1}{\Gamma (r-\nu )}\displaystyle \int _{0}^{\xi}(\xi -t)^{r- \nu -1}g^{(r)}(t)\, dt,\quad \nu >0 ,\quad \xi >0, \\& \,r-1< \nu \leq r,\quad r\in \mathbb{N}. \end{aligned}
(1)

In addition, the following properties apply

\begin{aligned} D^{\nu}C&=0,\ (\text{for a constant C}), \end{aligned}
(2)
\begin{aligned} D^{\nu}\,\xi ^{\ell}&= \textstyle\begin{cases} 0, & \text{if }\ell \in{\mathbb{N}}_{0} \quad and \quad \ell < \lceil \nu \rceil , \\ \frac{\ell \,!}{\Gamma (\ell +1-\nu )}\,\xi ^{\ell -\nu}, & \text{if } \ell \in{\mathbb{N}}_{0} \quad and \quad \ell \geq \lceil \nu \rceil , \end{cases}\displaystyle \end{aligned}
(3)

where $${\mathbb{N}}=\{1,2,\ldots \}$$ and $${\mathbb{N}}_{0}=\{0,1,2, \ldots \}$$.

### 2.2 A account on seventh-kind CPs

Consider the seventh-kind CPs $$\{\bar{Z}_{\ell}(\xi ),\ \ell \ge 0\}$$ defined as [31]

$$\bar{Z}_{\ell}(\xi )= \textstyle\begin{cases} \frac{(2)_{\frac{\ell}{2}} \, P_{\frac{\ell}{2}}^{\left (-\frac{1}{2},\frac{3}{2}\right )}\left (2 \xi ^{2}-1\right )}{\left (\frac{5}{2}\right )_{\frac{\ell}{2}}}, & \text{if \ell  even} , \\ \frac{x\, \Gamma \left (\frac{\ell +5}{2}\right ) \, P_{\frac{\ell -1}{2}}^{\left (-\frac{1}{2},\frac{5}{2}\right )}\left (2 \xi ^{2}-1\right )}{\left (\frac{5}{2}\right )_{\frac{\ell +1}{2}}}, & \text{if \ell  odd}, \end{cases}$$
(4)

where $$(a)_{n}$$ is the Pochhammer function defined as $$(a)_{n}=\frac{\Gamma (a+n)}{\Gamma (a)}$$, and $$P_{\ell}^{\left (a,b\right )}\left (\xi \right )$$ is the classical Jacobi polynomial of degree â„“.

$$\left \{\bar{Z}_{\ell}(\xi )\right \}_{\ell \ge 0}$$ is an orthogonal set on $$[-1,1]$$ with the following orthogonality relation:

$$\int _{-1}^{1}\displaystyle \frac{\xi ^{4}}{\sqrt{1-\xi ^{2}}}\, \bar{Z}_{\ell}( \xi )\, \, \bar{Z}_{s}(\xi )\,d\xi =\varrho _{\ell}\,\delta _{\ell ,s},$$
(5)

where

$$\varrho _{\ell}= \frac{9\, \pi }{8} \textstyle\begin{cases} \displaystyle \frac{1}{(\ell +1)\, (\ell +3)}, & \text{if \ \ell  even}, \\ \displaystyle \frac{(\ell +1)\, (\ell +3)}{\ell \, (\ell +2)^{2}\, (\ell +4)}, & \text{if \ \ell  odd}, \end{cases}$$
(6)

with

$$\delta _{\ell ,s}= \textstyle\begin{cases} 1, & \text{if \ell =s} , \\ 0, & \text{if \ell \neq s}. \end{cases}$$
(7)

### Lemma 1

[31] The following trigonometric expressions apply:

\begin{aligned} \bar{Z}_{2\ell}(\cos (\varphi ))=&\frac{3}{4} \left ( \frac{\cos ((2 \ell +1) \varphi )}{2 \ell +1}+ \frac{\cos ((2 \ell +3) \varphi )}{2 \ell +3}\right ) \sec ^{3}( \varphi ), \end{aligned}
(8)
\begin{aligned} \begin{aligned} \bar{Z}_{2\ell +1}(\cos (\varphi ))&= \frac{3\, \sec ^{4}(\varphi )}{8\, (2\ell +1)\, (2\ell +3)\, (2\ell +5)} \\ &\times \left ((2\ell +5)\, (2(\ell +2)\, \cos ((2\ell +1)\,\varphi )+2 \,(2\ell +1)\, \cos ((2\ell +3)\,\varphi ))\right . \\ &+\left .2(2\ell +1)\, (\ell +1) \,\cos ((2\ell +5)\,\varphi )\right ). \end{aligned} \end{aligned}
(9)

### Lemma 2

[31] Let $$\ell \in{\mathbb{N}}_{0}$$. $$\bar{Z}_{j}(\xi )$$ may be expressed as:

\begin{aligned} \bar{Z}_{2\ell}(\xi )=&\sum _{m=0}^{\ell } \frac{(-1)^{m}\, (2 \ell -m+1)!}{(\ell -m)!\, m!\, \left (\frac{5}{2}\right )_{\ell -m}} \, \xi ^{2 \ell -2 m}, \end{aligned}
(10)
\begin{aligned} \bar{Z}_{2\ell +1}(\xi )=&\sum _{m=0}^{\ell } \frac{(-1)^{m}\, (2 \ell -m+2)!}{(\ell -m)!\, m!\, \left (\frac{5}{2}\right )_{\ell -m+1}} \, \xi ^{2 \ell -2 m+1}. \end{aligned}
(11)

### Theorem 1

[31] Let $$\ell \in {\mathbb{N}}_{0}$$. $$\xi ^{2\, \ell}$$ and $$\xi ^{2\, \ell +1}$$ can be expressed in terms of $$\bar{Z}_{j}(\xi )$$ as follows:

\begin{aligned} \xi ^{2\ell}=& 2 \ell ! \sum _{m=0}^{\ell } \frac{(\ell -m+1) \left (\frac{5}{2}\right )_{\ell}}{m! (2 \ell -m+2)!} \, \bar{Z}_{2 \ell -2 m}(\xi ), \end{aligned}
(12)
\begin{aligned} \xi ^{2\ell +1}=& \ell ! \sum _{m=0}^{\ell } \frac{(3+2 \ell -2 m) \left (\frac{5}{2}\right )_{\ell +1}}{m! (2 \ell -m+3)!} \bar{Z}_{2 \ell -2 m+1}(\xi ). \end{aligned}
(13)

### Theorem 2

[31] It is possible to express $$\bar{Z}_{j}(\xi )$$ as a combination of the first-kind CPs $$T_{i}(\xi )$$ as

\begin{aligned} &\bar{Z}_{2\,\ell}(\xi )=\frac{6 }{(2 \,\ell +1)\,(2\, \ell +3)}\sum _{s=0}^{ \ell }(-1)^{s+1} \,c_{\ell -s}\,(s+1)\, (-2 \ell +s-1)\, T_{2\, \ell -2 s}(\xi ), \end{aligned}
(14)
\begin{aligned} &\bar{Z}_{2\,\ell +1}(\xi )= \frac{6 }{(2 \,\ell +1)\, (2\, \ell +3) \,(2\, \ell +5)} \\ &\phantom{\bar{Z}_{2\,\ell +1}(\xi )=}{}\times \sum _{s=0}^{ \ell }(-1)^{s}\, (s+1)\, (-2\, \ell +s-2)\, (-2\, \ell +2\, s-1)\,T_{2 \, \ell -2\, s+1}(\xi ), \end{aligned}
(15)

and

$$c_{\ell}= \textstyle\begin{cases} \frac{1}{2}, & \textit{if \ell =0}, \\ 1, & \textrm{otherwise}. \end{cases}$$
(16)

### Theorem 3

The second derivative of $$\bar{Z}_{\ell}(\xi )$$ can be expressed as

\begin{aligned} &\frac{d^{2} \,\bar{Z}_{2\, \ell}(\xi )}{d\,\xi ^{2}} \\ &\quad= \sum _{k=0}^{ \ell }8\, (k-\ell )\, \left ( \frac{2 (-1)^{k}\, \left (4\, (\ell -k)^{2}-1\right )}{4\, \ell \, (\ell +2)+3}-2 \, \ell \, (k+1)+k^{2}+1\right )\,\bar{Z}_{2\, \ell -2\, k-2}(\xi ), \end{aligned}
(17)
\begin{aligned} & \frac{d^{2} \,\bar{Z}_{2\, \ell +1}(\xi )}{d\,\xi ^{2}} \\ &\quad=\sum _{k=0}^{ \ell }4\, \left (-2 \,\ell \,(k+1)+ \frac{2\, (-1)^{k}\, (2\, \ell -2\, k-1)\, (2 \ell -2 k+1)\, (2 \,\ell -2\, k+3)}{(2\, \ell +1)\, (2\, \ell +3) \,(2\, \ell +5)}+k^{2}-k \right ) \\ & \qquad{} \times (-2\, \ell +2\, k-1)\,\bar{Z}_{2\, \ell -2\, k-1}( \xi ). \end{aligned}
(18)

### Proof

First, we prove (17). The series expression of $$\bar{Z}_{i}(t)$$ in (10) leads to the following expression for $$\frac{d^{2} \,\bar{Z}_{2\, \ell}(\xi )}{d\,\xi ^{2}}$$

$$\frac{d^{2} \,\bar{Z}_{2\, \ell}(\xi )}{d\,\xi ^{2}}=\sum _{k=0}^{ \ell } \frac{(-1)^{k}\, (2\, \ell -2\, k)\, (2\, \ell -2\, k-1)\, \Gamma (2\, \ell -k+2)}{k!\, (\ell -k)!\, \left (\frac{5}{2}\right )_{\ell -k}} \,\xi ^{2\, \ell -2 \,k-2},$$
(19)

which, by utilizing the inversion formula (1), can be expressed as

\begin{aligned} &\frac{d^{2} \,\bar{Z}_{2\, \ell}(\xi )}{d\,\xi ^{2}} \\ &\quad= \sum _{k=0}^{\ell }\sum _{s=0}^{\ell -k-1} \frac{8\, (-1)^{k} \,(-2 \,\ell +2\, k+1)\, (-\ell +k+s)\, (2\, \ell -k+1)!}{(2\, \ell -2\, k+3)\, k! \, s!\, (2\, \ell -2\, k-s)!} \,\bar{Z}_{2\, \ell -2\, k-2\, s-2}(\xi ). \end{aligned}
(20)

Upon expanding and rearranging the terms, the last relation can be transformed into

\begin{aligned} \frac{d^{2} \,\bar{Z}_{2\, \ell}(\xi )}{d\,\xi ^{2}}={}& \sum _{k=0}^{\ell }\sum _{s=0}^{k} \frac{8 (k-\ell ) (-1)^{k-s} (-2 \,\ell +2 k-2 s+1) \Gamma (2 \ell -k+s+2)}{\Gamma (s+1) (2 \ell -2 k+2 s+3) \Gamma (k-s+1) \Gamma (2 \ell -2 k+s+1)} \\ &{}\times \bar{Z}_{2\, \ell -2\, k-2}(\xi ). \end{aligned}
(21)

Now, setting

$$G_{k,\ell}=\sum _{s=0}^{k} \frac{8\,(-1)^{k-s} \,(-2\, \ell +2\, k-2\, s+1) \,(2 \,\ell -k+s+1)!}{s!\, (2\, \ell -2\, k+2\, s+3)\, (k-s)!\, (2\, \ell -2 \,k+s)!}.$$
(22)

By directly applying Zeilbergerâ€™s algorithm [43], we can derive the subsequent recurrence relation:

\begin{aligned} &\left ( 4\,k\ell -2\,{k}^{2}+6\,\ell -2\,k-1 \right ) \left ( -1+2\,\ell -2\,k \right ) ^{2}G_{ k+2,\ell} \\ &\quad{} -4\, \left ( 2\,\ell +5 \right ) \left ( - 1+2\,\ell \right ) \left ( -1+\ell -k \right ) G_{ k+1,\ell} \\ &\quad{} - \left ( 4 \,k\ell -2\,{k}^{2}+10\,\ell -6\,k-5 \right ) \left ( -3+2 \,\ell -2\,k \right ) ^{2} G_{k,\ell} =0, \end{aligned}
(23)

with the initial values:

$$G_{0,\ell}={\frac {-8\, \left (4\,{\ell}^{2}-1\right )}{2\,\ell +3}}, \quad G_{1,\ell}= \frac {-32\, \ell \, \left (4\,{\ell}^{2}+8\,\ell -5 \right )}{ \left ( 2\,\ell +1\right ) \left ( 2\,\ell +3 \right ) }.$$

The recursive formula (23) has the following exact solution

$$G_{k,\ell}=8\, \left ( \frac{2\, (-1)^{k}\, \left (4 \,(\ell -k)^{2}-1\right )}{4 \,\ell \, (\ell +2)+3}-2 \, \ell \,(k+1)+k^{2}+1\right ).$$

Therefore, relation (17) may be obtained.

To prove formula (18), we use formula (11) to get

$$\frac{d^{2} \,\bar{Z}_{2\, \ell +1}(\xi )}{d\,\xi ^{2}}=\sum _{k=0}^{ \ell } \frac{(-1)^{k}\, (2\, \ell -2 \,k+1) \,(2\, \ell -2\, k)\, \Gamma (2\, \ell -k+3)}{k!\, (\ell -k)! \,\left (\frac{5}{2}\right )_{\ell -k+1}} \,\xi ^{2 \ell -2 k-1}.$$
(24)

Using (13), we get

\begin{aligned} \frac{d^{2} \,\bar{Z}_{2\, \ell +1}(\xi )}{d\,\xi ^{2}}={}& \sum _{k=0}^{\ell }\sum _{s=0}^{\ell -k-1} \frac{4\,(-1)^{k}\, (2\, \ell -2\, k+1)\, (2\, \ell -2\, k-2\, s+1)\, (2\, \ell -k+2)!}{(2\,\ \ell -2\, k+5)\, k!\, s!\, \, (2\, \ell -2\, k-s+1)!} \\ &{}\times \bar{Z}_{2\, \ell -2\, k-2\, s-1}(\xi ), \end{aligned}
(25)

that may be turned into

\begin{aligned} \frac{d^{2} \,\bar{Z}_{2\, \ell +1}(\xi )}{d\,\xi ^{2}}={}& \sum _{k=0}^{\ell }\sum _{s=0}^{k} \frac{4\,(-2\, \ell +2\, k-1)\, (-1)^{k-s}\, (-2\, \ell +2\, k-2\, s-1)\, (2\, \ell -k+s+2)!}{s!\, (2\, \ell -2\, k+2 \,s+5) \,(k-s)!\, (2\, \ell -2\, k+s+1)!} \\ &{}\times \bar{Z}_{2\, \ell -2\, k-1}(\xi ). \end{aligned}
(26)

Now, set

$$\tilde{G}_{k,\ell}=\sum _{s=0}^{k} \frac{ 4\,(-1)^{k-s}\, (-2\, \ell +2\, k-2\, s-1)\, (2 \,\ell -k+s+2)!}{s!\, (2\, \ell -2\, k+2 \,s+5)\, (k-s)!\, (2\, \ell -2\, k+s+1)!}.$$
(27)

Apply Zeilbergerâ€™s algorithm to demonstrate that the function $$\tilde{G}_{k,\ell}$$ fulfills the subsequent recurrence relation:

\begin{aligned} &\left ( 1+2\,\ell -2\,k \right ) \left ( 4\,k\ell -2\,{k}^{2}+6 \,\ell +3 \right ) \left ( \ell -k \right ) \tilde{G}_{k+2,\ell} \\ & - \left ( -1+2\,\ell -2\,k\right ) \left ( 4\,{\ell}^{2}+4\,k\ell -2 \,{k}^{2}+20\,\ell -2\,k+3 \right ) \tilde{G}_{k+1,\ell} \\ & - \left ( -1+\ell -k \right ) \left ( 4\,k\ell -2\,{k}^{2}+ 10\, \ell -4\,k+1 \right ) \left ( -3+2\,\ell -2\,k \right )\, \tilde{G}_{k, \ell}=0, \end{aligned}
(28)

with:

$$\tilde{G}_{0,\ell}={ \frac {-8\, \left (2\,{\ell}^{2}+3\,\ell +1\right )}{5+2\,\ell}}, \quad \tilde{G}_{1,\ell}={ \frac {-8\left (8\,{\ell}^{3}+36\,{\ell}^{2}+22\,\ell +3\right )}{4\,{\ell}^{2}+16\,\ell +15}}.$$

The solution of (28) is

$$\tilde{G}_{k,\ell}=4\,\left (-2\, \ell \, (k+1)+ \frac{2\, (-1)^{k}\, (2\, \ell -2\, k-1)\, (2\, \ell -2 \,k+1)\, (2\, \ell -2 \,k+3)}{(2 \,\ell +1) \, (2\, \ell +3)\, (2\, \ell +5)}+k^{2}-k \right ).$$

Therefore, relation (18) may be obtained.â€ƒâ–¡

### 2.3 A account on the shifted CPs of seventh kind

The shifted seventh-kind CPs: $$C_{n}(t)$$ is defined on $$[0,\tau ],\, \tau >0$$ by

$$C_{n}(t)= \bar{Z}_{n}\left (\frac{2\,t}{\tau}-1\right ).$$
(29)

According to (5), the polynomials $$C_{n}(t),n\ge 0$$, are orthogonal on $$[0,\tau ]$$, with

$$\displaystyle \int _{0}^{\tau }\,\displaystyle \frac{(\frac{2\,t}{\tau}-1)^{4}}{\sqrt{t\,(\tau -t)}}\, C_{n}(t)\,C_{m}(t) \, dt=\varrho _{n}\,\delta _{n,m}.$$
(30)

### Theorem 4

[31] The analytic form of $$C_{i}(t)$$ is given as follows:

$$C_{i}(t)=\displaystyle \sum _{p=0}^{i}\lambda _{p,i}\, {\left (\frac{t}{\tau} \right )}^{p},$$
(31)

where

$$\lambda _{p,i}= \textstyle\begin{cases} \frac{3 \,(i+2)}{2\, (i+1)\, (i+3)}, & \textit{if p=0},\quad i \textit{ even,} \\ -\frac{3 \,(i+1) \,(i+3)}{2\, i \,(i+2)\, (i+4)}, & \textit{if p=0}, \quad i \textit{ odd}, \\ \sum _{j=\left \lfloor \frac{p+1}{2}\right \rfloor }^{\frac{i}{2}} \frac{3 \,(-1)^{i/2}\, j \,4^{p} \,(i-2 \,j+2)\, (i+2\, j+2) \,(-1)^{j+p} \, (2 \, j+p-1)!}{(i+1)\, (i+3)\, (2\, p)!\, (2\, j-p)!}, & \textit{if p\neq 0},\quad i\textit{ even} , \\ \sum _{j=\left \lfloor \frac{p}{2}\right \rfloor }^{\frac{i-1}{2}} \frac{3 \, (-1)^{\frac{i+1}{2}}\, (2 \,j+1)^{2}\, 2^{2\, p-1}\, (i-2\, j+1) \,(i+2\, j+3)\, (-1)^{j+p}\, (2\, j+p)!}{i\, (i+2)\, (i+4) \,(2\, p)!\, (2 \, j-p+1)!}, & \textit{if p\neq 0}, \quad i\textit{ odd.} \end{cases}$$
(32)

### Theorem 5

[31] $$t^{m}$$ can be expanded in terms of $$C_{i}(t)$$ as

$$t^{m}=\tau ^{m} \displaystyle \sum _{r=0}^{m}H_{r,m}\, C_{r}(t),$$
(33)

where

$$H_{r,m}= \textstyle\begin{cases} \sum _{L=1}^{\left \lfloor \frac{m-r}{2}\right \rfloor } \frac{2^{3-2\, m}\, (r+2)\, (2\, L+r)\, (2\,m)!\,\Gamma \left (\frac{r+5}{2}\right ) \, (L+r-1)!}{3\, \Gamma (3-L) \,L!\, \Gamma \left (\frac{r+1}{2}\right )\, \Gamma (L+r+3) \,(-2\, L+m-r)! \,(2\, L+m+r)!}+ \frac{4^{-m} \,(r+3) \,\Gamma (2\, m+1)}{3\, (m-r)!\, (m+r)!}, \\ \quad\textit{if r even,} \\ \sum _{L=0}^{\left \lfloor \frac{1}{2} (m-r+1)\right \rfloor } \frac{4^{2-m}\, (r+2) (2\, m)!\, \Gamma \left (\frac{r}{2}+3\right )\, (L+r-1)!}{3 \,\Gamma (3-L)\, L!\, \Gamma \left (\frac{r}{2}\right )\, (L+r+2)!\, (-2\, L+m-r)!\, (2\, L+m+r)!}, \\ \quad\textit{if r odd}. \end{cases}$$
(34)

### Corollary 1

It is possible to express $$C_{j}(t)$$ as a combination of the shifted first-kind CPs $$T^{*}_{i}(t)$$ as

\begin{aligned} &C_{2\,\ell}(t)=\frac{6 }{(2 \,\ell +1)\,(2\, \ell +3)}\sum _{s=0}^{ \ell }(-1)^{s+1} \,c_{\ell -s}\,(s+1)\, (-2 \ell +s-1)\, T^{*}_{2\, \ell -2 s}(t), \end{aligned}
(35)
\begin{aligned} &C_{2\,\ell +1}(t)= \frac{6 }{(2 \,\ell +1)\, (2\, \ell +3) \,(2\, \ell +5)}\sum _{s=0}^{ \ell }(-1)^{s}\, (s+1)\, (-2\, \ell +s-2)\, (-2\, \ell +2\, s-1) \\ &\phantom{C_{2\,\ell +1}(t)=}{}\times T^{*}_{2 \, \ell -2\, s+1}(t). \end{aligned}
(36)

### Proof

The proof is easily obtained by replacing Î¾ with $$(\frac{2\,t}{\tau}-1)$$ in Theorem 2.â€ƒâ–¡

### Theorem 6

For $$0<\alpha <1$$, we have

$$D^{\alpha}C_{i}(t)=t^{-\alpha}\,\left ( \sum _{p=0}^{i} \eta _{p,i}^{ \alpha}\, C_{p}(t)-\frac{\lambda _{0,i}}{\Gamma (1-\alpha )}\right ),$$
(37)

where

$$\eta _{p,i}^{\alpha}=\sum _{r=p}^{i} \frac{r!\, \lambda _{r,i}\, H_{p,r} }{\Gamma (r-\alpha +1)}.$$
(38)

### Proof

From Eq.Â (31), we have

$$C_{i}(t)=\displaystyle \sum _{p=0}^{i}\lambda _{p,i}\, {\left (\frac{1}{\tau} \right )}^{p} \,t^{p},$$
(39)

using (3), the last equation can be rewritten as

$$D^{\alpha}C_{i}(t)=\displaystyle \sum _{p=1}^{i}\lambda _{p,i}\, {\left ( \frac{1}{\tau}\right )}^{p} \,\frac{p!}{\Gamma (p-\alpha +1 )}\,t^{p- \alpha }.$$
(40)

In virtue of the inversion formula (33), Eq.Â (40) becomes

$$D^{\alpha}C_{i}(t)=t^{-\alpha}\,\displaystyle \sum _{p=1}^{i}\lambda _{p,i} \, \frac{p!}{\Gamma (p-\alpha +1 )}\,\sum _{r=0}^{p} H_{r,p}\,C_{p}(t).$$
(41)

The arrangement of Eq.Â (41) leads to the following formula

$$D^{\alpha}C_{i}(t)=t^{-\alpha}\,\left ( \sum _{p=0}^{i} \eta _{p,i}^{ \alpha}\, C_{p}(t)-\frac{\lambda _{0,i}\,H_{0,0}}{\Gamma (1-\alpha )} \,C_{0}(t)\right ),$$
(42)

which can be written in simple form as

$$D^{\alpha}C_{i}(t)=t^{-\alpha}\,\left ( \sum _{p=0}^{i} \eta _{p,i}^{ \alpha}\, C_{p}(t)-\frac{\lambda _{0,i}}{\Gamma (1-\alpha )}\right ),$$

where

$$\eta _{p,i}^{\alpha}=\sum _{r=p}^{i} \frac{r!\, \lambda _{r,i}\, H_{p,r} }{\Gamma (r-\alpha +1)}.$$

This completes the proof of the theorem.â€ƒâ–¡

## 3 Tau approach for solving the TFDE

Our main objective in this section is to handle the following TFDE [44]

$$\partial _{t}^{\alpha}u(\xi ,t)-\partial _{\xi}^{2}u(\xi ,t)=f(\xi ,t), \quad \forall \,(\xi ,t)\in \Omega ,$$
(43)

governed by the following conditions:

\begin{aligned} u(\xi ,0)=&p_{1}(\xi ),\quad \forall \, \xi \in (-1,1), \end{aligned}
(44)
\begin{aligned} u(-1,t)=&p_{2}(t),\quad u(1,t)=p_{3}(t),\quad \forall \, t\in (0,\tau ), \end{aligned}
(45)

where $$\alpha \in (0,1),\, \Omega =(-1,1)\times (0,\tau )$$. Moreover, $$p_{1}(\xi ),\,p_{2}(t),\,p_{3}(t)$$ are continuous functions and $$f(\xi ,t)$$ is a known continuous function.

Consider the space

\begin{aligned} &L^{2}_{\omega (\xi ,t)}(\Omega )={\mathrm{{span}}}\{{\bar{Z}_{i}( \xi )}\,{C_{j}(t)}:i=0,1,2,\ldots \,\text{and}\,j=0,1,2,\ldots \}, \\ & \end{aligned}
(46)

and assume that $$u(\xi ,t)\in L^{2}_{\omega (\xi ,t)}(\Omega )$$ is expanded as

$$u(\xi ,t)=\sum _{i=0}^{\infty}\sum _{j=0}^{\infty}b_{ij}\,\bar{Z}_{i}( \xi )\,C_{j}(t),$$
(47)

where $${\omega}(\xi ,t)= \frac{\xi ^{4}\,(\frac{2\,t}{\tau}-1)^{4}}{\sqrt{1-\xi ^{2}}\,\sqrt{t\,(\tau -t)}}$$. It is easy to see that $$b_{ij}$$ are given by

$$b_{ij}=\frac{1}{ \varrho _{i}\,\varrho _{j}}\,\displaystyle \int _{0}^{\tau} \int _{-1}^{1} \,\displaystyle \,u(\xi ,t)\, \bar{Z}_{i}(\xi )\,C_{j}(t)\,{ \omega}(\xi ,t)\,d\xi \, dt.$$
(48)

Therefore, $$u(\xi ,t)$$ can be approximated in $$L^{2}_{\omega (\xi ,t)}(\Omega )$$ as

$$u_{\mathsf{M}}(\xi ,t)=\sum _{i=0}^{\mathsf{M}}\sum _{j=0}^{ \mathsf{M}}b_{ij}\,\bar{Z}_{i}(\xi )\,C_{j}(t)=\boldsymbol{\bar{Z}}( \xi )\,\mathbf{B}\,\boldsymbol{{C}^{T}}(t),$$
(49)

where

\begin{aligned}& \boldsymbol{\bar{Z}}(\xi )=[\bar{Z}_{0}(\xi ),\bar{Z}_{1}(\xi ), \ldots ,\bar{Z}_{\mathsf{M}}(\xi )],\\& \boldsymbol{C}(t)=[C_{0}(t),C_{1}(t),\ldots ,C_{\mathsf{M}}(t)], \end{aligned}

and $$\mathbf{B}=(b_{ij})_{0\leq i,j\leq \mathsf{M}}$$ is a $$(\mathsf{M}+1)\times (\mathsf{M}+1)$$ matrix.

Now, the following residual $$\mathbf{R}(\xi ,t)$$ of Eq.Â (43) can be obtained after replacing $$u(\xi ,t)$$ with $$u_{\mathsf{M}}(\xi ,t)$$

$$\mathbf{R}(\xi ,t)=\partial _{t}^{\alpha}u_{\mathsf{M}}(\xi ,t)- \partial _{\xi}^{2}u_{\mathsf{M}}(\xi ,t)-f(\xi ,t).$$
(50)

Let us define

\begin{aligned} (\,u(\xi ,t),v(\xi ,t)\,)_{\omega (\xi ,t)}=&\int _{0}^{\tau}\int _{-1}^{1}u( \xi ,t)\,v(\xi ,t)\,\omega (\xi ,t)\,d\xi \,dt, \end{aligned}
(51)
\begin{aligned} (\,u(\xi ),v(\xi )\,)_{\bar{\omega}(\xi )}=&\int _{-1}^{1}u(\xi )\,v( \xi )\,\bar{\omega}(\xi )\,d\xi , \end{aligned}
(52)
\begin{aligned} (\,u(t),v(t)\,)_{\hat{\omega}(t)}=&\int _{0}^{\tau}u(t)\,v(t)\, \hat{\omega}(t)\,dt, \end{aligned}
(53)

where $$\bar{\omega}(\xi )=\frac{\xi ^{4}}{\sqrt{1-\xi ^{2}}}$$ and $$\hat{\omega}(t)= \frac{(\frac{2\,t}{\tau}-1)^{4}}{\sqrt{t\,(\tau -t)}}$$.

The application of tau method [30] leads to

$$(\,\mathbf{R}(\xi ,t),\bar{Z}_{r}(\xi )\,C_{s}(t) \,)_{\omega (\xi ,t)}=0, \quad r:0,1,\ldots,\mathsf{M}-2,\quad s:0,1,\ldots,\mathsf{M}-1,$$
(54)

which leads to the following matrix equation

$$\mathbf{A^{T}}\,\mathbf{B}\,\mathbf{Y}-\mathbf{G^{T}}\,\mathbf{B}\, \mathbf{E}=\mathbf{F},$$
(55)

where

$\begin{array}{rlrl}& \mathbf{A}={\left({a}_{i,r}\right)}_{\left(\mathsf{M}+1\right)Ã—\left(\mathsf{M}âˆ’1\right)},& & \phantom{\rule{-40pt}{0ex}}{a}_{i,r}={\left({\stackrel{Â¯}{Z}}_{r}\left(\mathrm{Î¾}\right),{\stackrel{Â¯}{Z}}_{i}\left(\mathrm{Î¾}\right)\right)}_{\stackrel{Â¯}{\mathrm{Ï‰}}\left(\mathrm{Î¾}\right)},\\ & \mathbf{Y}={\left({y}_{j,s}\right)}_{\left(\mathsf{M}+1\right)Ã—\mathsf{M}},& & \phantom{\rule{-40pt}{0ex}}{y}_{j,s}={\left({C}_{s}\left(t\right),{D}^{\mathrm{Î±}}{C}_{j}\left(t\right)\right)}_{\stackrel{Ë†}{\mathrm{Ï‰}}\left(t\right)},\\ & \mathbf{G}={\left({g}_{i,r}\right)}_{\left(\mathsf{M}+1\right)Ã—\left(\mathsf{M}âˆ’1\right)},& & \phantom{\rule{-40pt}{0ex}}{g}_{i,r}={\left({\stackrel{Â¯}{Z}}_{r}\left(\mathrm{Î¾}\right),{D}^{2}{\stackrel{Â¯}{Z}}_{i}\left(\mathrm{Î¾}\right)\right)}_{\stackrel{Â¯}{\mathrm{Ï‰}}\left(\mathrm{Î¾}\right)},\\ & \mathbf{E}={\left({e}_{j,s}\right)}_{\left(\mathsf{M}+1\right)Ã—\mathsf{M}},& & \phantom{\rule{-40pt}{0ex}}{e}_{j,s}={\left({C}_{s}\left(t\right),{C}_{j}\left(t\right)\right)}_{\stackrel{Ë†}{\mathrm{Ï‰}}\left(t\right)},\\ & \mathbf{F}={\left({f}_{r,s}\right)}_{\left(\mathsf{M}âˆ’1\right)Ã—\mathsf{M}},& & \phantom{\rule{-40pt}{0ex}}{f}_{r,s}={\left(f\left(\mathrm{Î¾},t\right),{\stackrel{Â¯}{Z}}_{r}\left(\mathrm{Î¾}\right)\phantom{\rule{0.2em}{0ex}}{C}_{s}\left(t\right)\right)}_{\mathrm{Ï‰}\left(\mathrm{Î¾},t\right)}.\end{array}$

### Theorem 7

The elements of matrices A, E, G and Y are given by

\begin{aligned} &a_{i,r}=\varrho _{r}\,\delta _{r,i}, \end{aligned}
(56)
\begin{aligned} &e_{j,s}=\varrho _{s}\,\delta _{s,j}, \end{aligned}
(57)
\begin{aligned} &g_{2\,i,r}=\sum _{r=0}^{i} 8\, (r-i)\, \left ( \frac{2 (-1)^{r}\, \left (4\, (i-r)^{2}-1\right )}{4\, i\, (i+2)+3}-2 \, i\, (r+1)+r^{2}+1\right ) \\ &\phantom{g_{2\,i,r}=}{}\times\varrho _{2\, j-2\, r-2}\,\delta _{2\, j-2 \, r-2,r}, \end{aligned}
(58)
\begin{aligned} &g_{2\,i+1,r}=\sum _{r=0}^{i} 4\, \left (-2 \,i \,(r+1)+ \frac{2\, (-1)^{r}\, (2\, i-2\, r-1)\, (2 i-2 r+1)\, (2 \,i-2\, r+3)}{(2\, i+1)\, (2\, i+3) \,(2\, i+5)}+r^{2}-r \right ) \\ & \phantom{g_{2\,i+1,r}=}{} \times (-2\, i+2\, r-1)\,\varrho _{2\, i-2\, r-1}\,\delta _{2 \, i-2\, r-1,r}, \end{aligned}
(59)
\begin{aligned} & y_{j,s} =\sum _{p=0}^{s}\sum _{r=0}^{j}\lambda _{p,s}\, \lambda _{r,j}\,\tau ^{\alpha} \,\left [16\,\beta \left (\alpha +p+r+ \frac{9}{2},\frac{1}{2}\right )-32\,\beta \left (\alpha +p+r+ \frac{7}{2},\frac{1}{2}\right )\right . \\ &\phantom{y_{j,s} =}{}+24\,\beta \left (\alpha +p+r+\frac{5}{2},\frac{1}{2}\right ) \\ &\phantom{y_{j,s} =}{}\left .-8 \,\beta \left (\alpha +p+r+\frac{3}{2},\frac{1}{2}\right )+\beta \left (\alpha +p+r+\frac{1}{2},\frac{1}{2}\right )\right ]. \end{aligned}
(60)

### Proof

By using the orthogonality relations (5), (30) and Theorem 3, the entries of A, E, and G may be easily found.

By virtue of Theorem 6, the elements $$y_{j,s}$$ can be written as

\begin{aligned} y_{j,s}=&(C_{s}(t),D^{\alpha}C_{j}(t))_{\hat{\omega}(t)} \\ & =\displaystyle \int _{0}^{\tau }\,\displaystyle \frac{(\frac{2\,t}{\tau}-1)^{4}}{\sqrt{t\,(\tau -t)}}\, C_{s}(t)\,D^{ \alpha}C_{j}(t)\, dt \\ & =\sum _{p=0}^{j} \eta _{p,j}^{\alpha}\,\displaystyle \int _{0}^{\tau }\,\displaystyle \frac{\left (\frac{16 t^{4}}{\tau ^{4}}-\frac{32 t^{3}}{\tau ^{3}}+\frac{24 t^{2}}{\tau ^{2}}-\frac{8 t}{\tau }+1\right )}{\sqrt{t\,(\tau -t)}} \, t^{-\alpha}\, C_{s}(t)\, C_{p}(t)\, dt \\ & -\frac{\lambda _{0,j}}{\Gamma (1-\alpha )}\displaystyle \int _{0}^{\tau }\, \displaystyle \frac{\left (\frac{16 t^{4}}{\tau ^{4}}-\frac{32 t^{3}}{\tau ^{3}}+\frac{24 t^{2}}{\tau ^{2}}-\frac{8 t}{\tau }+1\right )}{\sqrt{t\,(\tau -t)}} \,t^{-\alpha}\, C_{s}(t)\,dt. \end{aligned}
(61)

With the aid of formula (31), the last equation becomes

\begin{aligned} y_{j,s} =&\sum _{p=0}^{j}\sum _{n=0}^{s}\sum _{r=0}^{p} \frac{\eta _{p,j}^{\alpha}\,\lambda _{n,s}\,\lambda _{r,p}}{\tau ^{n+r}} \\ & \times \int _{0}^{\tau }\left ( \frac{16 \, t^{n+r-\alpha +\frac{7}{2}}}{\tau ^{4}}- \frac{32\, t^{n+r-\alpha +\frac{5}{2}}}{\tau ^{3}}+ \frac{24\, t^{n+r-\alpha +\frac{3}{2}}}{\tau ^{2}}- \frac{8 \,t^{n+r-\alpha +\frac{1}{2}}}{\tau }+t^{n+r-\alpha - \frac{1}{2}}\right ) \\ &{}\times\frac{1}{\sqrt{\tau -t}}dt -\frac{\lambda _{0,j}}{\Gamma (1-\alpha )}\,\sum _{n=0}^{s} \frac{\lambda _{n,s}}{\tau ^{n}} \\ & \times \int _{0}^{\tau }\left ( \frac{16 \, t^{n-\alpha +\frac{7}{2}}}{\tau ^{4}}- \frac{32\, t^{n-\alpha +\frac{5}{2}}}{\tau ^{3}}+ \frac{24\, t^{n-\alpha +\frac{3}{2}}}{\tau ^{2}}- \frac{8 \,t^{n-\alpha +\frac{1}{2}}}{\tau }+t^{n-\alpha -\frac{1}{2}} \right ) \frac{1}{\sqrt{\tau -t}}dt. \end{aligned}
(62)

The integration in the right-hand side of Eq.Â (62) can be evaluated to give the following result

\begin{aligned} y_{j,s} =&\sum _{p=0}^{j}\sum _{n=0}^{s}\sum _{r=0}^{p} \frac{\eta _{p,j}^{\alpha}\,\lambda _{n,s}\,\lambda _{r,p}}{\tau ^{n+r}} \,\left [16\,\beta \left (n+r-\alpha +\frac{9}{2},\frac{1}{2}\right )-32 \,\beta \left (n+r-\alpha +\frac{7}{2},\frac{1}{2}\right )\right . \\ & \left .+24\,\beta \left (n+r-\alpha +\frac{5}{2},\frac{1}{2}\right )-8 \,\beta \left (n+r-\alpha +\frac{3}{2},\frac{1}{2}\right )+\beta \left (n+r-\alpha +\frac{1}{2},\frac{1}{2}\right )\right ] \\ & -\frac{\lambda _{0,j}}{\Gamma (1-\alpha )}\,\sum _{n=0}^{s} \frac{\lambda _{n,s}}{\tau ^{n}}\,\left [16\,\beta \left (n-\alpha + \frac{9}{2},\frac{1}{2}\right )-32\,\beta \left (n-\alpha + \frac{7}{2},\frac{1}{2}\right )\right . \\ & \left .+24\,\beta \left (n-\alpha +\frac{5}{2},\frac{1}{2}\right )-8 \,\beta \left (n-\alpha +\frac{3}{2},\frac{1}{2}\right )+\beta \left (n- \alpha +\frac{1}{2},\frac{1}{2}\right )\right ], \end{aligned}
(63)

where $$\beta (\xi ,y)$$ is the beta function.â€ƒâ–¡

Moreover, we get the following initial and boundary conditions

\begin{aligned} &\sum _{i=0}^{\mathsf{M}}\sum _{j=0}^{\mathsf{M}}b_{ij} \,a_{i,r}\,C_{j}(0)=(p_{1}(\xi ),\,\bar{Z}_{r}(\xi ) )_{\bar{\omega}( \xi )},\quad r=0,1,\ldots \mathsf{M}, \\ & \sum _{i=0}^{\mathsf{M}}\sum _{j=0}^{\mathsf{M}}b_{ij}\,\bar{Z}_{i}(-1) \,e_{j,s}=(p_{2}(t),\,C_{s}(t) )_{\hat{\omega}(t)},\quad s=0,1,\ldots \mathsf{M}-1, \\ & \sum _{i=0}^{\mathsf{M}}\sum _{j=0}^{\mathsf{M}}b_{ij}\,\bar{Z}_{i}(1) \,e_{j,s}=(p_{3}(t),\,C_{s}(t) )_{\hat{\omega}(t)},\quad s=0,1,\ldots \mathsf{M}-1. \end{aligned}
(64)

Merging the $$\mathsf{M}^{2}-\mathsf{M}$$ equations in (55) with the $$(3\,\mathsf{M}+1)$$ constraints in (64), a system of order $$(\mathsf{M}+1)^{2}$$ is obtained, and thus the approximate solution can be derived using the Gaussian elimination technique.

### Remark 1

The AlgorithmÂ 1 specifies the necessary steps to obtain the desired approximate solution to the time-fractional diffusion equation.

## 4 Error analysis

This section is confined to investigating the double expansion used in approximation. The following two lemmas are useful for what comes next.

### Lemma 3

[31] The following inequality is satisfied in the interval $$[0,1]$$

$$|C_{k}(t)|< (k+3),\,\qquad \forall \ k\ge 0.$$
(65)

### Lemma 4

The following inequality is satisfied in the interval $$[0,\tau ]$$

$$|C_{k}(t)|< (k+3),\,\qquad \forall \ k\ge 0.$$
(66)

### Proof

The proof of Lemma 4 is similar to the proof of Lemma 3 after using $$|T^{*}_{\ell}(t)|\leq 1$$.â€ƒâ–¡

### Lemma 5

Let $$\ell \ge 0$$. The following inequality holds

$$|\bar{Z}_{\ell}(\xi )|< (\ell +3),\,\qquad \forall \ \xi \in [-1,1].$$
(67)

### Proof

The proof of this lemma is similar to the proof of Lemma 3 after using Eqs.Â (14) and (15) along with $$|T_{m}(t)|\leq 1$$.â€ƒâ–¡

### Theorem 8

Assume that a function $$u(\xi ,t)=\bar{h}(\xi )\,\bar{\bar{h}}(t)\in L^{2}_{\omega (\xi ,t)}( \Omega )$$, with $$\bar{h}(\xi )$$ and $$\bar{\bar{h}}(t)$$ have bounded fourth derivative satisfy the expansion in (47). Then the series in (49) converges uniformly to $$u(\xi ,t)$$. Moreover, the following bound on the expansion coefficients in (49) satisfy the inequality

$$|b_{ij}|\lesssim \frac{1}{i^{3}\,j^{3}},\quad \forall \ i,j>3,$$
(68)

where $$n\lesssim \hat{n}$$ means that there ia a constant c with $$n\leq c\,\hat{n}$$.

### Proof

From Eq.Â (48), we have

$$b_{ij}=\frac{1}{ \varrho _{i}\,\varrho _{j}}\,\displaystyle \int _{0}^{\tau} \int _{-1}^{1} \,\displaystyle \,u(\xi ,t)\, \bar{Z}_{i}(\xi )\,C_{j}(t)\,{ \omega}(\xi ,t)\,d\xi \, dt.$$
(69)

The previous equation will be transformed after using the following substitutions

$$t=\frac{\tau}{2}(1+\cos \varphi ),\quad \xi =\cos \psi ,$$

and the assumption $$u(\xi ,t)=\bar{h}(\xi )\,\bar{\bar{h}}(t)$$ along with the definition of $$C_{i}(t)$$ (29) into

\begin{aligned} b_{ij}={}&\frac{1}{ \varrho _{i}\,\varrho _{j}}\,\displaystyle \int _{0}^{\pi} \int _{0}^{\pi} \,\displaystyle \bar{h}(\cos \psi )\,\bar{\bar{h}}\left ( \frac{\tau}{2}(1+\cos \varphi )\right ) \\ &{}\times \bar{Z}_{i}(\cos \psi )\, \bar{Z}_{i}(\cos \varphi )\,\cos ^{4}\psi \,\cos ^{4}\varphi \,d \varphi \,d\psi , \end{aligned}
(70)

which can be written as

$$b_{ij}=I_{1}\times I_{2},$$
(71)

where

$$I_{1}=\frac{1}{ \varrho _{i}}\,\displaystyle \int _{0}^{\pi}\bar{h}(\cos \psi ) \bar{Z}_{i}(\cos \psi )\,\cos ^{4}\psi \,d\psi ,$$
(72)

and

$$I_{2}=\frac{1}{ \varrho _{j}}\,\int _{0}^{\pi}\bar{\bar{h}}\left ( \frac{\tau}{2}(1+\cos \varphi )\right )\, \bar{Z}_{i}(\cos \varphi ) \,\cos ^{4}\varphi \,d\varphi .$$
(73)

Now, we have four cases:

(i) If $$i,j\, \textrm{even}$$.

Firstly, we will find $$I_{1}$$.

Using Lemma 1, $$\varrho _{i}=\frac{9\, \pi }{8\,(i+1)\, (i+3)}$$ in Eq.Â (6) along with the following property

$$\cos a\,\cos b=\frac{1}{2}(\cos (a+b)+\cos (a-b))$$

and simplifying the result yields

\begin{aligned} I_{1}={}&\frac{2}{ 3\,\pi}\,\displaystyle \int _{0}^{\pi }\,\displaystyle \, \bar{h}(\cos \psi )\,\cos \psi \left [(i+3)\,\cos ((i+1)\,\psi )+(i+1) \,\cos ((i+3)\,\psi )\right ]\,d\psi \\ ={}&\frac{1}{ 3\,\pi}\,\displaystyle \int _{0}^{\pi }\,\displaystyle \,\bar{h}(\cos \psi ) \, \, [2\,(i+2)\,\cos ((i+2)\psi )\\ &{}+(i+3)\,\cos (i\psi )+(i+1)\, \cos ((i+4)\psi ) ]\,d\psi . \end{aligned}
(74)

If we integrate Eq.Â (74) by parts three times, we get the following relation

$$I_{1}=\frac{1}{24\,\pi} \displaystyle \int _{0}^{\pi}\bar{h}'''(\cos \psi )\, \Delta _{i}(\psi )\,d\psi ,$$
(75)

where

\begin{aligned} \Delta _{i}(\psi )={}&\frac{2}{(i+2)\,(i+3)}\,\left (\cos ((i+2) \psi )-\cos ((i+4)\psi )\right ) \\ &{}-\frac{2\,(i+1)}{(i+3)\,(i+4)\,(i+5} \,\left (\cos ((i+3)\psi )-\cos ((i+5)\psi )\right ) \\ &{} +\frac{(i+3)}{i\,(i-1)\,(i-2)}\,\left (\cos ((i-3)\psi )-\cos ((i-1) \psi )\right ) \\ &{}-\frac{8}{i\,(i-1)\,(i+1)}\,\left (\cos ((i-1)\psi )- \cos ((i+1)\psi )\right ) \\ &{} -\frac{2}{(i+1)\,(i+2)}\,\left (\cos (i\psi )-\cos ((i+2)\psi ) \right )\\ &{} +\frac{(i+1)}{(i+4)\,(i+5)\,(i+6}\,\left (\cos ((i+5)\psi )- \cos ((i+7)\psi )\right ) \\ &{} - \frac{2 \left (i^{2}-5\, i-18\right )}{i\, (i+1)\, (i+2)\, (i+3)\, (i+4)} \,\left (\cos ((i+1)\psi )-\cos ((i+3)\psi )\right ). \end{aligned}
(76)

Now, integrating Eq.Â (76) again by parts and taking into consideration that $$\bar{h}(\xi )$$ and $$\bar{\bar{h}}(t)$$ have bounded fourth derivative, then the following inequality is obtained

$$|I_{1}|\lesssim \frac{1}{i^{3}},\quad \forall \ i>3.$$
(77)

In addition, after applying the trigonometric representations to $$I_{2}$$, the integration by parts three times results in

$$I_{2}=\frac{\tau ^{3}}{192\,\pi} \displaystyle \int _{0}^{\pi}\bar{\bar{h}}''' \left (\frac{\tau}{2}(1+\cos \varphi )\right )\,\hat{\Delta}_{i}( \varphi )\,d\varphi ,$$
(78)

where

\begin{aligned} \hat{\Delta}_{i}(\varphi )={}& \frac{(i+3)}{i\,(i-1)\,(i-2)}\,\left (\cos ((i-3)\varphi )-\cos ((i-1) \varphi )\right ) \\ &{}-\frac{2\,(i+1)}{(i+3)\,(i+4)\,(i+5}\,\left (\cos ((i+3) \varphi )-\cos ((i+5)\varphi )\right ) \\ &{} -\frac{2}{(i+1)\,(i+2)}\,\left (\cos (i\varphi )-\cos ((i+2) \varphi )\right ) \\ &{} +\frac{(i+1)}{(i+4)\,(i+5)\,(i+6}\,\left (\cos ((i+5) \varphi )-\cos ((i+7)\varphi )\right ) \\ &{} -\frac{8}{i\,(i-1)\,(i+1)}\,\left (\cos ((i-1)\varphi )-\cos ((i+1) \varphi )\right ) \\ &{}+\frac{2}{(i+2)\,(i+3)}\,\left (\cos ((i+2)\varphi )- \cos ((i+4)\varphi )\right ) \\ &{} - \frac{2 \left (i^{2}-5\, i-18\right )}{i\, (i+1)\, (i+2)\, (i+3)\, (i+4)} \,\left (\cos ((i+1)\varphi )-\cos ((i+3)\varphi )\right ). \end{aligned}
(79)

And hence, we get the following inequality

$$|I_{2}|\lesssim \frac{1}{j^{3}},\quad \forall \ j>3,$$
(80)

after integrating Eq.Â (79) by parts and using the hypothesis of the theorem.

Finally, we have from Eqs.Â (71),Â (77) and (80)

$$|b_{ij}|=|I_{1}|\times |I_{2}|\lesssim \frac{1}{i^{3}\,j^{3}},\quad \forall \ i,j>3.$$
(81)

To prove the following three cases

(ii) $$i,j\, \textrm{odd}$$.

(iii) $$i\, \textrm{even},\,j \,\textrm{odd}$$.

(iiii) $$i\, \textrm{odd},\,j\, \textrm{even}$$.

Imitating similar steps to those given in case (i), we get the following inequality

$$|b_{ij}|\lesssim \frac{1}{i^{3}\,j^{3}},\quad \forall \ i,j>3.$$

This completes the proof of this theorem.â€ƒâ–¡

### Remark 2

The following inequalities will be obtained if we follow similar steps to those given in Theorem 8

$$|b_{0j}|\lesssim \frac{1}{j^{3}},\quad |b_{1j}|\lesssim \frac{1}{j^{3}}, \quad |b_{2j}|\lesssim \frac{1}{j^{3}},\quad |b_{3j}|\lesssim \frac{1}{j^{3}},\quad \forall j>3,$$
(82)

and

$$|b_{i0}|\lesssim \frac{1}{i^{3}},\quad |b_{i1}|\lesssim \frac{1}{i^{3}}, \quad |b_{i2}|\lesssim \frac{1}{i^{3}},\quad |b_{i3}|\lesssim \frac{1}{i^{3}},\quad \forall \ i>3.$$
(83)

### Lemma 6

[45] Consider a positive, decreasing, continuous function $$f(\xi )$$. If $$\xi \geq n$$ with $$f(k)=a_{k}$$ such that $$\sum{a_{n}}$$ is convergent and $$R_{n}=\sum _{k=n+1}^{\infty}a_{k}$$, then $$R_{n}\leq \int _{n}^{\infty}f(\xi )d\,\xi$$.

### Theorem 9

If $$u(\xi ,t)$$ satisfies the hypothesis of Theorem 8, then the following estimate of truncation error is fulfilled:

$${\|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{\omega (\xi ,t)}( \Omega )}\lesssim \frac{1}{\mathsf{M}^{3}}.$$
(84)

### Proof

We can write

\begin{aligned}& {\|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{\omega ( \xi ,t)}(\Omega )} \\ &\quad\leq \left \|\sum _{i=0}^{\mathsf{M}}\sum _{j=\mathsf{M}+1}^{\infty}b_{ij} \,\bar{Z}_{i}(\xi )\,C_{j}(t)\right \|_{L^{2}_{\omega (\xi ,t)}( \Omega )}+\left \|\sum _{i=\mathsf{M}+1}^{\infty}\sum _{j=0}^{\infty}b_{ij} \,\bar{Z}_{i}(\xi )\,C_{j}(t)\right \|_{L^{2}_{\omega (\xi ,t)}( \Omega )} \\ &\quad =\sum _{i=0}^{\mathsf{M}}\sum _{j=\mathsf{M}+1}^{\infty}|b_{ij}|\, \sqrt{\varrho _{i}\,\varrho _{j}}+\sum _{i=\mathsf{M}+1}^{\infty} \sum _{j=0}^{\infty}|b_{ij}|\,\sqrt{\varrho _{i}\,\varrho _{j}} \\ &\quad =\sum _{j=\mathsf{M}+1}^{\infty}\left (\,|b_{0j}|\,\sqrt{\varrho _{0}}+|b_{1j}| \,\sqrt{\varrho _{1}}+|b_{2j}|\,\sqrt{\varrho _{2}}+|b_{3j}|\,\sqrt{ \varrho _{3}}\,\right )\,\sqrt{\varrho _{j}} \\ &\qquad{}+\sum _{i=4}^{\mathsf{M}} \sum _{j=\mathsf{M}+1}^{\infty}|b_{ij}|\,\sqrt{\varrho _{i}\,\varrho _{j}} \\ &\qquad{} +\sum _{i=\mathsf{M}+1}^{\infty}\left (\,|b_{i0}|\,\sqrt{\varrho _{0}}+|b_{i1}| \,\sqrt{\varrho _{1}}+|b_{i2}|\,\sqrt{\varrho _{2}}+|b_{i3}|\,\sqrt{ \varrho _{3}}\,\right )\,\sqrt{\varrho _{i}}\\ &\qquad{} +\sum _{i=\mathsf{M}+1}^{ \infty}\sum _{j=4}^{\infty}|b_{ij}|\,\sqrt{\varrho _{i}\,\varrho _{j}}, \end{aligned}
(85)

Inserting Eqs.Â (82) and (83) into Eq.Â (85) and using the inequality $$|\varrho _{i}|\lesssim \frac{1}{i^{2}},\quad \forall i>0$$, one has

\begin{aligned} {\|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{\omega ( \xi ,t)}(\Omega )}\lesssim \sum _{j=\mathsf{M}+1}^{\infty} \frac{1}{j^{4}}+\sum _{i=4}^{\mathsf{M}}\sum _{j=\mathsf{M}+1}^{ \infty}\frac{1}{i^{4}\,j^{4}} +\sum _{i=\mathsf{M}+1}^{\infty} \frac{1}{i^{4}}+\sum _{i=\mathsf{M}+1}^{\infty}\sum _{j=4}^{\infty} \frac{1}{i^{4}\,j^{4}}. \end{aligned}
(86)

Using Lemma 6 together with the inequality

$$\sum _{j=n+1}^{b}f(j)\leq \int _{x=n}^{b}f(\xi )\,d\xi ,$$

where f is decreasing function, we get

$${\|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{\omega (\xi ,t)}( \Omega )}\lesssim \frac{1}{\mathsf{M}^{3}}.$$

This completes the proof of Theorem 9.â€ƒâ–¡

### Theorem 10

The truncation error estimate can be estimated as

$$|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)|\lesssim \frac{1}{\mathsf{M}}.$$
(87)

### Proof

First, we can write

\begin{aligned} |u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)|&\leq \left |\sum _{j= \mathsf{M}+1}^{\infty}(\,b_{0j}\,\bar{Z}_{0}(\xi )+b_{1j}\,\bar{Z}_{1}( \xi )+b_{2j}\,\bar{Z}_{2}(\xi )+b_{3j}\,\bar{Z}_{3}(\xi )\,)\,C_{j}(t) \right | \\ & +\left |\sum _{i=\mathsf{M}+1}^{\infty}(\,b_{i0}\,C_{0}(t)+b_{i1}\,C_{1}(t)+b_{i2} \,C_{2}(t)+b_{i3}\,C_{3}(t)\,)\,\bar{Z}_{i}(\xi )\right | \\ & +\left |\sum _{i=4}^{\mathsf{M}}\sum _{j=\mathsf{M}+1}^{\infty}b_{ij} \,\bar{Z}_{i}(\xi )\,C_{j}(t)\right |+\left |\sum _{i=\mathsf{M}+1}^{ \infty}\sum _{j=4}^{\infty}b_{ij}\,\bar{Z}_{i}(\xi )\,C_{j}(t)\right |. \end{aligned}
(88)

With the aid of Lemmas 4, 5, Theorem 8 and Eqs.Â (82), (83), the last equation becomes

\begin{aligned} |u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)|\lesssim & \sum _{i= \mathsf{M}+1}^{\infty}\sum _{j=4}^{\infty} \frac{(i+3)\,(j+3)}{i^{3}\,j^{3}}+\sum _{j=\mathsf{M}+1}^{\infty} \frac{j+3}{j^{3}} \\ & +\sum _{i=4}^{\mathsf{M}}\sum _{j=\mathsf{M}+1}^{\infty} \frac{(i+3)\,(j+3)}{i^{3}\,j^{3}}+\sum _{i=\mathsf{M}+1}^{\infty} \frac{i+3}{i^{3}}. \\ & \end{aligned}
(89)

Now, noting the inequality:

$$\frac{i+3}{i^{3}}< \frac{i+3}{i(i^{2}-1)},\quad \forall \ i>1,$$

we get

\begin{aligned} &\sum _{i=\mathsf{M}+1}^{\infty } \frac{i+3}{i \left (i^{2}-1\right )}= \frac{\mathsf{M}+2}{\mathsf{M}^{2}+\mathsf{M}}< \frac{2}{\mathsf{M}}, \quad \forall \ \mathsf{M}>0, \\ & \sum _{i=4}^{\mathsf{M}} \frac{i+3}{i \left (i^{2}-1\right )}= \frac{1}{\mathsf{M}+1}-\frac{2}{\mathsf{M}}+\frac{5}{12}< 1, \\ & \sum _{i=4}^{\infty } \frac{i+3}{i \left (i^{2}-1\right )}= \frac{5}{12}< 1. \end{aligned}
(90)

Inserting Eqs.Â (90) into Eq.Â (89), we get

$$|u(\xi ,t)-u_{\mathsf{M}}(\xi ,t)|\lesssim \frac{1}{\mathsf{M}}.$$

â€ƒâ–¡

### Theorem 11

(Stability) Under the assumptions of Theorem 8, we have

$${\|u_{\mathsf{M}+1}(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{\omega ( \xi ,t)}(\Omega )}\lesssim \frac{1}{\mathsf{M}^{3}}.$$

### Proof

We have

\begin{aligned} &{\|u_{\mathsf{M}+1}(\xi ,t)-u_{\mathsf{M}}(\xi ,t)\|}_{L^{2}_{ \omega (\xi ,t)}(\Omega )} \\ \begin{aligned} &\quad =\Biggl\| \displaystyle \sum _{i=0}^{\mathsf{M}} b_{i,\mathsf{M}+1}\,\bar{Z}_{i}( \xi )\,C_{\mathsf{M}+1}(t)\\ &\qquad{}+\displaystyle \sum _{j=0}^{\mathsf{M}} b_{\mathsf{M}+1,j} \,\bar{Z}_{\mathsf{M}+1}(\xi )\,C_{j}(t)+b_{\mathsf{M}+1,\mathsf{M}+1} \,\bar{Z}_{\mathsf{M}+1}(\xi )\,C_{\mathsf{M}+1}(t)\Biggr\| _{L^{2}_{ \omega (\xi ,t)}(\Omega )} \end{aligned} \\ &\quad \leq \displaystyle \sum _{i=0}^{\mathsf{M}} |b_{i,\mathsf{M}+1}|\,\sqrt{ \varrho _{i}}\,\sqrt{\varrho _{\mathsf{M}+1}}+\displaystyle \sum _{j=0}^{ \mathsf{M}} |b_{\mathsf{M}+1,j}|\,\sqrt{\varrho _{j}}\,\sqrt{\varrho _{ \mathsf{M}+1}}+|b_{\mathsf{M}+1,\mathsf{M}+1}|\,|\varrho _{\mathsf{M}+1}|. \end{aligned}
(91)

Now, with the aid of Theorem 8, Eqs.Â (82) and (83) and using similar steps to those followed in the proof of Theorem 9, we get the desired result.â€ƒâ–¡

## 5 Illustrative examples

The following section presents three examples to ensure the validity and accuracy of our presented algorithm.

### Example 1

Consider the following equation

$$\partial _{t}^{\alpha}u(\xi ,t)-\partial _{\xi}^{2}u(\xi ,t)=12\, t^{4} \, \xi ^{2} \,\left ( \frac{2\, \xi ^{2}\, t^{-\alpha}}{\Gamma (5-\alpha )}-1\right ),$$
(92)

governed by

\begin{aligned} &u(\xi ,0)=0,\quad \forall \, \xi \in (-1,1), \end{aligned}
(93)
\begin{aligned} & u(-1,t)=u(1,t)=t^{4},\quad \forall \, t\in (0,1), \end{aligned}
(94)

with the exact solution: $$u(\xi ,t)=t^{4} \,\xi ^{4}$$.

FigureÂ 1 shows the absolute errors (AE) at different values of Î± at $$\mathsf{M}=4$$. TableÂ 1 shows the AE at $$\alpha =0.5$$ and $$\mathsf{M}=4$$. In addition, TableÂ 2 shows the $$L_{\infty}$$ and $$L_{2}$$ errors for $$\alpha =0.6$$ at $$\mathsf{M}=4$$ and $$\mathsf{M}=5$$. The approximation of the exact solution is seen to be close.

### Example 2

Consider the following equation

$$\partial _{t}^{\alpha}u(\xi ,t)-\partial _{\xi}^{2}u(\xi ,t)=\xi ^{3} \, \alpha \, t^{1 -\alpha } E_{1,2-\alpha}(\alpha \, t)-6\, \xi \, e^{ \alpha t},$$
(95)

governed by

\begin{aligned} &u(\xi ,0)=\xi ^{3},\quad \forall \, \xi \in (-1,1), \end{aligned}
(96)
\begin{aligned} & u(-1,t)=-e^{\alpha \, t}, \quad u(1,t)=e^{\alpha \, t},\quad \forall \, t \in (0,1), \end{aligned}
(97)

where $$E_{p,q}(t)$$ is the Mittagâ€“Leffler function and the exact solution of this problem is $$u(\xi ,t)=\xi ^{3}\, e^{\alpha \, t}$$.

In Fig.Â 2, we plot the AE at different values of Î± when $$\mathsf{M}=18$$. These figures show the fast reduction of AE at small values of M to achieve very high accuracy. TableÂ 3 shows the $$L_{\infty}$$ and $$L_{2}$$ errors for $$\alpha =0.5$$ at $$\mathsf{M}=18$$ and $$\mathsf{M}=19$$. These results show the accuracy of our method.

### Example 3

Consider the following equation

$$\partial _{t}^{\alpha}u(\xi ,t)-\partial _{\xi}^{2}u(\xi ,t)=-2\, \sin (\pi \, t)+\frac{1}{2}\, \pi \, \xi ^{2}\, t^{1-\alpha }\, (E_{1,2- \alpha }(-i\, \pi \, t)+E_{1,2-\alpha }(i\, \pi \, t)),$$
(98)

governed by

\begin{aligned} &u(\xi ,0)=0,\quad \forall \, \xi \in (-1,1), \end{aligned}
(99)
\begin{aligned} & u(-1,t)=u(1,t)=\sin (\pi \, t),\quad \forall \, t\in (0,1), \end{aligned}
(100)

where $$E_{p,q}(t)$$ is the Mittagâ€“Leffler function, and the exact solution of this problem is $$u(\xi ,t)=\xi ^{2}\, \sin (\pi \, t)$$.

In Fig.Â 3, we sketch the AE (left) and Approximate solution (right) for $$\alpha =0.5$$ at $$\mathsf{M}=18$$. In TablesÂ 4 and 5, we list the $$L_{\infty}$$ and $$L_{2}$$ errors for different values of Î± at $$\mathsf{M}=18$$. TableÂ 6 presents the maximum AE at $$\alpha =0.3$$ and different values of M. These findings demonstrate how closely the approximate solution approaches the exact solution.

### Example 4

Consider the following equation [44]

\begin{aligned} &\partial _{t}^{\alpha}u(\xi ,t)-\partial _{\xi}^{2}u(\xi ,t)\\ &\quad=\pi ^{2} \,\sin (\pi \, t) \sin (\pi \, \xi )+\frac{1}{2}\, \pi \, \sin (\pi \, \xi )\, t^{1-\alpha }\, (E_{1,2-\alpha }(-i\, \pi \, t)+E_{1,2- \alpha }(i\, \pi \, t)), \end{aligned}
(101)

governed by

\begin{aligned} &u(\xi ,0)=0,\quad \forall \, \xi \in (-1,1), \end{aligned}
(102)
\begin{aligned} & u(-1,t)=u(1,t)=0,\quad \forall \, t\in (0,1), \end{aligned}
(103)

where $$E_{p,q}(t)$$ is the Mittagâ€“Leffler function and the exact solution of this problem is $$u(\xi ,t)=\sin (\pi \, \xi )\, \sin (\pi \, t)$$.

TableÂ 7 shows a comparison of the best $$L_{2}$$ errors between our at $$\mathsf{M}=18$$ method and method in [44] at $$M=11,\,N=15$$ when $$\alpha =0.9$$. Moreover, Fig.Â 4, illustrates the AE (left) and approximate solution (right) for $$\alpha =0.1$$ at $$\mathsf{M}=18$$. TableÂ 8 presents the maximum AE for $$\alpha =0.3$$ and different values of M. These results prove that the approximate solution is very close to the exact solution.

### Remark 3

In light of Figs.Â 1â€“4, we can conclude the following benefit:

Excellently precise approximations can be obtained by selecting modified sets of seventh-kind CPs as basis functions and taking specific terms of the retained modes.

## 6 Concluding remarks

In conclusion, our study introduces a novel application of the spectral tau methodology tailored to tackle the TFDE by leveraging seventh-kind CPs as basis functions. The choice of CPs stems from their rapid convergence, numerical stability, and orthogonality properties, allowing for constructing a robust basis expansion explicitly designed for solving TFDE. Through a comprehensive error analysis and numerical validation, we have demonstrated our proposed methodâ€™s efficacy and computational prowess in accurately capturing the complex dynamics of TFDE solutions. Moving forward, our work opens avenues for further exploration and refinement of spectral techniques for solving TFDE, offering a promising approach with diverse applications across scientific and engineering domains. Continued efforts in advancing numerical methods and innovative algorithmic developments can significantly enhance our understanding and computational capabilities in modeling complex dynamical systems governed by fractional differential equations.

## Data availability

No datasets were generated or analysed during the current study.

## References

1. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2022)

2. Shishkina, E., Sitnik, S.: Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics. Academic Press, San Diego (2020)

3. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, San Diego (1998)

4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol.Â 204. Elsevier, Amsterdam (2006)

5. Nisar, K.S., Farman, M., Abdel-Aty, M., Ravichandran, C.: A review of fractional order epidemic models for life sciences problems: past, present and future. Alex. Eng. J. 95, 283â€“305 (2024)

6. Arshad, S., Siddique, I., Nawaz, F., Shaheen, A., Khurshid, H.: Dynamics of a fractional order mathematical model for covid-19 epidemic transmission. Physica A 609, 128383 (2023)

7. Qayyum, M., Ahmad, E., Saeed, S.T., AkgÃ¼l, A., El Din, S.M.: New solutions of fractional 4D chaotic financial model with optimal control via He-Laplace algorithm. Ain Shams Eng. J. 15(3), 102503 (2024)

8. Yadav, P., Jahan, S., Nisar, K.S.: Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics. Ain Shams Eng. J. 15(1), 102299 (2024)

9. Mahdy, A.M.S.: Numerical solutions for solving model time-fractional Fokkerâ€“Planck equation. Numer. Methods Partial Differ. Equ. 37(2), 1120â€“1135 (2021)

10. Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H.: Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations. Appl. Numer. Math. 167, 237â€“256 (2021)

11. Eid, A., Khader, M.M., Megahed, A.M.: Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneoâ€“Christov model. Open Phys. 22(1), 20240001 (2024)

12. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26(2), 448â€“479 (2010)

13. Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974)

14. Kumar, Y., Singh, V.K.: Computational approach based on wavelets for financial mathematical model governed by distributed order fractional differential equation. Math. Comput. Simul. 190, 531â€“569 (2021)

15. Li, Z., Chen, Q., Wang, Y., Li, X.: Solving two-sided fractional super-diffusive partial differential equations with variable coefficients in a class of new reproducing kernel spaces. Fractal Fract. 6(9), 492 (2022)

16. Shah, R., Khan, H., Arif, M., Kumam, P.: Application of Laplaceâ€“Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 21(4), 335 (2019)

17. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, New York (2003)

18. BÃ©g, O.A., Hameed, M., BÃ©g, T.A.: Chebyshev spectral collocation simulation of nonlinear boundary value problems in electrohydrodynamics. Int. J. Comput. Methods Eng. Sci. Mech. 14(2), 104â€“115 (2013)

19. Shuman, D.I., Vandergheynst, P., Kressner, D., Frossard, P.: Distributed signal processing via Chebyshev polynomial approximation. IEEE Trans. Signal Inf. Process. Netw. 4(4), 736â€“751 (2018)

20. Montijano, E., Montijano, J.I., Sagues, C.: Chebyshev polynomials in distributed consensus applications. IEEE Trans. Signal Process. 61(3), 693â€“706 (2012)

21. Abdelhakem, M., Alaa-Eldeen, T., Baleanu, D., Alshehri, M.G., El-Kady, M.: Approximating real-life BVPs via Chebyshev polynomialsâ€™ first derivative pseudo-Galerkin method. Fractal Fract. 5(4), 165 (2021)

22. Ahmed, H.M.: Numerical solutions for singular Lane-Emden equations using shifted Chebyshev polynomials of the first kind. Contemp. Math. 4, 132â€“149 (2023)

23. Bezerra, F.D.M., Santos, L.A.: Chebyshev polynomials for higher order differential equations and fractional powers. Math. Ann. 388(1), 675â€“702 (2024)

24. Tural-Polat, S.N., Dincel, A.T.: Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind. Alex. Eng. J. 61(7), 5145â€“5153 (2022)

25. Hosseininia, M., Heydari, M.H., Razzaghi, M.: A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation. Math. Methods Appl. Sci. 46(18), 18768â€“18788 (2023)

26. Doha, E.H., Abd-Elhameed, W.M., Bassuony, M.A.: On using third and fourth kinds Chebyshev operational matrices for solving Lane-Emden type equations. Rom. J. Phys. 60(3â€“4), 281â€“292 (2015)

27. Xu, Y.: An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials. Adv. Appl. Math. 29(2), 328â€“343 (2002)

28. Draux, A., Sadik, M., Moalla, B.: Markovâ€“Bernstein inequalities for generalized Gegenbauer weight. Appl. Numer. Math. 61(12), 1301â€“1321 (2011)

29. Obeid, M., Abd El Salam, M.A., Younis, J.A.: Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials. Appl. Math. Sci. Eng. 31(1), 2187388 (2023)

30. Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H.: Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem. Math. Sci. 17(4), 415â€“429 (2023)

31. Abd-Elhameed, W.M., Youssri, Y.H., Atta, A.G.: Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials. J. Math. Model. 12(2), 277â€“299 (2024)

32. Abd-Elhameed, W.M., Youssri, Y.H., Amin, A.K., Atta, A.G.: Eighth-kind Chebyshev polynomials collocation algorithm for the nonlinear time-fractional generalized Kawahara equation. Fractal Fract. 7(9), 652 (2023)

33. Ahmed, H.M., Hafez, R.M., Abd-Elhameed, W.M.: A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices. Phys. Scr. 99(4), 045250 (2024)

34. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

35. Ahmed, H.F., Hashem, W.A.: Improved Gegenbauer spectral tau algorithms for distributed-order time-fractional telegraph models in multi-dimensions. Numer. Algorithms 93(3), 1013â€“1043 (2023)

36. Atta, A.G.: Two spectral Gegenbauer methods for solving linear and nonlinear time fractional Cable problems. Int. J. Mod. Phys. C 35(6), 2450070 (2024)

37. Amin, A.Z., Lopes, A.M., Hashim, I.: A space-time spectral collocation method for solving the variable-order fractional Fokker-Planck equation. J. Appl. Anal. Comput. 13, 969â€“985 (2023)

38. Abdelkawy, M.A.: A collocation method based on Jacobi and fractional order Jacobi basis functions for multi-dimensional distributed-order diffusion equations. Int. J. Nonlinear Sci. Numer. Simul. 19(7â€“8), 781â€“792 (2018)

39. Alsuyuti, M.M., Doha, E.H., Ezz-Eldien, S.S., Youssef, I.K.: Spectral Galerkin schemes for a class of multi-order fractional pantograph equations. J. Comput. Appl. Math. 384, 113157 (2021)

40. Abd-Elhameed, W.M., Al-Harbi, M.S., Atta, A.G.: New convolved Fibonacci collocation procedure for the Fitzhugh-Nagumo non-linear equation. Nonlinear Eng. 13, 20220332 (2024)

41. Azimi, R., Mohagheghy Nezhad, M., Pourgholi, R.: Legendre spectral tau method for solving the fractional integro-differential equations with a weakly singular kernel. Global Anal. Discrete Math. (2022)

42. Abdelghany, E.M., Abd-Elhameed, W.M., Moatimid, G.M., Youssri, Y.H., Atta, A.G.: A tau approach for solving time-fractional heat equation based on the shifted sixth-kind Chebyshev polynomials. Symmetry 15(3), 594 (2023)

43. Koepf, W.: Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Vieweg, Braunschweig (1998)

44. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108â€“2131 (2009)

45. Stewart, J.: Single Variable Calculus: Early Transcendentals. Cengage Learning, United States of America (2015)

Not applicable.

## Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

## Author information

Authors

### Contributions

All authors contributed equally to the work.

### Corresponding author

Correspondence to Y. H. Youssri.

## Ethics declarations

Not applicable.

### Consent for publication

The authors agreed to publish their work in Boundary Value Problems.

### Competing interests

The authors declare no competing interests.

### Publisherâ€™s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and permissions

Abd-Elhameed, W.M., Youssri, Y.H. & Atta, A.G. Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials. Bound Value Probl 2024, 102 (2024). https://doi.org/10.1186/s13661-024-01907-6