Collocation Method for the Time-Fractional Generalized Kawahara Equation Using a Certain Lucas Polynomial Sequence

Abstract

This paper proposes a numerical technique to solve the time-fractional generalized Kawahara differential equation (TFGKDE). Certain shifted Lucas polynomials are utilized as basis functions. We first establish some new formulas concerned with the introduced polynomials and then tackle the equation using a suitable collocation procedure. The integer and fractional derivatives of the shifted polynomials are used with the typical collocation method to convert the equation with its governing conditions into a system of algebraic equations. The convergence and error analysis of the proposed double expansion are rigorously investigated, demonstrating its accuracy and efficiency. Illustrative examples are provided to validate the effectiveness and applicability of the proposed algorithm.

1. Introduction

Fractional differential equations (FDEs) are essential in modern mathematics and science because they successfully represent complicated systems with memory and hereditary qualities that standard differential equations cannot handle by extending integer-order derivatives to non-integer orders. There are many types of fractional derivatives. The most popular way to define the fractional derivative is in the Caputo sense. According to [1,2], for instance, the Caputo definition is more mathematically rigorous than the Riemann–Liouville definition. Applied science and engineering especially recognize Caputo’s concept [3]. In addition, the Caputo derivative’s features aid in transforming higher-fractional-order differential systems into lower ones [4].

Fractional differential equations provide a versatile and exact framework for explaining phenomena in various domains, including physics, biology, economics, and engineering. They help investigate anomalous diffusion in porous media, viscoelastic materials, and control systems with long-term memory. FDEs may also be used to represent electromagnetic waves in dielectric materials, as well as for signal processing and population dynamics. These fractional-order models are critical for capturing the complex dynamics of real-world systems, allowing researchers to better understand and predict their behavior across time. For some applications of FDEs, one can consult [5,6,7].

The Kawahara differential equation (KDE), a fifth-order nonlinear partial differential equation, has applications in various scientific domains where higher-order dispersion effects are significant. It is often used to simulate the propagation of small-amplitude long waves in shallow water, especially in situations where surface tension substantially impacts wave dynamics. Several analytical and numerical techniques have been developed to solve the different versions of the KDE. For example, in [8], the authors used the Tanh method to find some exact traveling wave solutions to the KDE. In [9], some solitary solutions were proposed for the time-fractional extended KDE. The homotopy analysis method was used in [10] to solve two versions of Kawahara differential equations (DEs). In [11], the authors used the homotopy perturbation transform method to solve a modified Kawahara DE. The Galerkin method with splines was used to handle the time-fractional nonlinear KDE in [12]. The authors of [13] used a modified Laplace homotopy perturbation method to treat the damped modified KDE. The authors of [14] developed other analytical and approximate solutions for the modified KDE. A numerical method was used in [15] to solve the nonlinear fractional KDE. A numerical approach was followed in [16] for the nonlinear generalized time-fractional Kawahara-type equations. A Haar wavelet method was followed in [17] to solve the fractional KDE.

The Lucas polynomial sequence is a sequence of polynomials that generalize many celebrated sequences, such as Fibonacci and Lucas polynomials. Horadam first introduced these polynomials in [18]. Because of their distinct features and connections with orthogonal polynomials, they were vital in both theoretical and numerical aspects. They have important rules in number theory and combinatorics; see, for instance, [19,20]. From a numerical perspective, the different Lucas polynomial sequences were utilized to solve various types of DEs. In [21], the authors developed a numerical algorithm to solve stochastic FDEs using Pell polynomials. The Pell–Lucas polynomials were used in [22] to solve the nonlinear fractional Duffing equations. Fibonacci polynomials were used in [23] to solve FDEs. In [24], the author used the Fibonacci wavelets method to solve some singular boundary value problems. Vieta Fibonacci polynomials were used in [25] to solve some stochastic fractional integro-DEs. Certain Horadam polynomials were employed in [26] to solve the nonlinear fifth-order KdV equations.

Spectral methods are robust numerical approaches for solving DEs, especially partial DEs, by expressing the solution as a series expansion of basis functions, usually derived from special polynomials that may be orthogonal or non-orthogonal. These approaches are distinguished for their remarkable precision in addressing problems with smooth solutions since solutions converge exponentially. Spectral methods have applications in scientific and engineering disciplines such as fluid dynamics, climate modeling, quantum mechanics, and structural analysis, for example, [27,28,29]. Prominent variants of spectral methods comprise collocation methods, which impose the governing equations at designated discrete points within the domain; this approach is beneficial owing to its extensive application to all types of DEs. It can be used to solve ordinary DEs; see, for example, [30,31]. FDEs are treated by the collocation method in many contributions; see, for example, [32,33,34,35]. The Galerkin method is another spectral approach that enforces the residual of the given differential equation to be orthogonal to the basis function; see, for example, [36,37,38]. The tau method is an important spectral approach that differs from the Galerkin method in flexibility in choosing basis functions; see, for example, [39,40,41].

Operational matrices of derivatives (OMDs) are mathematical tools that enable the numerical solution of different DEs to be obtained. These OMDEs are utilized to convert the DEs with their governing conditions into algebraic systems of equations, which suitable solvers can solve. This approach is advantageous in spectral methods when approximate solutions are expressed as combinations of special functions. The applications of operational matrices of derivatives encompass many domains, such as the numerical solution of ordinary DEs, partial DEs, and FDEs. For example, the authors of [42] used certain operational matrices of derivatives to treat some specific initial value problems. The authors of [43] utilized some Legendre polynomials’ operational matrices to deal with some BVPs. Regarding the partial DEs, there are many contributions regarding the utilization of operational matrices of derivatives to deal with such problems. For example, a matrix-based strategy was implemented in [44] to treat the FitzHugh–Nagumo nonlinear equation based on an operational matrix of derivatives of the convolved Fibonacci polynomials. The authors of [45] derived an operational matrix of fractional derivatives of Legendre polynomials and utilized them to solve FDEs. The authors of [46] established a Jacobi operational matrix of derivatives to solve variable-order FDEs. A finite class of classical orthogonal polynomial operational matrices was introduced in [47] and utilized to solve FDEs. OMDs of the eighth kind of Chebyshev polynomials were established and used to treat the nonlinear time-fractional generalized KDE. For some other contributions, see [48,49,50].

This paper aims to introduce and use a class of shifted Lucas polynomial sequences to treat the TFGKDE. We will develop many new theoretical results that will be pivotal in designing the proposed collocation algorithm. To test our presented algorithm, we compare our method with the reproducing kernel method that was developed in [51].

We think that the novelty of this paper can be summarized in the following items:

• The employment of the introduced Lucas polynomial sequence in numerical analysis is new.

• Derivations of some new theoretical results, such as the high-order and operational matrices of derivatives of the utilized Horadam sequence of polynomials.

• A new study for the convergence analysis of the proposed double expansion.

The parts of this paper are divided as follows: Section 2 gives an account of a certain Lucas polynomial sequence. In addition, a certain shifted Lucas polynomial sequence will be introduced. Some new formulas that will be useful in designing our numerical algorithm are established in Section 3. Section 4 designs the numerical algorithm to solve the TFGKDE. Some numerical experiments will be given in Section 6. Section 7 concludes the discussions.

2. An Overview of Lucas Polynomial Sequences

Horadam’s key publication [18] introduced several generalized polynomials that the following recursive formula can generate:

(1)${\varphi }_n\left(z\right)=r\left(z\right){\varphi }_{n-1}\left(z\right)+q\left(z\right){\varphi }_{n-2}\left(z\right),{\varphi }_0\left(z\right)=2,{\varphi }_1\left(z\right)=r\left(z\right).$

The Binet form of the polynomials ${\varphi }_n\left(z\right)$ is

(2)${\varphi }_n\left(z\right)={\displaystyle \frac{{\left[r\left(z\right)+\sqrt{r^2\left(z\right)+4q\left(z\right)}\right]}^n-{\left[r\left(z\right)-\sqrt{r^2\left(z\right)+4q\left(z\right)}\right]}^n}{2^n\sqrt{r^2\left(z\right)+4q\left(z\right)}}},n\ge 0.$

Several celebrated polynomials can be considered as particular ones of ${\varphi }_n\left(z\right)$. Of these important polynomials are the generalized Lucas polynomials $L_k^{a,b}\left(z\right)$ that can be generated by the following recursive formula:

(3)$\begin{array}{c}\hfill L_n^{a,b}\left(z\right)=az{L}_{n-1}^{a,b}\left(z\right)+b{L}_{n-2}^{a,b}\left(z\right),L_0^{a,b}\left(z\right)=2,L_1^{a,b}\left(z\right)=az,n\ge 2,\end{array}$

where a and b are non-zero real numbers.

The above generalized Lucas polynomials include important polynomials, such as Lucas, Pell–Lucas, Fermat, and first-kind Chebyshev polynomials. The following formulas hold:

(4)$\begin{array}{cccc}\hfill L_i\left(z\right)& =L_i^{(1,1)}\left(z\right),\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{Q}_i\left(z\right)& =L_i^{(2,1)}\left(z\right),\hfill \end{array}$

(5)$\begin{array}{cccc}\hfill {\mathcal{F}}_i\left(z\right)& =L_i^{(3,-2)}\left(z\right),\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{T}_i\left(z\right)& =2L_i^{(2,-1)}\left(z\right).\hfill \end{array}$

Recently, the authors of [52] found useful formulas for the shifted Lucas polynomials, defined as

$L_i^{*}\left(z\right)=L_i(2z-1),$

and they employed them to solve the time-fractional FitzHugh–Nagumo differential equation. This paper aims to introduce and employ another class of shifted Horadam polynomials. More precisely, we will consider the sequence ${\left\{{\mathcal{L}}_n\left(z\right)\right\}}_{n\ge 0}$ defined as

(6)${\mathcal{L}}_n\left(z\right)=L_n^{-2,-1}(2z-1).$

Providing some useful formulas for the shifted polynomials ${\mathcal{L}}_n\left(z\right)$ is the focus of the next section.

3. Some New Formulas of ${\mathcal{L}}_{\mathit{n}}\left(\mathit{z}\right)$

In this section, we aim to derive the following key formulas that will be essential for designing our proposed numerical algorithm:

• The analytic form of ${\mathcal{L}}_n\left(z\right)$.

• The inversion formula of ${\mathcal{L}}_n\left(z\right)$.

• The expressions of the high-order derivatives of ${\mathcal{L}}_n\left(z\right)$ as combinations of their original ones.

Theorem 1.

For every positive integer i, ${\mathcal{L}}_i\left(z\right)$ can be expressed in the following form:

(7) ${\mathcal{L}}_i\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }(i-1)!}}\sum _{s=0}^i\left({\displaystyle \genfrac{}{}{0pt}{}{i}{i-s}}\right)(s+i-1)!\Gamma \left({\displaystyle \frac{1}{2}}-s\right)z^s.$

Proof. 

Formula (7) will be proved by induction. Assume that it is true for all $k<i$, that is,

(8)${\mathcal{L}}_k\left(z\right)=\sum _{s=0}^k{B}_{s,k}{z}^s,k<i,$

with

$B_{s,k}={\displaystyle \frac{2\Gamma \left(\frac{1}{2}-s\right)(k+s-1)!\left(\genfrac{}{}{0pt}{}{k}{k-s}\right)}{\sqrt{\pi }(k-1)!}},$

and we have to prove (7).

If we use the recursive formula of ${\mathcal{L}}_i\left(z\right)$ given by

${\mathcal{L}}_i\left(z\right)=-2(2z-1){\mathcal{L}}_{i-1}\left(z\right)-{\mathcal{L}}_{i-2}\left(z\right),$

and apply the induction hypothesis (8), we obtain the following equation:

(9)${\mathcal{L}}_i\left(z\right)=-2(2z-1)\sum _{s=0}^{i-1}{B}_{s,i-1}{z}^s-\sum _{s=0}^{i-2}{B}_{s,i-2}{z}^s,$

then, ${\mathcal{L}}_i\left(z\right)$ can be rewritten as

(10)${\mathcal{L}}_i\left(z\right)=-4\sum _{s=1}^i{B}_{s-1,i-1}{z}^s+2\sum _{s=0}^{i-1}{B}_{s,i-1}{z}^s-\sum _{s=0}^{i-2}{B}_{s,i-2}{z}^s.$

If we note the following identities:

$B_{-1,i-1}=B_{i,i-1}=B_{i,i-2}=B_{i-1,i-2}=0,$

then it is easy to see that Formula (10) can be converted into

(11)${\mathcal{L}}_i\left(z\right)=\sum _{s=0}^i(-4B_{s-1,i-1}+2B_{s,i-1}-B_{s,i-2})z^s.$

Noting the identity

$-4B_{s-1,i-1}+2B_{s,i-1}-B_{s,i-2}=B_{s,i},$

then Formula (7) can be obtained. □

The inversion formula to Formula (7) is also important in the sequel. To derive this inversion formula, the following lemma is first needed.

Lemma 1.

The following identity holds for every non-negative integer m:

(12) $\sum _{r=0}^m{\displaystyle \frac{{(-1)}^{j+r}(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{j-r}{m-r}\right)(2j-m-r-1)!}{(2j-r)!}}=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^j}{2},}\hfill & m=0,\hfill \\ 0,\hfill & m>0.\hfill \end{array}\right.$

Proof. 

The formula holds for $m=0$. It remains to show its validity for $m\ge 1$. To show this, it is sufficient to prove the following identity:

(13)$\sum _{r=0}^{m-1}{\displaystyle \frac{{(-1)}^{j+r}(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{j-r}{m-r}\right)(2j-m-r-1)!}{(2j-r)!}}={\displaystyle \frac{{(-1)}^{j+m+1}\left(\genfrac{}{}{0pt}{}{j}{m}\right)(2j-2m)!}{2(2j-m)!}}.$

The left-hand side of the last identity can be written as

$\begin{array}{cc}\hfill G_{m,j}=& \sum _{r=0}^{m-1}{\displaystyle \frac{{(-1)}^{j+r}(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{j-r}{m-r}\right)(2j-m-r-1)!}{(2j-r)!}}\hfill \\ \hfill =& \sum _{r=-1}^{m-1}{\displaystyle \frac{{(-1)}^{j+r}(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{j-r}{m-r}\right)(2j-m-r-1)!}{(2j-r)!}},\hfill \end{array}$

which is equivalent to

(14)$G_{m,j}=\sum _{s=0}^m{\displaystyle \frac{{(-1)}^{1+j+s}(1+j-s)\left(\genfrac{}{}{0pt}{}{j}{s-1}\right)\left(\genfrac{}{}{0pt}{}{1+j-s}{1-s+m}\right)(2j-s-m)!}{(2j-s+1)!}}.$

We can find $G_{m,j}$ in a closed form. Using a suitable symbolic computation, particularly Zeilberger’s algorithm [53], it can be demonstrated that $G_{m,j}$ fits the following recursive formula:

$G_{m+1,j}={\displaystyle \frac{m-2j}{2(m+1)(-2m+2j-1)}}{G}_{m,j},G_{1,j}={\displaystyle \frac{{(-1)}^{j}j}{2(2j-1)}},$

which can be solved to obtain

$G_{m,j}={\displaystyle \frac{{(-1)}^{j+m+1}\left(\genfrac{}{}{0pt}{}{j}{m}\right)(2j-2m)!}{2(2j-m)!}}.$

This proof is now complete. □

Theorem 2.

The following inversion formula holds for ${\mathcal{L}}_j\left(z\right),j\ge 0$:

(15) $z^j={\displaystyle \frac{\Gamma \left(\frac{1}{2}+j\right)}{\sqrt{\pi }}}\sum _{r=0}^j{\displaystyle \frac{{(-1)}^{j+r}\left(\genfrac{}{}{0pt}{}{j}{r}\right)c_{j-r}}{{(1+j-r)}_j}}{\mathcal{L}}_{j-r}\left(z\right),$

with

(16) $c_k=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & k=0,\hfill \\ 1,\hfill & k>0.\hfill \end{array}\right.$

Note that ${\left(z\right)}_r$ is the Pochhammer function defined as

${\displaystyle {\left(z\right)}_r=\frac{\Gamma (z+r)}{\Gamma \left(z\right)}.}$

Proof. 

Assume that

(17)$R_j\left(z\right)={\displaystyle \frac{\Gamma \left(\frac{1}{2}+j\right)}{\sqrt{\pi }}}\sum _{r=0}^j{\displaystyle \frac{{(-1)}^{j+r}\left(\genfrac{}{}{0pt}{}{j}{r}\right)c_{j-r}}{{(1+j-r)}_j}}{\mathcal{L}}_{j-r}\left(z\right),$

and we will prove that

$R_j\left(z\right)=z^j.$

We will use the analytic form of ${\mathcal{L}}_i\left(z\right)$ in (7) to obtain

(18)$\begin{array}{cc}\hfill R_j\left(z\right)=& {\displaystyle \frac{2\Gamma \left(\frac{1}{2}+j\right)}{\pi }}\sum _{r=0}^j{\displaystyle \frac{{(-1)}^{j+r}{c}_{j-r}\left(\genfrac{}{}{0pt}{}{j}{r}\right)}{{(1+j-r)}_j(j-r-1)!}}\sum _{s=0}^{j-r-1}\left({\displaystyle \genfrac{}{}{0pt}{}{j-r}{s}}\right)(2j-s-2r-1)!\times \hfill \\ \hfill & \Gamma \left({\displaystyle \frac{1}{2}}-j+s+r\right)z^{j-r-s},\hfill \end{array}$

which can be written alternatively as

(19)$R_j\left(z\right)={\displaystyle \frac{2\Gamma \left(\frac{1}{2}+j\right)}{\pi }}\sum _{m=0}^{j-1}\Gamma \left({\displaystyle \frac{1}{2}}-j+m\right)\sum _{r=0}^m{\displaystyle \frac{{(-1)}^{j+r}(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{j-r}{m-r}\right)\Gamma (2j-m-r)}{(2j-r)!}}{z}^{j-m}.$

The above formula’s right-hand side, when expanded and rearranged, yields

$R_j\left(z\right)=2\sum _{r=0}^i{\left(i+\sigma +1-r\right)}_r\left(\sum _{s=0}^r{\displaystyle \frac{{(-1)}^{r-s}(i-s+1)\left(\genfrac{}{}{0pt}{}{i}{s}\right)\left(\genfrac{}{}{0pt}{}{i-s}{r-s}\right)(2i-s-r+1)!}{(2i-s+2)!}}\right)z^{i-r}.$

Now, the application of Formula (12) of Lemma 1 yields

$R_j\left(z\right)=z^j.$

This ends the proof. □

Based on the analytic Formula (7) and its inversion form in (15), we can derive an expression for the q-th derivative of ${\mathcal{L}}_i\left(z\right)$. This expression is exhibited in the following theorem.

Theorem 3.

Let $i,q$ be two non-negative integers with $i\ge q$. We have

(20) ${\displaystyle D^q{\mathcal{L}}_i\left(z\right)=\sum _{k=0}^{i-q}{S}_{k,i,q}{\mathcal{L}}_k\left(z\right),}$

where

$S_{k,i,q}={\eta }_{k,i,q}{\displaystyle \frac{{(-1)}^{i-k}{4}^{q}i{c}_k{\left(\frac{1}{2}(2+i+k-q)\right)}_{q-1}{\left(q\right)}_{\frac{1}{2}(i-k-q)}}{\left(\frac{1}{2}(i-k-q)\right)!}},$

with

${\eta }_{k,i,q}=\left\{\begin{array}{cc}1,\hfill & (i-q-k)\mathrm{even},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

Proof. 

In virtue of the analytic form in (7), we can write $D^q{\mathcal{L}}_i\left(z\right)$ in the following form:

(21)$D^q{\mathcal{L}}_i\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }(k-1)!}}\sum _{s=0}^{k-q}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{s}}\right)(2k-s-1)!\Gamma \left({\displaystyle \frac{1}{2}}-k+s\right){(1+k-q-s)}_q{z}^{k-s-q}.$

The inversion Formula (15) can be applied to (21) to obtain the following formula:

(22)$\begin{array}{cc}\hfill D^q{\mathcal{L}}_i\left(z\right)=& {\displaystyle \frac{2}{\sqrt{\pi }(k-1)!}}\sum _{s=0}^{k-q}{4}^{-k+q+s}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{s}}\right)(2k-s-1)!\Gamma \left({\displaystyle \frac{1}{2}}-k+s\right){(1+k-q-s)}_q\hfill \\ \hfill & \times \sum _{r=0}^{k-s-q}{\displaystyle \frac{{(-1)}^{k-q-s+r}{c}_{k-q-s-r}(2k-2q-2s)!}{r!(2k-2q-2s-r)!}}{\mathcal{L}}_{k-s-q-r}\left(z\right).\hfill \end{array}$

Some algebraic computations transform the last formula into

(23)$\begin{array}{cc}\hfill D^q{\mathcal{L}}_i\left(z\right)=& {\displaystyle \frac{2k}{\pi }}\sum _{k=0}^{k-q}{(-1)}^{k+k-q}{c}_{k-k-q}\left(z\right)\times \hfill \\ \hfill & \sum _{s=0}^k{\displaystyle \frac{(2k-s-1)!\Gamma \left(\frac{1}{2}+k-q-s\right)\Gamma \left(\frac{1}{2}-k+s\right)}{s!(k-s)!(2k-k-2q-s)!}}{\mathcal{L}}_{k-q-k}\left(z\right),\hfill \end{array}$

that can written in the following alternative form:

(24)$\begin{array}{cc}\hfill D^q{\mathcal{L}}_i\left(z\right)=& {\displaystyle \frac{\Gamma \left(\frac{1}{2}-i\right)\left(2i\right)!\Gamma \left(\frac{1}{2}(1+2i-2q)\right)}{\pi }}\sum _{k=0}^{i-q}{\displaystyle \frac{{(-1)}^{i+k-q}c(i-k-q)}{k!(2i-k-2q)!}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}{\displaystyle \frac{1}{2}}-i,-k,-2i+k+2q\\ 1-2i,{\displaystyle \frac{1}{2}}-i+q\end{array}|1\right){\mathcal{L}}_{i-q-k}\left(z\right).\hfill \end{array}$

The ${}_{3}F_2\left(1\right)$ in (24) may be simplified into the following form with the help of Watson’s identity [54]:

(25)${}_{3}F_2\left(\begin{array}{c}{\displaystyle \frac{1}{2}}-i,-k,-2i+k+2q\\ 1-2i,{\displaystyle \frac{1}{2}}-i+q\end{array}|1\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\left(i-\frac{k}{2}-1\right)!\Gamma \left(\frac{1+k}{2}\right){\left(q\right)}_{\frac{k}{2}}}{\sqrt{\pi }(i-1)!{\left(\frac{1}{2}+i-\frac{k}{2}-q\right)}_{\frac{k}{2}}},}\hfill & k\mathrm{even},\hfill \\ 0,\hfill & k\mathrm{odd}.\hfill \end{array}\right.$

Inserting (25) into (24) yields the following formula:

(26)$\begin{array}{c}\hfill D^q{\mathcal{L}}_i\left(z\right)={\displaystyle \frac{{(-1)}^q{2}^{2q}i}{(q-1)!}}\sum _{k=0}^{⌊{\displaystyle \frac{i-q}{2}}⌋}{\displaystyle \frac{c_{i-2k-q}(i-k-1)!(k+q-1)!}{k!(i-k-q)!}}{\mathcal{L}}_{i-q-2k}\left(z\right),\end{array}$

which can be written in the form

(27)${\displaystyle D^q{\mathcal{L}}_i\left(z\right)=\sum _{k=0}^{i-q}{S}_{k,i,q}{\mathcal{L}}_k\left(z\right),}$

where

$S_{k,i,q}={\eta }_{k,i,q}{\displaystyle \frac{{(-1)}^{i-k}{4}^{q}i{c}_k{\left(\frac{1}{2}(2+i+k-q)\right)}_{q-1}{\left(q\right)}_{\frac{1}{2}(i-k-q)}}{\left(\frac{1}{2}(i-k-q)\right)!}},$

with

${\eta }_{k,i,q}=\left\{\begin{array}{cc}1,\hfill & (i-q-k)\mathrm{even},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

Theorem 3 is now proved. □

The following specific derivative formulas can be obtained as direct consequences of Theorem 3.

Corollary 1.

For $i\ge 1$, one has

(28) ${\displaystyle \frac{d{\mathcal{L}}_i\left(z\right)}{dz}}=\sum _{k=0}^{i-1}{\beta }_{k,i}^1{\mathcal{L}}_k\left(z\right),$

where

(29) ${\beta }_{k,i}^1=4i{c}_k{(-1)}^{i-k}{\eta }_{k,i,1}.$

Corollary 2.

For $i\ge 2$, one has

(30) ${\displaystyle \frac{d^2{\mathcal{L}}_i\left(z\right)}{d{z}^2}}=\sum _{k=0}^{i-2}{\beta }_{k,i}^2{\mathcal{L}}_k\left(z\right),$

where

(31) ${\beta }_{k,i}^2=4i{c}_k{(-1)}^{i-k}(i-k)(i+k){\eta }_{k,i,2}.$

Corollary 3.

For $i\ge 3$, one has

(32) ${\displaystyle \frac{d^3{\mathcal{L}}_i\left(z\right)}{d{z}^3}}=\sum _{k=0}^{i-3}{\beta }_{k,i}^3{\mathcal{L}}_k\left(z\right),$

where

(33) ${\beta }_{k,i}^3=2i{c}_k{(-1)}^{i-k}(i-k-1)(i-k+1)(i+k-1)(i+k+1){\eta }_{k,i,3}.$

Corollary 4.

For $i\ge 5$, one has

(34) ${\displaystyle \frac{d^5{\mathcal{L}}_i\left(z\right)}{d{z}^5}}=\sum _{k=0}^{i-5}{\beta }_{k,i}^4{\mathcal{L}}_k\left(z\right),$

where

(35) ${\beta }_{k,i}^4={\displaystyle \frac{8i{c}_k{(-1)}^{i-k}(i+k-3)(i+k-1)(i+k+1)(i+k+3)\Gamma \left(\frac{1}{2}(i-k+5)\right){\eta }_{k,i,5}}{3\Gamma \left(\frac{1}{2}(i-k-3)\right)}}.$

Proof. 

They are direct consequences of Theorem 3 only by setting $q=1,2,3,5$, respectively, in (20). □

Corollary 5.

If we define the following vector:

(36) $\mathcal{L}\left(z\right)={[{\mathcal{L}}_0\left(z\right),{\mathcal{L}}_1\left(z\right),\dots ,{\mathcal{L}}_M\left(z\right)]}^T,$

then we can have the following expressions:

(37) $\begin{array}{cc}\hfill & {\displaystyle \frac{d\mathcal{L}\left(z\right)}{dz}}=\mathit{A}\mathcal{L}\left(z\right),\hfill \\ & {\displaystyle \frac{d^2\mathcal{L}\left(z\right)}{d{z}^2}}=\mathit{B}\mathcal{L}\left(z\right),\hfill \\ & {\displaystyle \frac{d^3\mathcal{L}\left(z\right)}{d{z}^3}}=\mathit{F}\mathcal{L}\left(z\right),\hfill \\ & {\displaystyle \frac{d^5\mathcal{L}\left(z\right)}{d{z}^5}}=\mathit{G}\mathcal{L}\left(z\right),\hfill \end{array}$

where $\mathcal{L}\left(z\right)$ is as defined in (36). Also, $\mathit{A}=\left({\beta }_{k,i}^1\right)$, $\mathit{B}=\left({\beta }_{k,i}^2\right)$, $\mathit{F}=\left({\beta }_{k,i}^3\right)$, $\mathit{G}=\left({\beta }_{k,i}^4\right)$ are the operational matrices of derivatives, each of which has order ${(M+1)}^2$. In addition, their entries are given in (29), (31), (33) and (35), respectively.

Now, we give an explicit expression for the fractional derivatives $D_t^{\alpha }{\mathcal{L}}_j\left(t\right),0<\alpha <1$.

Theorem 4.

For $\alpha \in (0,1)$, we have the following expression:

(38) $D_t^{\alpha }\mathcal{L}\left(t\right)=t^{-\alpha }\left(\sum _{s=0}^j{\mathcal{U}}_{s,j}^{\alpha }{\mathcal{L}}_s\left(t\right)-{\rho }^{\alpha }\right),$

where

(39) ${\mathcal{U}}_{s,j}^{\alpha }={\displaystyle \frac{j{(-1)}^s{2}^{1-2s}{c}_s\Gamma \left(\frac{1}{2}-s\right)\left(2s\right)!(j+s-1)!}{\sqrt{\pi }(j-s)!}}{}_3\tilde{F}_2\left(\begin{array}{c}s-j,s+1,j+s\\ s+{\displaystyle \frac{1}{2}},2s+1\end{array}|1\right),$

(40) ${\rho }^{\alpha }={\displaystyle \frac{2}{\Gamma (1-\alpha )}},$

and ${}_3\tilde{F}_2\left(z\right)$ is the regularized hypergeometric function.

Proof. 

The power form formula of ${\mathcal{L}}_j\left(t\right)$ in (7) allows us to write $D_t^{\alpha }{\mathcal{L}}_j\left(t\right)$ as

(41)$D_t^{\alpha }{\mathcal{L}}_j\left(t\right)={\displaystyle \frac{2j}{\sqrt{\pi }}}\sum _{s=1}^j{\displaystyle \frac{\Gamma \left(\frac{1}{2}-s\right)(j+s-1)!}{\Gamma \left(s+\frac{1}{2}\right)(j-s)!}}{t}^{s-\alpha }.$

Applying (15) to (41) yields the following relation:

(42)$D_t^{\alpha }{\mathcal{L}}_j\left(t\right)={\displaystyle \frac{2t^{-\alpha }}{\pi \Gamma \left(j\right)}}\sum _{s=1}^j\sum _{m=0}^s{\displaystyle \frac{{(-1)}^{m}s!c_m\Gamma \left(\frac{1}{2}-s\right)\left(\genfrac{}{}{0pt}{}{j}{j-s}\right)\left(\genfrac{}{}{0pt}{}{s}{s-m}\right)(j+s-1)!}{{(m+1)}_s}}{\mathcal{L}}_m\left(t\right),$

which can be written alternatively as

(43)$D_t^{\alpha }{\mathcal{L}}_j\left(t\right)=t^{-\alpha }\left({\displaystyle \frac{2}{\pi \Gamma \left(j\right)}}\sum _{s=0}^j\sum _{m=s}^j{\displaystyle \frac{{(-1)}^{s}m!c_s\Gamma \left(\frac{1}{2}-m\right)\left(\genfrac{}{}{0pt}{}{j}{j-m}\right)\left(\genfrac{}{}{0pt}{}{m}{m-s}\right)(j+m-1)!}{{(s+1)}_p}}{\mathcal{L}}_s\left(t\right)-{\rho }^{\alpha }\right),$

where

(44)${\rho }^{\alpha }={\displaystyle \frac{2}{\Gamma (1-\alpha )}}.$

Now, ${\displaystyle \sum _{m=s}^j\frac{m!\Gamma \left(\frac{1}{2}-m\right)\left(\genfrac{}{}{0pt}{}{j}{j-m}\right)\left(\genfrac{}{}{0pt}{}{m}{m-s}\right)(j+m-1)!}{{(s+1)}_p}}$ can be expressed as

(45)$\begin{array}{cc}\hfill \sum _{m=s}^j{\displaystyle \frac{m!\Gamma \left(\frac{1}{2}-m\right)\left(\genfrac{}{}{0pt}{}{j}{j-m}\right)\left(\genfrac{}{}{0pt}{}{m}{m-s}\right)(j+m-1)!}{{(s+1)}_p}}=& {\displaystyle \frac{\sqrt{\pi }{4}^{-s}j!\Gamma \left(\frac{1}{2}-s\right)\left(2s\right)!(j+s-1)!}{\Gamma (j-s+1)}}\times \hfill \\ \hfill & {}_3\tilde{F}_2\left(\begin{array}{c}s-j,s+1,j+s\\ s+{\displaystyle \frac{1}{2}},2s+1\end{array}|1\right).\hfill \end{array}$

Therefore, we obtain the following result:

(46)$D_t^{\alpha }\mathcal{L}\left(t\right)=t^{-\alpha }\left(\sum _{s=0}^j{\mathcal{U}}_{s,j}^{\alpha }{\mathcal{L}}_s\left(t\right)-{\rho }^{\alpha }\right),$

where

(47)${\mathcal{U}}_{s,j}^{\alpha }={\displaystyle \frac{j{(-1)}^s{2}^{1-2s}{c}_s\Gamma \left(\frac{1}{2}-s\right)\left(2s\right)!(j+s-1)!}{\sqrt{\pi }(j-s)!}}{}_3\tilde{F}_2\left(\begin{array}{c}s-j,s+1,j+s\\ s+{\displaystyle \frac{1}{2}},2s+1\end{array}|1\right).$

This proves the theorem. □

Remark 1.

The fractional derivative of $\mathcal{L}\left(t\right)$ can be expressed in matrix form as follows:

(48) $D_t^{\alpha }\mathcal{L}\left(t\right)=t^{-\alpha }(\mathcal{U}\mathcal{L}\left(t\right)-{\rho }^{\alpha }),$

where $\mathcal{L}\left(t\right)={[{\mathcal{L}}_0\left(t\right),{\mathcal{L}}_1\left(t\right),\dots ,{\mathcal{L}}_M\left(t\right)]}^T$ and $\mathcal{U}=\left({\mathcal{U}}_{L,j}^{\alpha }\right)$ are the operational matrixes of the fractional derivative of order ${(M+1)}^2$, whose entries are given, respectively, in (39) and (40).

4. A Collocation Approach for the Time-Fractional Generalized KDE

Consider the following time-fractional generalized KDE [51,55]:

(49)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }\xi (z,t)}{\partial t^{\alpha }}}-{\displaystyle \frac{{\partial }^5\xi (z,t)}{\partial z^5}}& +{\displaystyle \frac{{\partial }^3\xi (z,t)}{\partial z^3}}+\xi (z,t){\displaystyle \frac{\partial \xi (z,t)}{\partial z}}+{\lambda }_1(z,t){\displaystyle \frac{\partial \xi (z,t)}{\partial z}}\hfill \\ & +{\lambda }_2(z,t)\xi (z,t)={\lambda }_3(z,t),0\le z,t\le 1,\hfill \end{array}$

controlled by the following initial and boundary conditions:

(50)$\begin{array}{cc}\hfill & \xi (z,0)=0,\hfill \\ & \xi (0,t)={\displaystyle \frac{\partial \xi (0,t)}{\partial z}}=0,\hfill \\ & \xi (1,t)={\displaystyle \frac{\partial \xi (1,t)}{\partial z}}={\displaystyle \frac{{\partial }^2\xi (1,t)}{\partial z^2}}=0,\hfill \end{array}$

where $0<\alpha \le 1$ and ${\lambda }_1(z,t),{\lambda }_2(z,t),{\lambda }_3(z,t)$ are continuous functions.

Now, assume that

(51)${\mathcal{Z}}^N=\mathrm{span}\{{\mathcal{L}}_m\left(z\right){\mathcal{L}}_n\left(t\right):0\le i,j\le M\},$

consequently, any function ${\xi }^M(z,t)\in {\mathcal{Z}}^N$ can be represented as

(52)${\xi }^M(z,t)=\sum _{i=0}^M\sum _{j=0}^M{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)=\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right),$

where $\widehat{\mathit{\xi }}={\left({\widehat{\xi }}_{ij}\right)}_{0\le i,j\le M}$ is the matrix of unknowns with the order ${(M+1)}^2$.

Now, the residual $\mathcal{R}(z,t)$ of Equation (49) can be written as

(53)$\begin{array}{cc}\hfill \mathcal{R}(z,t)=& {\displaystyle \frac{{\partial }^{\alpha }{\xi }^M(z,t)}{\partial t^{\alpha }}}-{\displaystyle \frac{{\partial }^5{\xi }^M(z,t)}{\partial z^5}}+{\displaystyle \frac{{\partial }^3{\xi }^M(z,t)}{\partial z^3}}+{\xi }^M(z,t){\displaystyle \frac{\partial {\xi }^M(z,t)}{\partial z}}\hfill \\ & +{\lambda }_1(z,t){\displaystyle \frac{\partial {\xi }^M(z,t)}{\partial z}}+{\lambda }_2(z,t){\xi }^M(z,t)-{\lambda }_3(z,t).\hfill \end{array}$

Thanks to Remarks 5 and 1 along with the expansion (52), the following expressions can be obtained:

(54)$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{\alpha }{\xi }^M(z,t)}{\partial t^{\alpha }}}=\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\left(t^{-\alpha }(\mathcal{U}\mathit{\theta }\left(t\right)-{\rho }^{\alpha })\right),\hfill \\ & {\displaystyle \frac{{\partial }^5{\xi }^M(z,t)}{\partial z^5}}={\left(\mathit{G}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right),\hfill \\ & {\displaystyle \frac{{\partial }^3{\xi }^M(z,t)}{\partial z^3}}={\left(\mathit{F}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right),\hfill \\ & {\xi }^M(z,t){\displaystyle \frac{\partial {\xi }^M(z,t)}{\partial z}}=\left[\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right]\left[{\left(\mathit{A}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right],\hfill \\ & {\lambda }_1(z,t){\displaystyle \frac{\partial {\xi }^M(z,t)}{\partial z}}={\lambda }_1(z,t)\left[{\left(\mathit{A}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right],\hfill \\ & {\lambda }_2(z,t){\xi }^M(z,t)={\lambda }_2(z,t)\left[\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right].\hfill \end{array}$

Consequently, $\mathcal{R}(z,t)$ can be written as

(55)$\begin{array}{cc}\hfill \mathcal{R}(z,t)=& \mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\left(t^{-\alpha }(\mathcal{U}\mathit{\theta }\left(t\right)-{\rho }^{\alpha })\right)-{\left(\mathit{G}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)+{\left(\mathit{F}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\hfill \\ \hfill & +\left[\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right]\left[{\left(\mathit{A}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right]{\lambda }_1(z,t)\left[{\left(\mathit{A}\mathcal{L}\left(z\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right]\hfill \\ & +{\lambda }_2(z,t)\left[\mathcal{L}{\left(z\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(t\right)\right]-{\lambda }_3(z,t).\hfill \end{array}$

The expansion coefficients $c_{ij}$ can be found using the collocation approach, which enforces the residual $\mathcal{R}(z,t)$ to be zero at certain collocation points $\left({\displaystyle \frac{i}{M+1}},{\displaystyle \frac{j}{M+1}}\right)$.

(56)$\mathcal{R}\left({\displaystyle \frac{i}{M+1}},{\displaystyle \frac{j}{M+1}}\right)=0,1\le i\le M-4,1\le j\le M.$

Moreover, the governing conditions in (50) lead to the following equations:

(57)$\begin{array}{cc}\hfill & \mathit{\theta }{\left({\displaystyle \frac{i}{M+1}}\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left(0\right)=f\left({\displaystyle \frac{i}{M+1}}\right),1\le i\le M+1,\hfill \\ & \mathit{\theta }{\left(0\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left({\displaystyle \frac{j}{M+1}}\right)=g_0\left({\displaystyle \frac{j}{M+1}}\right),1\le j\le M,\hfill \\ & \mathit{\theta }{\left(1\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left({\displaystyle \frac{j}{M+1}}\right)=g_1\left({\displaystyle \frac{j}{M+1}}\right),1\le j\le M,\hfill \\ & {\left(\mathit{A}\mathit{\theta }\left(0\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left({\displaystyle \frac{j}{M+1}}\right)=g_2\left({\displaystyle \frac{j}{M+1}}\right),1\le j\le M,\hfill \\ & {\left(\mathit{A}\mathit{\theta }\left(1\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left({\displaystyle \frac{j}{M+1}}\right)=g_3\left({\displaystyle \frac{j}{M+1}}\right),1\le i\le M,\hfill \\ & {\left(\mathit{B}\mathit{\theta }\left(1\right)\right)}^T\widehat{\mathit{\xi }}\mathit{\theta }\left({\displaystyle \frac{j}{M+1}}\right)=g_4\left({\displaystyle \frac{j}{M+1}}\right),1\le i\le M.\hfill \end{array}$

As a result, we obtain a nonlinear system of equations with ${(M+1)}^2$ equations, which can be solved using a suitable numerical solver, like Newton’s iterative method.

5. The Convergence and Error Analysis

Lemma 2.

The following inequality holds [56]:

(58) $|I_n\left(z\right)|\le {\displaystyle \frac{z^{n}cosh\left(z\right)}{2^{n}n!}},z>0,$

where $I_n\left(z\right)$ represents the modified Bessel function of order n of the first kind.

Lemma 3.

Consider the infinitely differentiable function $f\left(z\right)$ at the origin. We have

(59) $f\left(z\right)=\sum _{m=0}^{\infty }\sum _{r=m}^{\infty }{\displaystyle \frac{{(-1)}^m{f}^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)c_m}{\sqrt{\pi }(r-m)!(m+r)!}}{\mathcal{L}}_m\left(z\right).$

Proof. 

Assume that $f\left(z\right)$ can be expanded as

(60)$f\left(z\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{f^{\left(m\right)}\left(0\right)}{m!}}{z}^m.$

Due to Equation (15), the last equation is transformed into

(61)$f\left(z\right)=\sum _{m=0}^{\infty }\sum _{s=0}^m{\displaystyle \frac{f^{\left(m\right)}\left(0\right)c_s\Gamma \left(\frac{1}{2}+m\right){(-1)}^s\left(\genfrac{}{}{0pt}{}{m}{m-s}\right)}{\sqrt{\pi }m!{(s+1)}_m}}{\mathcal{L}}_s\left(z\right),$

The last formula can be written alternatively as

(62)$f\left(z\right)=\sum _{m=0}^{\infty }\sum _{r=m}^{\infty }{\displaystyle \frac{{(-1)}^m{f}^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)c_m}{\sqrt{\pi }(r-m)!(m+r)!}}{\mathcal{L}}_m\left(z\right).$

This proves Lemma 3. □

Lemma 4.

The following inequality holds for ${\mathcal{L}}_m\left(z\right)$:

(63) $|{\mathcal{L}}_m\left(z\right)|\le 2,z\in [0,1].$

Proof. 

Using Formula in (7) together with the simple inequality: $\left|z\right|<1$, we obtain

(64)$\begin{array}{cc}\hfill \left|{\mathcal{L}}_m\left(z\right)\right|=& \left|{\displaystyle \frac{2}{\sqrt{\pi }(m-1)!}}\sum _{s=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{m-s}}\right)(s+m-1)!\Gamma \left({\displaystyle \frac{1}{2}}-s\right)z^s\right|\hfill \\ & \le \left|{\displaystyle \frac{2}{\sqrt{\pi }(m-1)!}}\sqrt{\pi }{(-1)}^m(m-1)!\right|\hfill \\ & =2.\hfill \end{array}$

This proves the lemma. □

Theorem 5.

If $f\left(z\right)$ is defined on $[0, 1]$ and $|f^{\left(i\right)}\left(0\right)|\le {\varrho }^i,i>0,$ where ϱ is a positive constant, and ${\displaystyle f\left(z\right)=\sum _{m=0}^{\infty }{a}_m{\mathcal{L}}_m\left(z\right)}$, we obtain

(65) $|a_m|\le {\displaystyle \frac{\left(e^{\varrho }+1\right)2^{-2m-1}{\varrho }^m}{m!}}.$

Moreover, the series converges absolutely.

Proof. 

Making use of Lemma 3 together with the assumptions of the theorem yields

(66)$\begin{array}{cc}\hfill |a_m|& =\left|\sum _{r=m}^{\infty }{\displaystyle \frac{{(-1)}^m{f}^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)c_m}{\sqrt{\pi }(r-m)!(m+r)!}}\right|\hfill \\ & \le \sum _{r=m}^{\infty }{\displaystyle \frac{{\varrho }^r\Gamma \left(r+\frac{1}{2}\right)}{\sqrt{\pi }(r-m)!(m+r)!}}\hfill \\ & =e^{\varrho /2}{I}_m\left({\displaystyle \frac{\varrho }{2}}\right).\hfill \end{array}$

With the help of Lemma 2, we can express the previous inequality as

(67)$|a_m|\le {\displaystyle \frac{e^{\varrho /2}\left(cosh\left(\frac{\varrho }{2}\right){\left(\frac{\varrho }{2}\right)}^m\right)}{2^{m}m!}},$

which can be written as

(68)$|a_m|\le {\displaystyle \frac{\left(e^{\varrho }+1\right)2^{-2m-1}{\varrho }^m}{m!}}.$

Now, we demonstrate the theorem’s second part. Since

(69)$\sum _{m=0}^{\infty }\left|a_m{\mathcal{L}}_m\left(z\right)\right|\le \sum _{m=0}^{\infty }{\displaystyle \frac{2^{-2m}\left(e^{\varrho }+1\right){\varrho }^m}{m!}}=e^{\varrho /4}\left(e^{\varrho }+1\right),$

so the series converges absolutely. □

Theorem 6.

If $f\left(z\right)$ satisfies the hypothesis of Theorem 5, and ${\displaystyle e_M\left(z\right)=\sum _{m=M+1}^{\infty }{a}_m{\mathcal{L}}_m\left(z\right)}$, then the following error estimation is satisfied:

(70) $|e_M\left(z\right)|<{\displaystyle \frac{e^{\varrho /4}\left(e^{\varrho }+1\right){\left(\frac{\varrho }{4}\right)}^{M+1}}{M!}}.$

Proof. 

The definition of $e_M\left(z\right)$ enables us to write

(71)$\begin{array}{cc}\hfill |e_M\left(z\right)|& =\sum _{m=M+1}^{\infty }\left|c_m\right|\left|{\mathcal{L}}_m\left(z\right)\right|\hfill \\ & \le \sum _{m=M+1}^{\infty }{\displaystyle \frac{2^{-2m}\left(e^{\varrho }+1\right){\varrho }^m}{m!}}\hfill \\ & ={\displaystyle \frac{e^{\varrho /4}\left(e^{\varrho }+1\right)\left(M!-\Gamma \left(M+1,\frac{\varrho }{4}\right)\right)}{M!}}\hfill \\ & <{\displaystyle \frac{e^{\varrho /4}\left(e^{\varrho }+1\right){\left(\frac{\varrho }{4}\right)}^{M+1}}{M!}},\hfill \end{array}$

where $\Gamma (.,.)$ represents the upper incomplete gamma functions [57]. □

Theorem 7.

If a function ${\displaystyle \xi (z,t)=v_1\left(z\right)v_2\left(t\right)=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)}$, with $|v_1^{\left(i\right)}\left(0\right)|\le {\varsigma }_1^i$ and $|v_2^{\left(i\right)}\left(0\right)|\le {\varsigma }_2^i$, where ${\varsigma }_1$ and ${\varsigma }_2$ are positive constants. Then, one has

(72) $|{\widehat{\xi }}_{ij}|\le {\displaystyle \frac{\left(e^{{\varsigma }_1}+1\right)\left(e^{{\varsigma }_2}+1\right)2^{-2(i+j+1)}{\varsigma }_1^i{\varsigma }_2^j}{i!j!}}.$

Moreover, the series converges absolutely.

Proof. 

Based on Lemma 3 and the assumption $\xi (z,t)=v_1\left(z\right)v_2\left(t\right)$, we obtain

(73)${\widehat{\xi }}_{ij}=\sum _{r=i}^{\infty }\sum _{r=j}^{\infty }{\displaystyle \frac{{(-1)}^{i+j}{v}_1^{\left(r\right)}\left(0\right)v_2^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)\Gamma \left(r+\frac{1}{2}\right)c_j{c}_i}{\pi (i-r)!(i+r)!(r-j)!(j+r)!}}.$

Using the assumption $|v_1^{\left(i\right)}\left(0\right)|\le {\varsigma }_1^i$ and $|v_2^{\left(i\right)}\left(0\right)|\le {\varsigma }_2^i$, one obtains

(74)$|{\widehat{\xi }}_{ij}|\le \sum _{r=i}^{\infty }{\displaystyle \frac{{(-1)}^i{v}_1^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)c_i}{\sqrt{\pi }(r-i)(i+r)!}}\times \sum _{r=j}^{\infty }{\displaystyle \frac{{(-1)}^j{v}_2^{\left(r\right)}\left(0\right)\Gamma \left(r+\frac{1}{2}\right)c_j}{\sqrt{\pi }(r-j)!(j+r)!}}.$

We can now obtain the desired outcome by following the same procedures as in the proof of Theorem 5. □

Theorem 8.

Let $\xi (z,t)$ satisfy the assumptions of Theorem 7. The following upper estimate on the truncation error can be obtained:

$|\xi (z,t)-{\xi }^M(z,t)|<{\displaystyle \frac{e^{\frac{{\varsigma }_1+{\varsigma }_2}{4}}\left(e^{{\varsigma }_1}+1\right)\left(e^{{\varsigma }_2}+1\right)\left[{\left(\frac{{\varsigma }_2}{4}\right)}^{M+1}+{\left(\frac{{\varsigma }_2}{4}\right)}^{M+1}\right]}{M!}}.$

Proof. 

From definitions of $\xi (z,t)$ and ${\xi }^M(z,t),$ we obtain

(75)$\begin{array}{cc}\hfill |\xi (z,t)-{\xi }^M(z,t)|& =\left|\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)-\sum _{i=0}^M\sum _{j=0}^M{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)\right|\hfill \\ & \le \left|\sum _{i=0}^M\sum _{j=M+1}^{\infty }{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)\right|+\left|\sum _{i=M+1}^{\infty }\sum _{j=0}^{\infty }{\widehat{\xi }}_{ij}{\mathcal{L}}_i\left(z\right){\mathcal{L}}_j\left(t\right)\right|.\hfill \end{array}$

If we use Theorem 7, Lemma 4 and the following inequalities

(76)$\begin{array}{cc}\hfill & \sum _{i=0}^M{\displaystyle \frac{2^{-2i}\left(e^{{\varsigma }_1}+1\right){\varsigma }_1^i}{i!}}={\displaystyle \frac{e^{{\varsigma }_1/4}\left(e^{{\varsigma }_1}+1\right)\Gamma \left(M+1,\frac{{\varsigma }_1}{4}\right)}{M!}}<e^{{\varsigma }_1/4}\left(e^{{\varsigma }_1}+1\right),\hfill \\ & \sum _{i=M+1}^{\infty }{\displaystyle \frac{2^{-2i}\left(e^{{\varsigma }_1}+1\right){\varsigma }_1^i}{i!}}={\displaystyle \frac{e^{{\varsigma }_1/4}\left(e^{{\varsigma }_1}+1\right)\left(M!-\Gamma \left(M+1,\frac{{\varsigma }_1}{4}\right)\right)}{M!}}<{\displaystyle \frac{e^{{\varsigma }_1/4}\left(e^{{\varsigma }_1}+1\right){\left(\frac{{\varsigma }_1}{4}\right)}^{M+1}}{M!}},\hfill \\ & \sum _{i=0}^{\infty }{\displaystyle \frac{2^{-2i}\left(e^{{\varsigma }_1}+1\right){\varsigma }_1^i}{i!}}=e^{{\varsigma }_1/4}\left(e^{{\varsigma }_1}+1\right),\hfill \end{array}$

we obtain the following desired estimation

(77)$|\xi (z,t)-{\xi }^M(z,t)|<{\displaystyle \frac{e^{\frac{{\varsigma }_1+{\varsigma }_2}{4}}\left(e^{{\varsigma }_1}+1\right)\left(e^{{\varsigma }_2}+1\right)\left[{\left(\frac{{\varsigma }_2}{4}\right)}^{M+1}+{\left(\frac{{\varsigma }_2}{4}\right)}^{M+1}\right]}{M!}}.$

This ends the proof of this theorem. □

6. Illustrative Examples

Example 1 ([51]).

Consider the following equation:

(78) $\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }\xi (z,t)}{\partial t^{\alpha }}}& -{\displaystyle \frac{{\partial }^5\xi (z,t)}{\partial z^5}}+{\displaystyle \frac{{\partial }^3\xi (z,t)}{\partial z^3}}+\xi (z,t){\displaystyle \frac{\partial \xi (z,t)}{\partial z}}\hfill \\ & +(z^{2}t+1){\displaystyle \frac{\partial \xi (z,t)}{\partial z}}+(z-t)\xi (z,t)=f(z,t),0\le z,t\le 1,\hfill \end{array}$

governed by (50), and $f(z,t)$ is calculated so that the exact solution is

$\xi (z,t)=t^{1+\alpha }{z}^2\left({\displaystyle \frac{z^3}{6}}-{\displaystyle \frac{z^2}{2}}+{\displaystyle \frac{z}{2}}-{\displaystyle \frac{1}{6}}\right).$

The absolute errors (AEs) for different values of α values when $M=9$ are shown in Table 1, demonstrating the accuracy of the proposed method. For $M=9$, Table 2 compares our method to the method in [51] at various values of z when $0<t<1.$ This table shows that for small values of M, the results are accurate. The comparison shows that our strategy is more accurate than the method in [51]. Table 3 shows the AEs at $\alpha =0.2$ when $M=9$. Figure 1 plots the AEs at different values of M when $\alpha =0.85$, while Figure 2 plots the AEs when $\alpha =0.7$ and $M=10$. These figures demonstrate a good agreement between the exact and approximate solutions.

Example 2 ([51]).

Consider the following equation:

(79) $\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }\xi (z,t)}{\partial t^{\alpha }}}& -{\displaystyle \frac{{\partial }^5\xi (z,t)}{\partial z^5}}+{\displaystyle \frac{{\partial }^3\xi (z,t)}{\partial z^3}}+\xi (z,t){\displaystyle \frac{\partial \xi (z,t)}{\partial z}}+{\displaystyle \frac{\partial \xi (z,t)}{\partial z}}=f(z,t),0\le z,t\le 1,\hfill \end{array}$

governed by (50), and $f(z,t)$ is determined so that the exact solution is

$\xi (z,t)={\displaystyle \frac{1}{12}}{t}^{1+\alpha }{z}^2(z^4-2z^3+2z-1).$

Table 4 compares the maximum AEs for $M=9$ at various values of z with the method proposed in [51]. For relatively small values of M, the results are accurate, as can be seen in the table. Furthermore, the comparison shows that our method is more accurate than the method in [51]. In addition, Table 5 displays the AEs at $\alpha =0.5$, for different values of t when $M=9$. The accuracy of the suggested approach is demonstrated by Figure 3, which plots the AEs at different values of t when $\alpha =0.95$ and $M=10$. In addition, Figure 4 plots the AEs at different values of M when $\alpha =0.85$. These figures show a good agreement of the approximate solution with the exact one.

7. Discussion

We have successfully developed a numerical collocation method based on a class of specific Lucas polynomial sequences for the numerical solution of the TFGKDE. Some new theoretical results we obtained using algebraic computations were the foundations for obtaining the operational matrix of this polynomial’s integer and fractional derivatives, which helped design our proposed numerical algorithm. We think that our approach can be applied to other fractional differential equations. We also expect that other Lucas polynomial sequences can be introduced and used to solve different differential equations.

In conclusion, this study adds an adaptable framework for solving some Kawahara equations using certain types of Horadam polynomials. This will open new ideas for solving other types of differential equations.

Author Contributions

Conceptualization, W.M.A.-E. and A.G.A.; Methodology, W.M.A.-E., A.K.A.-H., O.M.A. and A.G.A.; Software, A.G.A.; Validation, W.M.A.-E., A.K.A.-H., O.M.A., M.H.A. and A.G.A.; Formal analysis, W.M.A.-E. and A.G.A.; Investigation, W.M.A.-E., A.K.A.-H., O.M.A., M.H.A. and A.G.A.; Writing—original draft, W.M.A.-E. and A.G.A.; Writing—review and editing, W.M.A.-E., A.K.A.-H., O.M.A., M.H.A. and A.G.A.; Visualization, W.M.A.-E. and A.G.A.; Supervision, W.M.A.-E. and O.M.A. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to express their gratitude to the editor and the anonymous reviewers for their insightful comments and constructive suggestions, which have significantly contributed to improving the quality of this manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The AEs at [Forumla omitted. See PDF.] and different values of M for Example 1.

Figure 2. The AEs at [Forumla omitted. See PDF.] for Example 1.

Figure 3. The AEs at [Forumla omitted. See PDF.] for Example 2.

Figure 4. The AEs at [Forumla omitted. See PDF.] for Example 2.

Table 1

The AEs of Example 1.

$(\mathit{z},\mathit{t})$	$\mathit{\alpha }=0.7$	$\mathit{\alpha }=0.8$	$\mathit{\alpha }=0.9$
(0.1, 0.1)	1.38172 $\times {10}^{-10}$	9.39553 $\times {10}^{-11}$	4.40408 $\times {10}^{-11}$
(0.2, 0.2)	5.6496 $\times {10}^{-12}$	1.45969 $\times {10}^{-11}$	1.25069 $\times {10}^{-11}$
(0.3, 0.3)	3.8616 $\times {10}^{-11}$	2.30332 $\times {10}^{-11}$	9.78697 $\times {10}^{-12}$
(0.4, 0.4)	5.51946 $\times {10}^{-12}$	1.13216 $\times {10}^{-12}$	2.85139 $\times {10}^{-12}$
(0.5, 0.5)	1.5361 $\times {10}^{-11}$	9.11035 $\times {10}^{-12}$	4.00171 $\times {10}^{-12}$
(0.6, 0.6)	1.1336 $\times {10}^{-12}$	2.9774 $\times {10}^{-12}$	2.52177 $\times {10}^{-12}$
(0.7, 0.7)	8.73609 $\times {10}^{-12}$	6.51013 $\times {10}^{-12}$	3.62756 $\times {10}^{-12}$
(0.8, 0.8)	8.17378 $\times {10}^{-12}$	7.40989 $\times {10}^{-12}$	4.7107 $\times {10}^{-12}$
(0.9, 0.9)	1.09754 $\times {10}^{-11}$	9.7457 $\times {10}^{-12}$	6.21363 $\times {10}^{-12}$

Table 2

Comparison of maximum AEs for $0<t<1$ of Example 1.

	$\mathit{\alpha }=0.7$		$\mathit{\alpha }=0.8$		$\mathit{\alpha }=0.9$
$(\mathit{z},\mathit{t})$	Method in [51]	Our Method	Method in [51]	Our Method	Method in [51]	Our Method
(0.1, t)	6.36 $\times {10}^{-6}$	1.38172 $\times {10}^{-11}$	5.66 $\times {10}^{-6}$	9.39553 $\times {10}^{-11}$	4.92 $\times {10}^{-6}$	5.00897 $\times {10}^{-11}$
(0.2, t)	1.79 $\times {10}^{-5}$	3.8989 $\times {10}^{-10}$	1.59 $\times {10}^{-5}$	2.65122 $\times {10}^{-10}$	1.38 $\times {10}^{-5}$	1.4134 $\times {10}^{-10}$
(0.3, t)	2.70 $\times {10}^{-5}$	5.88305 $\times {10}^{-10}$	2.41 $\times {10}^{-5}$	4.00043 $\times {10}^{-10}$	2.09 $\times {10}^{-5}$	2.13262 $\times {10}^{-10}$
(0.4, t)	3.03 $\times {10}^{-5}$	6.58352 $\times {10}^{-10}$	2.70 $\times {10}^{-5}$	4.47677 $\times {10}^{-10}$	2.34 $\times {10}^{-5}$	2.38648 $\times {10}^{-10}$
(0.5, t)	2.74 $\times {10}^{-5}$	5.95898 $\times {10}^{-10}$	2.44 $\times {10}^{-5}$	4.0521 $\times {10}^{-10}$	2.12 $\times {10}^{-5}$	2.16101 $\times {10}^{-10}$
(0.6, t)	2.02 $\times {10}^{-5}$	4.421075 $\times {10}^{-10}$	1.80 $\times {10}^{-5}$	3.00633 $\times {10}^{-10}$	1.56 $\times {10}^{-5}$	1.60246 $\times {10}^{-10}$
(0.7, t)	1.16 $\times {10}^{-5}$	2.58006 $\times {10}^{-10}$	1.03 $\times {10}^{-5}$	1.75445 $\times {10}^{-10}$	8.99 $\times {10}^{-6}$	9.35108 $\times {10}^{-11}$
(0.8, t)	4.50 $\times {10}^{-5}$	1.03006 $\times {10}^{-10}$	4.00 $\times {10}^{-6}$	7.0045 $\times {10}^{-11}$	3.48 $\times {10}^{-6}$	3.73307 $\times {10}^{-11}$
(0.9, t)	7.12 $\times {10}^{-7}$	1.71464 $\times {10}^{-11}$	6.34 $\times {10}^{-7}$	1.16598 $\times {10}^{-11}$	5.51 $\times {10}^{-7}$	6.21363 $\times {10}^{-12}$

Table 3

The AEs of Example 1 at $\alpha =0.2$.

z	$\mathit{t}=0.1$	$\mathit{t}=0.4$	$\mathit{t}=0.7$	$\mathit{t}=0.9$	$\mathit{t}=0.98$	$\mathit{t}=0.99$
0.1	6.56903 $\times {10}^{-11}$	5.74638 $\times {10}^{-12}$	3.72786 $\times {10}^{-12}$	1.64911 $\times {10}^{-11}$	3.39914 $\times {10}^{-6}$	4.54984 $\times {10}^{-6}$
0.2	1.85363 $\times {10}^{-10}$	1.62151 $\times {10}^{-11}$	1.05193 $\times {10}^{-11}$	4.65317 $\times {10}^{-11}$	9.54927 $\times {10}^{-6}$	1.2782 $\times {10}^{-5}$
0.3	2.79693 $\times {10}^{-10}$	2.44669 $\times {10}^{-11}$	1.58727 $\times {10}^{-11}$	4.65317 $\times {10}^{-11}$	1.43938 $\times {10}^{-5}$	1.92666 $\times {10}^{-5}$
0.4	3.12993 $\times {10}^{-10}$	2.73802 $\times {10}^{-11}$	1.77627 $\times {10}^{-11}$	7.85731 $\times {10}^{-11}$	1.61144 $\times {10}^{-5}$	2.15696 $\times {10}^{-5}$
0.5	2.833 $\times {10}^{-10}$	2.47828 $\times {10}^{-11}$	1.60778 $\times {10}^{-11}$	7.11206 $\times {10}^{-11}$	1.4571 $\times {10}^{-5}$	1.95037 $\times {10}^{-5}$
0.6	2.10183 $\times {10}^{-10}$	1.83867 $\times {10}^{-11}$	1.19285 $\times {10}^{-11}$	5.27661 $\times {10}^{-11}$	1.07429 $\times {10}^{-5}$	1.43797 $\times {10}^{-5}$
0.7	1.22658 $\times {10}^{-10}$	1.07302 $\times {10}^{-11}$	6.96172 $\times {10}^{-12}$	3.07955 $\times {10}^{-11}$	6.16875 $\times {10}^{-6}$	8.25706 $\times {10}^{-6}$
0.8	4.89695 $\times {10}^{-11}$	4.28383 $\times {10}^{-12}$	2.77937 $\times {10}^{-12}$	1.22961 $\times {10}^{-11}$	2.38728 $\times {10}^{-6}$	3.19545 $\times {10}^{-6}$
0.9	8.15143 $\times {10}^{-12}$	7.13133 $\times {10}^{-13}$	4.64319 $\times {10}^{-13}$	2.05545 $\times {10}^{-12}$	3.7767 $\times {10}^{-6}$	5.05522 $\times {10}^{-7}$

Table 4

Comparison of maximum AEs for $0<t<1$ of Example 2.

	$\mathit{\alpha }=0.7$		$\mathit{\alpha }=0.8$		$\mathit{\alpha }=0.9$
$(\mathit{z},\mathit{t})$	Method in [51]	Our Method	Method in [51]	Our Method	Method in [51]	Our Method
(0.1, t)	7.73 $\times {10}^{-6}$	8.70227 $\times {10}^{-11}$	7.98 $\times {10}^{-6}$	5.91726 $\times {10}^{-11}$	8.26 $\times {10}^{-6}$	3.15499 $\times {10}^{-11}$
(0.2, t)	1.57 $\times {10}^{-5}$	2.43683 $\times {10}^{-10}$	1.66 $\times {10}^{-5}$	1.65698 $\times {10}^{-10}$	1.75 $\times {10}^{-5}$	8.83338 $\times {10}^{-11}$
(0.3, t)	1.95 $\times {10}^{-5}$	3.63484 $\times {10}^{-10}$	2.10 $\times {10}^{-5}$	2.47164 $\times {10}^{-10}$	2.26 $\times {10}^{-5}$	1.31732 $\times {10}^{-10}$
(0.4, t)	2.17 $\times {10}^{-5}$	3.99944 $\times {10}^{-10}$	2.12 $\times {10}^{-5}$	2.71964 $\times {10}^{-10}$	2.32 $\times {10}^{-5}$	1.44899 $\times {10}^{-10}$
(0.5, t)	2.09 $\times {10}^{-5}$	3.53402 $\times {10}^{-10}$	1.86 $\times {10}^{-5}$	2.40324 $\times {10}^{-10}$	2.00 $\times {10}^{-5}$	1.27976 $\times {10}^{-10}$
(0.6, t)	1.64 $\times {10}^{-5}$	2.53687 $\times {10}^{-10}$	1.46 $\times {10}^{-5}$	1.72524 $\times {10}^{-10}$	1.44 $\times {10}^{-5}$	9.18057 $\times {10}^{-11}$
(0.7, t)	1.00 $\times {10}^{-5}$	1.41757 $\times {10}^{-10}$	8.90 $\times {10}^{-6}$	9.64113 $\times {10}^{-11}$	8.20 $\times {10}^{-6}$	5.12531 $\times {10}^{-11}$
(0.8, t)	4.10 $\times {10}^{-6}$	5.35954 $\times {10}^{-11}$	3.64 $\times {10}^{-6}$	3.64548 $\times {10}^{-11}$	3.18 $\times {10}^{-6}$	1.93545 $\times {10}^{-11}$
(0.9, t)	6.83 $\times {10}^{-7}$	8.36451 $\times {10}^{-12}$	6.07 $\times {10}^{-7}$	5.6901  $\times {10}^{-12}$	5.27 $\times {10}^{-7}$	3.01595 $\times {10}^{-12}$

Table 5

The AEs of Example 2 at $\alpha =0.5$.

z	$\mathit{t}=0.2$	$\mathit{t}=0.4$	$\mathit{t}=0.6$	$\mathit{t}=0.8$
0.1	6.81302 $\times {10}^{-12}$	3.61427 $\times {10}^{-12}$	1.47142 $\times {10}^{-12}$	4.53549 $\times {10}^{-12}$
0.2	1.90786 $\times {10}^{-11}$	1.01209 $\times {10}^{-11}$	4.12041 $\times {10}^{-12}$	1.26997 $\times {10}^{-11}$
0.3	2.84596 $\times {10}^{-11}$	1.50968 $\times {10}^{-11}$	6.14636 $\times {10}^{-12}$	1.89416 $\times {10}^{-11}$
0.4	3.13167 $\times {10}^{-11}$	1.66115 $\times {10}^{-11}$	6.76329 $\times {10}^{-12}$	2.08391 $\times {10}^{-11}$
0.5	2.76754 $\times {10}^{-11}$	1.4679 $\times {10}^{-11}$	5.97675 $\times {10}^{-12}$	1.84108 $\times {10}^{-11}$
0.6	1.98696 $\times {10}^{-11}$	1.05377 $\times {10}^{-11}$	4.29084 $\times {10}^{-12}$	1.3213 $\times {10}^{-11}$
0.7	1.11052 $\times {10}^{-11}$	5.88874 $\times {10}^{-12}$	2.39807 $\times {10}^{-12}$	7.38075 $\times {10}^{-12}$
0.8	4.19985 $\times {10}^{-12}$	2.22663 $\times {10}^{-12}$	9.06843 $\times {10}^{-13}$	2.78951 $\times {10}^{-12}$
0.9	6.55692 $\times {10}^{-13}$	3.47544 $\times {10}^{-13}$	1.41483 $\times {10}^{-13}$	4.35778 $\times {10}^{-13}$

References

1. Li, C.; Qian, D.; Chen, Y. On Riemann-Liouville and Caputo derivatives. Discret. Dyn. Nat. Soc.; 2011; 2011, 562494. [DOI: https://dx.doi.org/10.1155/2011/562494]

2. Li, C.; Zhao, Z. Introduction to fractional integrability and differentiability. Eur. Phys. J. Spec. Top.; 2011; 193, pp. 5-26. [DOI: https://dx.doi.org/10.1140/epjst/e2011-01378-2]

3. Li, C.; Dao, X.; Guo, P. Fractional derivatives in complex planes. Nonlinear Anal.; 2009; 71, pp. 1857-1869. [DOI: https://dx.doi.org/10.1016/j.na.2009.01.021]

4. Li, C.; Deng, W. Remarks on fractional derivatives. Appl. Math. Comput.; 2007; 187, pp. 777-784. [DOI: https://dx.doi.org/10.1016/j.amc.2006.08.163]

5. Magin, R. Fractional Calculus in Bioengineering, part 1. Crit. Rev. Biomed. Eng.; 2004; 32, [DOI: https://dx.doi.org/10.1615/CritRevBiomedEng.v32.i1.10]

6. Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and MEDIA; Springer: Berlin/Heidelberg, Germany, 2011.

7. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204.

8. Gasmi, B.; Moussa, A.A.; Mati, Y.; Alhakim, L.A.; Akgül, A. New exact traveling wave solutions to the Kawahara equation using the tanh (ξ) expansion method. Int. J. Appl. Comput. Math.; 2023; 9, 98. [DOI: https://dx.doi.org/10.1007/s40819-023-01568-6]

9. Varol, D. Solitary and periodic wave solutions of the space-time fractional extended Kawahara equation. Fractal Fract.; 2023; 7, 539. [DOI: https://dx.doi.org/10.3390/fractalfract7070539]

10. Zafar, H.; Ali, A.; Khan, K.; Sadiq, M.N. Analytical solution of time fractional Kawahara and modified Kawahara equations by homotopy analysis method. Int. J. Appl. Comput. Math.; 2022; 8, 94. [DOI: https://dx.doi.org/10.1007/s40819-022-01296-3]

11. Karunakar, P.; Chakraverty, S. Solution of interval-modified Kawahara differential equations using homotopy perturbation transform method. Wave Dynamics; World Scientific: Singapore, 2022; pp. 193-202.

12. Arifeen, S.U.; Ali, I.; Ahmad, I.; Shaheen, S. Computational study of time-fractional non-linear Kawahara equations using quintic B-spline and Galerkin’s method. Partial Differ. Equ. Appl. Math.; 2024; 12, 100779. [DOI: https://dx.doi.org/10.1016/j.padiff.2024.100779]

13. Aljahdaly, N.H.; Alweldi, A.M. On the modified Laplace homotopy perturbation method for solving damped modified Kawahara equation and its application in a fluid. Symmetry; 2023; 15, 394. [DOI: https://dx.doi.org/10.3390/sym15020394]

14. Wang, X. Analytical and numerical studies of the modified Kawahara equation with dual-power law nonlinearities. Numer. Algorithms; 2024; pp. 1-33. [DOI: https://dx.doi.org/10.1007/s11075-024-01828-6]

15. Al-Essa, L.A.; Ur Rahman, M. A survey on fractal fractional nonlinear Kawahara equation: Theoretical and computational analysis. Sci. Rep.; 2024; 14, 6990. [DOI: https://dx.doi.org/10.1038/s41598-024-57389-0] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/38523147]

16. Nirmala, A.N.; Kumbinarasaiah, S. A rapid numerical method for nonlinear generalized time-fractional Kawahara equations via domination polynomials of complete graph. Phys. Scr.; 2024; 99, 125264. [DOI: https://dx.doi.org/10.1088/1402-4896/ad9182]

17. Kumar, R.; Gupta, J. Numerical investigation of fractional Kawahara equation via Haar scale wavelet method. Contemp. Math.; 2024; 5, pp. 478-491. [DOI: https://dx.doi.org/10.37256/cm.5120242510]

18. Horadam, A.F. Extension of a synthesis for a class of polynomial sequences. Fibonacci Quart.; 1996; 34, pp. 68-74. [DOI: https://dx.doi.org/10.1080/00150517.1996.12429098]

19. Keskin, R.; Siar, Z. Some new identities concerning the Horadam sequence and its companion sequence. Commun. Korean Math. Soc.; 2019; 34, pp. 1-16.

20. Djordjevic, S.S.; Djordjevic, G.B. Generalized Horadam polynomials and numbers. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat.; 2018; 26, pp. 91-101. [DOI: https://dx.doi.org/10.2478/auom-2018-0005]

21. Singh, P.K.; Ray, S.S. A numerical approach based on Pell polynomial for solving stochastic fractional differential equations. Numer. Algorithms; 2024; 97, pp. 1513-1534. [DOI: https://dx.doi.org/10.1007/s11075-024-01760-9]

22. El-Sayed, A.A. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation. Demonstr. Math.; 2023; 56, 20220220. [DOI: https://dx.doi.org/10.1515/dema-2022-0220]

23. Postavaru, O. An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations. Math. Comput. Simul.; 2023; 212, pp. 406-422. [DOI: https://dx.doi.org/10.1016/j.matcom.2023.04.028]

24. Angadi, L.M. Wavelet based Galerkin method for the numerical solution of singular boundary value problems using Fibonacci wavelets. J. Sci. Res.; 2025; 17, pp. 227-234. [DOI: https://dx.doi.org/10.3329/jsr.v17i1.75341]

25. Gupta, R.; Saha Ray, S. A new effective coherent numerical technique based on shifted Vieta–Fibonacci polynomials for solving stochastic fractional integro-differential equations. Comput. Appl. Math.; 2023; 42, 256. [DOI: https://dx.doi.org/10.1007/s40314-023-02398-4]

26. Abd-Elhameed, W.M.; Alqubori, O.M.; Atta, A.G. A collocation approach for the nonlinear fifth-order KdV equations using certain shifted Horadam polynomials. Mathematics; 2025; 13, 300. [DOI: https://dx.doi.org/10.3390/math13020300]

27. Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods in Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 1988.

28. Hesthaven, J.; Gottlieb, S.; Gottlieb, D. Spectral Methods for Time-Dependent Problems; Cambridge University Press: Cambridge, UK, 2007; Volume 21.

29. Boyd, J.P. Chebyshev and Fourier Spectral Methods; Courier Corporation: Chelmsford, MA, USA, 2001.

30. Abd-Elhameed, W.M.; Al-Harbi, M.S.; Amin, A.K.; Ahmed, H.M. Spectral treatment of high-order Emden–Fowler equations based on modified Chebyshev polynomials. Axioms; 2023; 12, 99. [DOI: https://dx.doi.org/10.3390/axioms12020099]

31. Abdelhakem, M.; Baleanu, D.; Agarwal, P.; Moussa, H. Approximating system of ordinary differential-algebraic equations via derivative of Legendre polynomials operational matrices. Int. J. Mod. Phys. C; 2023; 34, 2350036. [DOI: https://dx.doi.org/10.1142/S0129183123500365]

32. Abdelkawy, M.A.; Lopes, A.M.; Babatin, M.M. Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order. Chaos Solitons Fract.; 2020; 134, 109721. [DOI: https://dx.doi.org/10.1016/j.chaos.2020.109721]

33. Hafez, R.M. Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method. Comput. Appl. Math.; 2018; 37, pp. 5253-5273. [DOI: https://dx.doi.org/10.1007/s40314-018-0635-1]

34. Abd-Elhameed, W.M.; Alqubori, O.M.; Al-Harbi, A.K.; Alharbi, M.H.; Atta, A.G. Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem. Electron. Res. Arch.; 2024; 32, pp. 6200-6224. [DOI: https://dx.doi.org/10.3934/era.2024288]

35. Haq, S.; Noreen, A.; Akbar, T.; Arifeen, S.U.; Ghafoor, A.; Khan, Z.A. Numerical solution of seventh order KdV equations via quintic B-splines collocation method. Alex. Eng. J.; 2025; 114, pp. 497-506. [DOI: https://dx.doi.org/10.1016/j.aej.2024.11.098]

36. Abd-Elhameed, W.M.; Al-Sady, A.M.; Alqubori, O.M.; Atta, A.G. Numerical treatment of the fractional Rayleigh–Stokes problem using some orthogonal combinations of Chebyshev polynomials. AIMS Math.; 2024; 9, pp. 25457-25481. [DOI: https://dx.doi.org/10.3934/math.20241243]

37. 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.; 2021; 384, 113157. [DOI: https://dx.doi.org/10.1016/j.cam.2020.113157]

38. Abd-Elhameed, W.M.; Alsuyuti, M.M. New spectral algorithm for fractional delay pantograph equation using certain orthogonal generalized Chebyshev polynomials. Commun. Nonlinear Sci. Numer. Simul.; 2024; 141, 108479. [DOI: https://dx.doi.org/10.1016/j.cnsns.2024.108479]

39. Abd-Elhameed, W.M.; Ahmed, H.M. Tau and Galerkin operational matrices of derivatives for treating singular and Emden–Fowler third-order-type equations. Int. J. Mod. Phys. C; 2022; 33, 2250061. [DOI: https://dx.doi.org/10.1142/S0129183122500619]

40. Yang, X.; Jiang, X.; Zhang, H. A time–space spectral tau method for the time fractional cable equation and its inverse problem. Appl. Numer. Math.; 2018; 130, pp. 95-111. [DOI: https://dx.doi.org/10.1016/j.apnum.2018.03.016]

41. El-Sayed, A.A.; Boulaaras, S.; Sweilam, N.H. Numerical solution of the fractional-order logistic equation via the first-kind Dickson polynomials and spectral tau method. Math. Methods Appl. Sci.; 2023; 46, pp. 8004-8017. [DOI: https://dx.doi.org/10.1002/mma.7345]

42. Napoli, A.; Abd-Elhameed, W.M. An innovative harmonic numbers operational matrix method for solving initial value problems. Calcolo; 2017; 54, pp. 57-76. [DOI: https://dx.doi.org/10.1007/s10092-016-0176-1]

43. Abdelhakem, M.; Moussa, H. Pseudo-spectral matrices as a numerical tool for dealing BVPs, based on Legendre polynomials’ derivatives. Alex. Eng. J.; 2023; 66, pp. 301-313. [DOI: https://dx.doi.org/10.1016/j.aej.2022.11.006]

44. 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.; 2024; 13, 20220332. [DOI: https://dx.doi.org/10.1515/nleng-2022-0332]

45. Saadatmandi, A.; Dehghan, M. Legendre operational matrix of fractional derivatives and its applications in solving the fractional differential equations. Appl. Math. Model.; 2010; 34, pp. 768-779.

46. Ahmed, H.M. Enhanced shifted Jacobi operational matrices of derivatives: Spectral algorithm for solving multiterm variable-order fractional differential equations. Bound. Value Probl.; 2023; 2023, 108. [DOI: https://dx.doi.org/10.1186/s13661-023-01796-1]

47. Ahmed, H.M. A new first finite class of classical orthogonal polynomials operational matrices: An application for solving fractional differential equations. Contemp. Math.; 2023; pp. 974-994. [DOI: https://dx.doi.org/10.37256/cm.4420232716]

48. Biswas, C.; Das, S.; Singh, A.; Altenbach, H. Solution of variable-order partial integro-differential equation using Legendre wavelet approximation and operational matrices. ZAMM-J. Appl. Math. Mech.; 2023; 103, e202200222. [DOI: https://dx.doi.org/10.1002/zamm.202200222]

49. Shloof, A.M.; Senu, N.; Ahmadian, A.; Nouh, M.I.; Salahshour, S. A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method. Phys. Scr.; 2023; 98, 034004. [DOI: https://dx.doi.org/10.1088/1402-4896/acba5d]

50. Pourbabaee, M.; Saadatmandi, A. The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications. Int. J. Comput. Math.; 2021; 98, pp. 2310-2329. [DOI: https://dx.doi.org/10.1080/00207160.2021.1895988]

51. Saldır, O.; Sakar, M.G.; Erdogan, F. Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate. Comput. Appl. Math.; 2019; 38, 198. [DOI: https://dx.doi.org/10.1007/s40314-019-0979-1]

52. Abd-Elhameed, W.M.; Alqubori, O.M.; Atta, A.G. A collocation procedure for treating the time-fractional FitzHugh–Nagumo differential equation using shifted Lucas polynomials. Mathematics; 2024; 12, 3672. [DOI: https://dx.doi.org/10.3390/math12233672]

53. Koepf, W. Hypergeometric Summation; 2nd ed. Springer Universitext Series; Springer: Berlin/Heidelberg, Germany, 2014.

54. Andrews, G.E.; Askey, R.; Roy, R. Special Functions; Cambridge University Press: Cambridge, UK, 1999.

55. 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.; 2023; 7, 652. [DOI: https://dx.doi.org/10.3390/fractalfract7090652]

56. Luke, Y.L. Inequalities for generalized hypergeometric functions. J. Approx. Theory; 1972; 5, pp. 41-65. [DOI: https://dx.doi.org/10.1016/0021-9045(72)90028-7]

57. Jameson, G.J.O. The incomplete gamma functions. Math. Gaz.; 2016; 100, pp. 298-306. [DOI: https://dx.doi.org/10.1017/mag.2016.67]