Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

Abstract

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

1. Introduction

Fractional calculus [1,2,3,4,5,6,7,8,9,10] generalizes the standard differential and integral operators to non-integer orders, which, due to their non-local properties, were proven adequate for modeling and controlling systems with memory. Fractional integro-differential equations with weakly singular kernel (FIDE-WSK) are effective for modeling physical phenomena in science and engineering fields (see [11] and references therein). However, often their solutions cannot be obtained analytically and, thus, numerical techniques were developed. Nemati et al. [12] adopted a procedure based on second kind Chebyshev polynomials and operational matrix. Wang and Zhu [13] used second kind Chebyshev wavelets (SCW) and operational matrix of fractional order integration, and Zhao et al. [14] proposed collocation methods. Recently, Mokhtary [15] applied the operational Jacobi Tau method, while others developed improved distinct techniques [16,17,18,19,20].

Variable-order fractional integro-differential equations (V-O-FIDE) generalize the standard FIDE and obtaining their solutions is more challenging. Xu and Ertuark [21] used the finite differences method, Chen et al. [22] adopted Legendre wavelets and Sun and Zhu [23] proposed Chebyshev polynomials. We can also cite Chen et al. [24], who derived a solution for the variable-order fractional linear cable equation using polynomials of Bernstein type, and Bhrawy and Zaky [25], who used a collocation method for the two-dimensional variable-order fractional nonlinear cable equation. Tavares et al. [26] proposed to use the Caputo derivative, while other numerical techniques can be found in the literature [27,28,29,30,31,32,33,34,35,36,37,38].

Spectral methods [39,40,41,42,43,44,45,46,47,48,49,50] have been widely adopted for solving different types of equations [51,52,53,54]. Their main goal is to approximate the solution by a finite sum of basis functions, where the coefficients are selected in order to minimize the error between the exact and approximate solutions. For spectral collocation methods, the approximation must satisfy the exact solution at the collocation points, meaning that the residuals must be zero at those points.

In this paper we extend the shifted Jacobi–Gauss collocation method to solve variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK). Using the fractional-order shifted Jacobi–Gauss collocation (FSJ-G-C) method with the Riemann–Liouville fractional (R-LF) derivative and integral, and fractional-order shifted Jacobi polynomials (FO-SJP), we reduce the original problem to N algebraic equations that, together with the initial conditions, yield a system of $N+1$ equations. We apply the technique to solve several examples and prove its efficiency and accuracy.

The structure of the paper is presented in the sequel. Section 2 briefly introduces some concepts of fractional calculus and FO-SJP. Section 3 and Section 4 present the new algorithm for solving VO-FIDE-WSK subject to initial and to nonlocal boundary conditions, respectively. Section 5 applies the method to some problems and illustrates its accuracy and effectiveness. Section 6 discusses the results obtained. Finally, Section 7 outlines the conclusions.

2. Mathematical Preliminaries

2.1. Basic Tools

We recall the Riemann–Liouville definitions of fractional integral and derivative of constant and variable-order, $\delta >0$ and $\delta \left(t\right)>0$.

Definition 1.

The R-LF integral operator $I^{\delta }$ is:

$$\begin{array}{c}\hfill I^{\delta }\psi \left(t\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_0^t{(t-\zeta )}^{\delta -1}\psi \left(\zeta \right)d\zeta ,\delta >0,x>0,I^0\psi \left(t\right)=\psi \left(t\right),\end{array}$$

(1)

which satisfies:

$$\begin{array}{c}\hfill I^{\gamma }{I}^{\delta }\psi \left(t\right)=I^{\gamma +\delta }\psi \left(t\right),I^{\gamma }{I}^{\delta }\psi \left(t\right)=I^{\delta }{I}^{\gamma }\psi \left(t\right),I^{\delta }{t}^{\rho }={\displaystyle \frac{\Gamma (\rho +1)}{\Gamma (\rho +1+\delta )}}{t}^{\rho +\delta },\end{array}$$

(2)

where $\Gamma \left(\delta \right)={\int }_0^{\infty }{t}^{\delta -1}{e}^{-t}dt$.

Definition 2.

The R-LF derivative operator $D^{\delta }$ is:

$$D^{\delta }\psi \left(t\right)=\frac{1}{\Gamma (m-\delta )}\frac{d^m}{d{t}^m}\left({\int }_0^t{(t-s)}^{m-\delta -1}\psi \left(s\right)ds\right),m-1<\delta \le m,t>0,$$

(3)

where m stands for the ceiling function applied on δ.

Definition 3.

The R-LF integral operator of variable order $I^{\delta \left(t\right)}$ is:

$$I^{\delta \left(t\right)}\psi =\frac{1}{\Gamma \left(\delta \left(t\right)\right)}{\int }_0^t{(t-s)}^{\delta \left(t\right)-1}\psi \left(s\right)ds.$$

(4)

Definition 4.

The R-LF derivative operator of variable order [55,56] $D^{\delta \left(t\right)}$ is:

$$D^{\delta \left(t\right)}\psi \left(t\right)=\frac{1}{\Gamma (m-\delta \left(t\right))}{\left[\frac{d^m}{d{\tau }^m}{\int }_0^{\tau }{(\tau -s)}^{m-\delta \left(t\right)-1}\psi \left(s\right)ds\right]}_{\tau =t},m-1<\delta \left(t\right)\le m,t>0.$$

(5)

2.2. The Shifted Jacobi Polynomials

Considering the properties of the Jacobi polynomials, ${\mathcal{P}}_k^{(\sigma ,\rho )}\left(t\right)$, we can write [57,58]:

$$\begin{array}{cc}& {\mathcal{P}}_{k+1}^{(\sigma ,\rho )}\left(t\right)=(a_k^{(\sigma ,\rho )}t-b_k^{(\sigma ,\rho )}){\mathcal{P}}_k^{(\sigma ,\rho )}\left(t\right)-c_k^{(\sigma ,\rho )}{\mathcal{P}}_{k-1}^{(\sigma ,\rho )}\left(t\right),k\ge 1,\hfill \\ & {\mathcal{P}}_0^{(\sigma ,\rho )}\left(t\right)=1,{\mathcal{P}}_1^{(\sigma ,\rho )}\left(t\right)=\frac{1}{2}(\sigma +\rho +2)t+\frac{1}{2}(\sigma -\rho ),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\mathcal{P}}_k^{(\sigma ,\rho )}(-t)={(-1)}^k{\mathcal{P}}_k^{(\sigma ,\rho )}\left(t\right),{\mathcal{P}}_k^{(\sigma ,\rho )}(-1)=\frac{{(-1)}^k\Gamma (k+\rho +1)}{k!\Gamma (\rho +1)},\end{array}$$

(6)

where the parameters $\sigma ,\rho >-1$, $x\in [-1,1]$ and

$$\begin{array}{cc}\hfill a_k^{(\sigma ,\rho )}& =\frac{(2k+\sigma +\rho +1)(2k+\sigma +\rho +2)}{2(k+1)(k+\sigma +\rho +1)},\hfill \\ \hfill b_k^{(\sigma ,\rho )}& =\frac{({\rho }^2-{\sigma }^2)(2k+\sigma +\rho +1)}{2(k+1)(k+\sigma +\rho +1)(2k+\sigma +\rho )},\hfill \\ \hfill c_k^{(\sigma ,\rho )}& =\frac{(k+\sigma )(k+\rho )(2k+\sigma +\rho +2)}{(k+1)(k+\sigma +\rho +1)(2k+\sigma +\rho )}.\hfill \end{array}$$

The derivative of order $r\in \mathbb{N}$ of ${\mathcal{P}}_j^{(\sigma ,\rho )}\left(t\right)$ is:

$$\begin{array}{c}\hfill D^r{\mathcal{P}}_j^{(\sigma ,\rho )}\left(t\right)=\frac{\Gamma (j+\sigma +\rho +q+1)}{2^r\Gamma (j+\sigma +\rho +1)}{\mathcal{P}}_{j-r}^{(\sigma +r,\rho +r)}\left(t\right).\end{array}$$

(7)

For the shifted Jacobi polynomial ${\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(t\right)={\mathcal{P}}_k^{(\sigma ,\rho )}(\frac{2t}{\mathcal{L}}-1),\mathcal{L}>0$, we have the analytic expression:

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(t\right)& =\sum _{j=0}^k{(-1)}^{k-j}\frac{\Gamma (k+\rho +1)\Gamma (j+k+\sigma +\rho +1)}{\Gamma (j+\rho +1)\Gamma (k+\sigma +\rho +1)(k-j)!j!{\mathcal{L}}^j}{t}^j\hfill \\ \hfill & =\sum _{j=0}^k\frac{\Gamma (k+\sigma +1)\Gamma (k+j+\sigma +\rho +1)}{j!(k-j)!\Gamma (j+\sigma +1)\Gamma (k+\sigma +\rho +1){\mathcal{L}}^j}{(t-\mathcal{L})}^j\hfill \end{array}$$

(8)

and, therefore, it yields:

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(0\right)& ={(-1)}^k\frac{\Gamma (k+\rho +1)}{\Gamma (\rho +1)k!},\hfill \\ \hfill {\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(\mathcal{L}\right)& =\frac{\Gamma (k+\sigma +1)}{\Gamma (\sigma +1)k!},\hfill \end{array}$$

(9)

$$D^r{\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(0\right)=\frac{{(-1)}^{k-r}\Gamma (k+\rho +1){(k+\sigma +\rho +1)}_r}{L^r\Gamma (k-r+1)\Gamma (r+\rho +1)},$$

(10)

$$D^r{\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(\mathcal{L}\right)=\frac{\Gamma (k+\sigma +1){(k+\sigma +\rho +1)}_r}{L^r\Gamma (k-r+1)\Gamma (r+\sigma +1)},$$

(11)

$$D^r{\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}\left(t\right)=\frac{\Gamma (r+k+\sigma +\rho +1)}{{\mathcal{L}}^r\Gamma (k+\sigma +\rho +1)}{\mathcal{P}}_{\mathcal{L},k-r}^{(\sigma +r,\rho +r)}\left(t\right).$$

(12)

Taking $w_{\mathcal{L}}^{(\sigma ,\rho )}\left(t\right)={(\mathcal{L}-t)}^{\sigma }{t}^{\rho }$, the inner product and norm in the weighted space $L_{w_{\mathcal{L}}^{(\sigma ,\rho )}}^2[0,\mathcal{L}]$ are:

$${(u,v)}_{w_{\mathcal{L}}^{(\sigma ,\rho )}}=\underset{0}{\overset{\mathcal{L}}{\int }}u\left(t\right)v\left(t\right)w_{\mathcal{L}}^{(\sigma ,\rho )}\left(t\right)dt,{\parallel v\parallel }_{w_{\mathcal{L}}^{(\sigma ,\rho )}}={(v,v)}_{w_{\mathcal{L}}^{(\sigma ,\rho )}}^{\frac{1}{2}}.$$

(13)

A complete $L_{w_{\mathcal{L}}^{(\sigma ,\rho )}}^2[0,\mathcal{L}]$-orthogonal system is made of a set of shifted Jacobi polynomials, where:

$$\parallel {\mathcal{P}}_{\mathcal{L},k}^{(\sigma ,\rho )}{\parallel }_{w_{\mathcal{L}}^{(\sigma ,\rho )}}^2={\left(\frac{\mathcal{L}}{2}\right)}^{\sigma +\rho +1}{h}_k^{(\sigma ,\rho )}=h_{\mathcal{L},k}^{(\sigma ,\rho )}.$$

(14)

Let the nodes and Christoffel numbers of the standard Jacobi–Gauss interpolation in the interval $[-1,1]$ be denoted by $t_{\mathcal{N},j}^{(\sigma ,\rho )}$ and ${\varpi }_{\mathcal{N},j}^{(\sigma ,\rho )},0⩽j⩽\mathcal{N}$, respectively. For the shifted Jacobi–Gauss interpolation in the interval $[0,\mathcal{L}]$ we have:

$$t_{\mathcal{L},\mathcal{N},j}^{(\sigma ,\rho )}=\frac{\mathcal{L}}{2}(t_{\mathcal{N},j}^{(\sigma ,\rho )}+1),$$

$${\varpi }_{\mathcal{L},\mathcal{N},j}^{(\sigma ,\rho )}={\left(\frac{\mathcal{L}}{2}\right)}^{\sigma +\rho +1}{\varpi }_{\mathcal{N},j}^{(\sigma ,\rho )},0⩽j⩽\mathcal{N}.$$

For $N>0$, $\varphi \in S_{2\mathcal{N}+1}[0,\mathcal{L}]$ and the Jacobi–Gauss quadrature property, we obtain:

$$\begin{array}{cc}\hfill \underset{0}{\overset{\mathcal{L}}{\int }}{(\mathcal{L}-t)}^{\sigma }{t}^{\rho }\varphi \left(t\right)dt& ={\left(\frac{\mathcal{L}}{2}\right)}^{\sigma +\rho +1}\underset{-1}{\overset{1}{\int }}{(1-t)}^{\sigma }{(1+t)}^{\rho }\varphi \left(\frac{\mathcal{L}}{2}(t+1)\right)dt\hfill \\ & ={\left(\frac{\mathcal{L}}{2}\right)}^{\sigma +\rho +1}\sum _{j=0}^{\mathcal{N}}{\varpi }_{\mathcal{N},j}^{(\sigma ,\rho )}\varphi \left(\frac{\mathcal{L}}{2}(t_{\mathcal{N},j}^{(\sigma ,\rho )}+1)\right)\hfill \\ & =\sum _{j=0}^{\mathcal{N}}{\varpi }_{\mathcal{L},\mathcal{N},j}^{(\sigma ,\rho )}\varphi \left(t_{\mathcal{L},\mathcal{N},j}^{(\sigma ,\rho )}\right).\hfill \end{array}$$

(15)

2.3. Fractional-Order Shifted Jacobi Polynomials

We present some results related with FO-SJP.

Definition 5.

The FO-SJP are given by:

$$\begin{array}{c}\hfill {\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)={\mathcal{P}}_j^{(\sigma ,\rho )}\left(2{\left(\frac{t}{\mathcal{T}}\right)}^{\lambda }-1\right),0<\lambda <1,j=0,1,\cdots ,0\le t\le \mathcal{T}.\end{array}$$

(16)

Theorem 1.

For ${\mathcal{W}}_{\mathcal{T},f}^{(\sigma ,\rho ,\lambda )}\left(t\right)=\lambda {({\mathcal{T}}^{\lambda }-t^{\lambda })}^{\sigma }{t}^{\lambda \rho +\lambda -1}$, the set of FO-SJP forms a complete $L_{{\mathcal{W}}_f^{(\sigma ,\rho ,\lambda )}}^2[0,\mathcal{T}]$-orthogonal system:

$$\underset{0}{\overset{\mathcal{T}}{\int }}{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{W}}_{\mathcal{T},f}^{(\sigma ,\rho ,\lambda )}\left(t\right)dx={\delta }_{ij}{h}_{\mathcal{T},k}^{(\sigma ,\rho ,\lambda )},$$

(17)

where $h_{\mathcal{T},k}^{(\sigma ,\rho ,\lambda )}={\left({\frac{\mathcal{T}}{2}}^{\lambda }\right)}^{\sigma +\rho +1}{h}_i^{(\sigma ,\rho )}$.

Proof. 

The orthogonality of the Jacobi polynomials allows to write:

$$\underset{-1}{\overset{1}{\int }}{\mathcal{P}}_i^{(\sigma ,\rho )}\left(x\right){\mathcal{P}}_j^{(\sigma ,\rho )}\left(x\right){\mathcal{W}}_f^{(\sigma ,\rho )}\left(x\right)dx={\delta }_{ij}{h}_i^{(\sigma ,\rho )}.$$

(18)

If we let $x=2{\left(\frac{t}{\mathcal{T}}\right)}^{\lambda }-1$, then we obtain:

$$\begin{array}{c}\hfill \underset{-1}{\overset{1}{\int }}{\mathcal{P}}_i^{(\sigma ,\rho )}\left(x\right){\mathcal{P}}_j^{(\sigma ,\rho )}\left(x\right){\mathcal{W}}_f^{(\sigma ,\rho )}\left(x\right)dx\\ \hfill =\frac{2\lambda }{{\mathcal{T}}^{\lambda }}\underset{0}{\overset{\mathcal{T}}{\int }}{t}^{\lambda -1}{\mathcal{P}}_i^{(\sigma ,\rho )}\left(2{\left(\frac{t}{\mathcal{T}}\right)}^{\lambda }-1\right){\mathcal{P}}_j^{(\sigma ,\rho )}\left(2{\left(\frac{t}{\mathcal{T}}\right)}^{\lambda }-1\right){\mathcal{W}}_f^{(\sigma ,\rho )}\left(2{\left(\frac{t}{\mathcal{T}}\right)}^{\lambda }-1\right)dt\\ \hfill ={\left(\frac{2}{{\mathcal{T}}^{\lambda }}\right)}^{\sigma +\rho +1}\underset{0}{\overset{\mathcal{T}}{\int }}{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{W}}_{\mathcal{T},f}^{(\sigma ,\rho ,\lambda )}\left(t\right)dt\\ \hfill ={\delta }_{ij}{h}_i^{(\sigma ,\rho )},\end{array}$$

(19)

yielding:

$$\begin{array}{cc}\hfill \underset{0}{\overset{\mathcal{T}}{\int }}{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right){\mathcal{W}}_{\mathcal{T},f}^{(\sigma ,\rho ,\lambda )}\left(t\right)dx=& {\delta }_{ij}{\left({\frac{\mathcal{T}}{2}}^{\lambda }\right)}^{\sigma +\rho +1}{h}_i^{(\sigma ,\rho )}\hfill \\ \hfill =& {\delta }_{ij}{h}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}.\hfill \end{array}$$

(20)

□

Corollary 1.

Let the finite-dimensional fractional-polynomial space be denoted by ${\mathcal{F}}_N=span\{{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}:0\le i\le N\}$. The orthogonality property (20), allows to express $\zeta \left(t\right)\in L_{{\mathcal{W}}_f^{(\sigma ,\rho ,\lambda )}}^2[0,\mathcal{T}]$ as:

$$\zeta \left(t\right)=\sum _{i=0}^{\infty }{\gamma }_i{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}\left(t\right),{\gamma }_i=\frac{1}{h_{\mathcal{T},k}^{(\sigma ,\rho ,\lambda )}}\underset{0}{\overset{\mathcal{T}}{\int }}{\mathcal{P}}_{\mathcal{T},i}^{(\sigma ,\rho ,\lambda )}\left(t\right)\zeta \left(t\right){\mathcal{W}}_{\mathcal{T},f}^{(\sigma ,\rho ,\lambda )}\left(t\right)dt.$$

Theorem 2.

The δ-order derivative of the FO-SJP, $D_t^{\delta }{\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)$, can be expressed in terms of the FO-SJP as:

$$D_t^{\delta }{\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)=\sum _{n=0}^{\mathcal{N}}{\epsilon }_{\delta }^{(n,j,\sigma ,\rho ,\lambda )}{\mathcal{P}}_{\mathcal{T},n}^{(\sigma ,\rho ,\lambda )}\left(t\right),$$

(21)

where

$$\begin{array}{c}{\epsilon }_{\delta }^{(n,j,\sigma ,\rho ,\lambda )}\hfill \\ =\sum _{k=1}^j\sum _{s=0}^n\frac{E_k^{(\sigma ,\rho ,\lambda ,j)}{E}_s^{(\sigma ,\rho ,\lambda ,n)}}{h_{\mathcal{T},n}^{(\sigma ,\rho ,\lambda )}}\frac{k\lambda \Gamma \left(k\lambda \right)\Gamma (\sigma +1)\Gamma \left(k+s+\rho -\frac{\delta }{\lambda }+1\right)T^{\lambda (\sigma +\rho +k+s+1)-\delta }}{\Gamma (k\lambda -\delta +1)\Gamma \left(k+s+\sigma +\rho -\frac{\delta }{\lambda }+2\right)}.\hfill \end{array}$$

Proof. 

The analytical expression of ${\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)$ is:

$${\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)=\sum _{k=0}^j{E}_k^{(\sigma ,\rho ,\lambda ,j)}{t}^{\lambda k},$$

where

$$E_k^{(\sigma ,\rho ,\lambda ,j)}=\frac{{(-1)}^{j-k}\left(\Gamma (j+\rho +1)\Gamma (j+k+\sigma +\rho +1)\right)}{k!(j-k)!\Gamma (k+\rho +1)\Gamma (j+\sigma +\rho +1){\mathcal{T}}^{\lambda k}}.$$

By means of Equation (3), we have:

$$D_t^{\delta }\left(t^{\lambda k}\right)=\frac{k\lambda \Gamma \left(k\lambda \right)t^{k\lambda -\delta }}{\Gamma (k\lambda -\delta +1)}$$

and, thus, if follows:

$$\begin{array}{cc}\hfill D_t^{\delta }\left({\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right)=& {\mathcal{P}}_{\mathcal{T},j}^{(\delta ,\sigma ,\rho ,\lambda )}\left(t\right)\hfill \\ \hfill =& \sum _{k=1}^j{E}_k^{(\sigma ,\rho ,\lambda ,j)}\frac{k\lambda \Gamma \left(k\lambda \right)t^{k\lambda -\delta }}{\Gamma (k\lambda -\delta +1)}.\hfill \end{array}$$

Using the Corollary 1, we can write:

$$t^{k\lambda -\delta }=\sum _{n=0}^{\mathcal{N}}{b}_{k,n}{\mathcal{P}}_{\mathcal{T},n}^{(\sigma ,\rho ,\lambda )}\left(t\right),$$

(22)

where

$$b_{k,n}=\frac{1}{h_{\mathcal{T},n}^{(\sigma ,\rho ,\lambda )}}\sum _{s=0}^n\frac{\Gamma (\sigma +1)\Gamma \left(k+s+\rho -\frac{\delta }{\lambda }+1\right)T^{\lambda (\sigma +\rho +k+s+1)-\delta }}{\Gamma \left(k+s+\sigma +\rho -\frac{\delta }{\lambda }+2\right)}{E}_s^{(\sigma ,\rho ,\lambda ,n)}.$$

Thence, we conclude that:

$$\begin{array}{cc}\hfill D_t^{\delta }\left({\mathcal{P}}_{\mathcal{T},j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right)=& \sum _{n=0}^{\mathcal{N}}{\epsilon }_{\delta }^{(n,j,\sigma ,\rho ,\lambda )}{\mathcal{P}}_{\mathcal{T},n}^{(\sigma ,\rho ,\lambda )}\left(t\right),\hfill \end{array}$$

(23)

where

$${\epsilon }_{\delta }^{(n,j,\sigma ,\rho ,\lambda )}=\sum _{k=1}^j{E}_k^{(\sigma ,\rho ,\lambda ,j)}\frac{k\lambda \Gamma \left(k\lambda \right)}{\Gamma (k\lambda -\delta +1)}{b}_{k,n}.$$

□

3. Algorithm for Solving VO-FIDE-WSK

The VO-FIDE-WSK to be solved are:

$$D^{\delta \left(t\right)}\chi \left(t\right)={\eta }_1{\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\alpha }}ds+{\eta }_2{\int }_0^{1}k(t,s)\chi \left(s\right)ds+h\left(t\right),0<\alpha <1,$$

(24)

with initial condition:

$$\chi \left(0\right)=d,$$

(25)

where $D^{\delta \left(t\right)}$ denotes the $\delta \left(t\right)$ variable-order fractional derivative, $\chi \left(t\right)$ is an unknown function and $h\left(t\right)$ is given.

The approximate solution of Equation (24) is given by:

$${\chi }_N\left(t\right)=\sum _{j=0}^N{e}_j{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)$$

(26)

and the variable-order fractional derivative $D^{\delta \left(t\right)}$ of the approximate solution ${\chi }_N\left(t\right)$ is estimated as:

$$\begin{array}{c}\hfill D^{\delta \left(t\right)}{\chi }_N\left(t\right)=\sum _{j=0}^N{e}_j{D}^{\delta \left(t\right)}\left({\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right).\end{array}$$

(27)

Since we have:

$$\begin{array}{c}\hfill D^{\delta \left(t\right)}{t}^{\lambda k}=\frac{1}{\Gamma (1-\delta (t\left)\right)}{\left[\frac{\partial }{\partial \tau }\left(\underset{0}{\overset{\tau }{\int }}\frac{{\chi }^{\lambda k}}{{(\tau -\chi )}^{\delta \left(t\right)}}d\chi \right)\right]}_{\tau =t}=\frac{t^{\lambda k-\delta \left(t\right)}\Gamma (1+\lambda k)}{\Gamma (1+\lambda k-\delta (t\left)\right)}.\end{array}$$

(28)

Then,

$$\begin{array}{cc}\hfill & D^{\delta \left(t\right)}{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)={\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\hfill \\ \hfill & =\sum _{k=0}^j\frac{{(-1)}^{j-k}\Gamma (j+\rho +1)\Gamma (j+k+\sigma +\rho +1)}{\Gamma (k+\rho +1)\Gamma (j+\sigma +\rho +1)(j-k)!k!L^k}{D}^{\delta \left(t\right)}{t}^{\lambda k}\hfill \\ \hfill & =\sum _{k=\lceil \delta \left(t\right)\rceil }^j\frac{{(-1)}^{i-k}\Gamma (1+\lambda k)\Gamma (j+\rho +1)\Gamma (j+k+\sigma +\rho +1)}{\Gamma (k+\rho +1)\Gamma (j+\sigma +\rho +1)(j-k)!k!\Gamma (\lambda k-\delta \left(t\right)+1)L^k}{t}^{\lambda k-\delta \left(t\right)}.\hfill \end{array}$$

(29)

Accordingly,

$$\begin{array}{c}\hfill D^{\delta \left(t\right)}{\chi }_N\left(t\right)=\sum _{j=0}^N{e}_j{D}^{\delta \left(t\right)}\left({\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right)=\sum _{j=0}^N{e}_j{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right).\end{array}$$

(30)

Comparing the term ${\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\alpha }}ds$ in Equations (1) and (24) we may write:

$${\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\alpha }}ds=\Gamma \left(\alpha \right)I^{1-\alpha }\chi \left(t\right),$$

(31)

with the approximation of $I^{1-\alpha }\chi \left(t\right)$ as:

$$\begin{array}{c}\hfill I^{1-\alpha }{\chi }_N\left(t\right)=\sum _{j=0}^N{e}_j{I}^{1-\alpha }\left({\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right).\end{array}$$

(32)

Given that:

$$\begin{array}{cc}\hfill I^{1-\alpha }{t}^{\lambda k}& =\frac{1}{\Gamma (1-\alpha )}{\int }_0^t\frac{t^{\lambda k}}{{(t-s)}^{\alpha }}ds\hfill \\ \hfill =& \frac{t^{\lambda k+1-\alpha }\Gamma (1+\lambda k)}{\Gamma (2+\lambda k-\alpha )},\hfill \end{array}$$

(33)

then

$$\begin{array}{c}\hfill I^{1-\alpha }{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)={\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\\ \hfill =\sum _{k=0}^j\frac{{(-1)}^{j-k}\Gamma (j+\rho +1)\Gamma (j+k+\sigma +\rho +1)}{\Gamma (k+\rho +1)\Gamma (j+\sigma +\rho +1)(j-k)!k!L^k}{I}^{1-\alpha }{t}^{\lambda k}\\ \hfill =\sum _{k=\lceil \delta \left(t\right)\rceil }^j\frac{{(-1)}^{i-k}\Gamma (1+\lambda k)\Gamma (j+\rho +1)\Gamma (j+k+\sigma +\rho +1)}{\Gamma (k+\rho +1)\Gamma (j+\sigma +\rho +1)(j-k)!k!\Gamma (2+\lambda k-\alpha )L^k}{t}^{\lambda k+1-\alpha }.\end{array}$$

(34)

Accordingly,

$$\begin{array}{c}\hfill I^{1-\alpha }{\chi }_N\left(t\right)=\sum _{j=0}^N{e}_j{I}^{1-\alpha }\left({\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\right)=\sum _{j=0}^N{e}_j{\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\end{array}$$

(35)

and we rewrite Equation (24) by using Equations (30) and (35) as:

$$\begin{array}{c}\hfill \sum _{j=0}^N{e}_j{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)\\ \hfill =h\left(t\right)+{\eta }_1\Gamma \left(\alpha \right)\sum _{j=0}^N{e}_j{\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t\right)+{\eta }_2{\int }_0^{1}k(t,s)\sum _{j=0}^N{e}_j{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(s\right)ds.\end{array}$$

(36)

The FSJ-G-C technique requires that the residual of (36) be zero at the $N+1$ shifted Jacobi Gauss points. Therefore, using (26) and (36), we can rewrite (24) as:

$$\begin{array}{c}\hfill \sum _{j=0}^N{e}_j{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)\\ \hfill =h\left(t_{L,N,i}^{(\sigma ,\rho )}\right)+{\eta }_1\Gamma \left(\alpha \right)\sum _{j=0}^N{e}_j{\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)+{\eta }_2{\int }_0^{1}k(t_{L,N,i}^{(\sigma ,\rho )},s)\sum _{j=0}^N{e}_j{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(s\right)ds\end{array}$$

(37)

and

$$\sum _{j=0}^N{e}_j\left[{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)-{\eta }_1\Gamma \left(\alpha \right){\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)-{\eta }_2{\int }_0^{1}k(t_{L,N,i}^{(\sigma ,\rho )},s){\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(s\right)ds\right]=h\left(t_{L,N,i}^{(\sigma ,\rho )}\right).$$

(38)

Joining Equations (25) and (26), we can write:

$$\begin{array}{c}\hfill \sum _{j=0}^N{e}_j{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(0\right)=\sum _{j=0}^N{(-1)}^j\frac{\Gamma (j+\rho +1)}{\Gamma (\rho +1)j!}{e}_j=d.\end{array}$$

(39)

Finally, from Equations (38) and (39) we get a system of $N+1$ algebraic equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \sum _{j=0}^N}{(-1)}^j\frac{\Gamma (j+\rho +1)}{\Gamma (\rho +1)j!}{e}_j=d,\hfill \\ {\displaystyle \sum _{j=0}^N}{e}_j\left[{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)-{\eta }_1\Gamma \left(\alpha \right){\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)-{\eta }_2{\int }_0^{1}k(t_{L,N,i}^{(\sigma ,\rho )},s){\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(s\right)ds\right]=h\left(t_{L,N,i}^{(\sigma ,\rho )}\right)\hfill \end{array}\right.\end{array}$$

(40)

to compute ${\chi }_N\left(t\right)$.

4. Algorithm for Solving VO-FIDE-WSK with Nonlocal Boundary Conditions

In this section, we present a modified spectral algorithm for solving the VO-FIDE-WSK

$$D^{\delta \left(t\right)}\chi \left(t\right)=h\left(t\right)+{\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\alpha }}ds,$$

(41)

with the nonlocal boundary conditions:

$$\chi \left(0\right)+\gamma \chi \left(1\right)+\lambda \underset{a}{\overset{b}{\int }}\varphi \left(s\right)\chi \left(s\right)ds=d_1.$$

(42)

Following the information included in the previous subsections, we obtain the system of algebraic equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \sum _{j=0}^N}{e}_j\left[{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(0\right)+{\gamma }_j{\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(1\right)+\lambda \left(\underset{a}{\overset{b}{\int }}\varphi \left(s\right){\mathcal{P}}_{L,j}^{(\sigma ,\rho ,\lambda )}\left(s\right)\right)\right]=d_1\hfill \\ \hfill & {\displaystyle \sum _{j=0}^N}{e}_j\left[{\Lambda }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)-{\eta }_1\Gamma \left(\alpha \right){\Delta }_{L,j}^{(\sigma ,\rho ,\lambda )}\left(t_{L,N,i}^{(\sigma ,\rho )}\right)\right]=h\left(t_{L,N,i}^{(\sigma ,\rho )}\right)\hfill \end{array}\right..\end{array}$$

(43)

After the coefficients $e_j$ are determined, it is straightforward to compute the approximate solution ${\chi }_N\left(t\right)$ at any value of t in the given domain.

5. Illustrative Examples

The proposed method is compared with alternative approaches to solve VO-FIDE-WSK equations. Its accuracy and effectiveness is illustrated. We must note that in all examples given below we take $L=1$.

Example 1.

We consider the VO-FIDE-WSK:

$$\begin{array}{cc}\hfill D^{\delta \left(t\right)}\chi \left(t\right)=& \frac{1}{2}{\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\frac{1}{2}}}ds+\frac{1}{3}{\int }_0^1(t-s)\chi \left(s\right)ds-\frac{2t^{2-\delta \left(t\right)}(\delta \left(t\right)-3t-3)}{\Gamma (4-\delta (t\left)\right)}-\hfill \\ & \frac{1}{105}8(6t+7)t^{5/2}-\frac{7t}{36}+\frac{3}{20},\hfill \end{array}$$

(44)

with supplementary condition:

$$\chi \left(0\right)=0.$$

(45)

The exact solution [13,19] is $\chi \left(t\right)=t^2+t^3$, when $\delta \left(t\right)=sin\left(t\right)$ and $\lambda =\frac{1}{2}$.

Table 1. The $E\left(t\right)$ of Example 1 for $(\sigma ,\rho )=(0,\frac{1}{2})$, $\lambda =\frac{1}{2}$.

	SCW [13]	CAS [19]	New Method
$\mathit{t}$	$\mathit{k}=\mathbf{4}$	$\mathit{N}=\mathbf{24}$	$\mathit{N}=\mathbf{6}$
0	$1.4395\times {10}^{-4}$	$6.3491\times {10}^{-3}$	$1.3878\times {10}^{-17}$
$\frac{1}{6}$	$2.2617\times {10}^{-4}$	$1.1460\times {10}^{-2}$	$1.0408\times {10}^{-16}$
$\frac{2}{6}$	$5.9826\times {10}^{-4}$	$9.6982\times {10}^{-3}$	$8.3267\times {10}^{-17}$
$\frac{3}{6}$	$1.1274\times {10}^{-3}$	$2.9504\times {10}^{-3}$	$2.7756\times {10}^{-17}$
$\frac{4}{6}$	$1.4961\times {10}^{-3}$	$9.6733\times {10}^{-3}$	$1.8041\times {10}^{-16}$
$\frac{5}{6}$	$2.2056\times {10}^{-3}$	$2.9484\times {10}^{-2}$	$1.9429\times {10}^{-16}$

summarizes and compares the absolute error, $E\left(t\right)$, at $t=\{0,\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6}\}$, between the exact and approximate solutions of the proposed method (for $N=6$) with those reported in references [13] (for $k=4$) and [19] (for $N=24$).

Figure 1 and Figure 2 depict $E\left(t\right)$ and the exact and approximate solutions, respectively, obtained by the proposed method for $N=6$. The results show good accuracy with a limited number of nodes.

Example 2.

We solve the VO-FIDE-WSK:

$$\begin{array}{cc}\hfill D^{\delta \left(t\right)}\chi \left(t\right)=& \frac{1}{4}{\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\frac{1}{4}}}ds+\frac{1}{7}{\int }_0^1{e}^{(t+s)}\chi \left(s\right)ds+\frac{t^{1-\delta \left(t\right)}(\delta \left(t\right)+2t-2)}{\Gamma (3-\delta (t\left)\right)}-\hfill \\ & \frac{4}{231}(11-8t)t^{7/4}-\frac{1}{7}(e-3)e^t,\hfill \end{array}$$

(46)

with condition:

$$\chi \left(0\right)=0.$$

(47)

The exact solution of the equation is $\chi \left(t\right)=t^2-t$ for $\lambda =\frac{1}{2}$ and $\delta \left(t\right)=0.15t$.

Table 2. The $E\left(t\right)$ of Example 2 obtained with $N=4$ for $(\sigma ,\rho )=(0,-\frac{1}{2})$, $(\sigma ,\rho )=(0,0)$ and $\lambda =\frac{1}{2}$.

	New Method at $\mathit{N}=4$
$\mathit{t}$	$\mathbf{\sigma }=\mathbf{0},\mathbf{\rho }=-\frac{\mathbf{1}}{\mathbf{2}}$	$\mathbf{\sigma }=\mathbf{\rho }=\mathbf{0}$
0	$5.2042\times {10}^{-18}$	$2.0817\times {10}^{-17}$
$0.1$	$1.3488\times {10}^{-15}$	$9.2548\times {10}^{-16}$
$0.2$	$5.8287\times {10}^{-16}$	$4.5103\times {10}^{-16}$
$0.3$	$3.3307\times {10}^{-16}$	$1.2490\times {10}^{-16}$
$0.4$	$2.7756\times {10}^{-16}$	$5.5511\times {10}^{-17}$
$0.5$	$5.2736\times {10}^{-16}$	$8.3267\times {10}^{-17}$
$0.6$	$8.3961\times {10}^{-16}$	$2.4980\times {10}^{-16}$
$0.7$	$1.2438\times {10}^{-15}$	$1.4572\times {10}^{-16}$
$0.8$	$1.2906\times {10}^{-15}$	$6.3144\times {10}^{-16}$
$0.9$	$1.1657\times {10}^{-15}$	$8.0491\times {10}^{-16}$

lists the values of $E\left(t\right)$ obtained with $N=4$ for $(\sigma ,\rho )=(0,-\frac{1}{2})$, $(\sigma ,\rho )=(0,0)$ and $\lambda =\frac{1}{2}$. Figure 3 and Figure 4 illustrate the $E\left(t\right)$ dynamics and compare the exact and approximate solutions, respectively. We can conclude that the method leads to very accurate solutions.

Example 3.

We solve the VO-FIDE-WSK:

$$\begin{array}{cc}\hfill D^{\delta \left(t\right)}\chi \left(t\right)=& {\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\frac{1}{2}}}ds+{\int }_0^1(t^2+cos\left(s\right))\chi \left(s\right)ds-\frac{1}{15}\left(16\sqrt{t}+5\right)t^2+\hfill \\ & \frac{2t^{2-tsin\left(t\right)}}{\Gamma (3-tsin(t\left)\right)}+sin\left(1\right)-2cos\left(1\right),\hfill \end{array}$$

(48)

with initial condition:

$$\chi \left(0\right)=0.$$

(49)

The exact solution is $\chi \left(t\right)=t^2$, when $\lambda =\frac{1}{2}$.

The solution of (48) is obtained by applying the new technique. The $E\left(t\right)$ is shown in

Table 3. The $E\left(t\right)$ of Example 3 obtained with $N=4$ for $(\sigma ,\rho )=(0,0)$ and $(\sigma ,\rho )=(\frac{1}{2},0)$, and $\lambda =\frac{1}{2}$.

	New Method at $\mathit{N}=4$
$\mathit{t}$	$\mathbf{\sigma }=\mathbf{\rho }=\mathbf{0}$	$\mathbf{\sigma }=\frac{\mathbf{1}}{\mathbf{2}},\mathbf{\rho }=\mathbf{0}$
0	0	0
$\frac{1}{6}$	$2.7756\times {10}^{-17}$	$7.6328\times {10}^{-17}$
$\frac{2}{6}$	$2.7756\times {10}^{-17}$	$1.9429\times {10}^{-16}$
$\frac{3}{6}$	$6.9389\times {10}^{-17}$	$4.4409\times {10}^{-16}$
$\frac{4}{6}$	$1.2837\times {10}^{-16}$	$5.2736\times {10}^{-16}$
$\frac{5}{6}$	$1.3878\times {10}^{-16}$	$6.8001\times {10}^{-16}$

when $N=4$ and for $(\sigma ,\rho )=(0,0)$ and $(\sigma ,\rho )=(\frac{1}{2},0)$. Figure 5 and Figure 6 depict the $E\left(t\right)$ dynamics and compare the approximate and exact solutions, respectively. As before, we verify the good accuracy provided by the method.

Example 4.

We solve the VO-FIDE-WSK:

$$\begin{array}{cc}\hfill D^{\delta \left(t\right)}\chi \left(t\right)=& \mu \left(t\right)\chi \left(t\right)+{\int }_0^t\frac{\chi \left(s\right)}{{(t-s)}^{\frac{1}{2}}}ds+\frac{1}{35}{t}^{11/6}\left(22-\frac{35\sqrt{\pi }\Gamma \left(\frac{7}{3}\right)}{\Gamma \left(\frac{17}{6}\right)}\right)+\hfill \\ & t^{-\delta \left(t\right)}\left(\frac{t^{4/3}\Gamma \left(\frac{7}{3}\right)}{\Gamma \left(\frac{7}{3}-\delta \left(t\right)\right)}+\frac{6t^3}{\Gamma (4-\delta (t\left)\right)}\right),\hfill \end{array}$$

(50)

with supplementary condition:

$$\chi \left(0\right)=0.$$

(51)

The exact solution of the equation [12,15] is $\chi \left(t\right)=t^3+t^{\frac{4}{3}}$ for $\delta \left(t\right)=sin\left(t\right)$, $\lambda =\frac{1}{2}$, and $\mu =-\frac{22}{35}{t}^{\frac{1}{2}}$.

To assess the rate of convergence, we list maximum absolute errors ($MAE$) in

Table 4. The $MAE$ of Example 4 for various values of N, $\lambda =\frac{1}{2}$, $\lambda =\frac{2}{3}$ and $(\sigma ,\rho )=(0,\frac{1}{2})$, and their comparison with those obtained in [12,15].

	Operational Matrix [12]	Operational Tau Method [15]	New Method
N			$\lambda =\frac{1}{2}$	$\lambda =\frac{2}{3}$
2	$1.34\times {10}^{-2}$	$1.34\times {10}^{-2}$	$2.54\times {10}^{-1}$	$1.11\times {10}^{-1}$
4	$5.14\times {10}^{-4}$	$5.14\times {10}^{-4}$	$8.67\times {10}^{-3}$	$1.14\times {10}^{-3}$
6	$1.56\times {10}^{-4}$	$1.56\times {10}^{-4}$	$2.76\times {10}^{-5}$	$8.28\times {10}^{-6}$
8	$6.46\times {10}^{-5}$	$6.46\times {10}^{-5}$	$5.93\times {10}^{-6}$	$3.99\times {10}^{-7}$
16	−	$7.38\times {10}^{-6}$	$4.91\times {10}^{-8}$	$5.54\times {10}^{-10}$
18	−	$4.62\times {10}^{-6}$	$3.69\times {10}^{-8}$	$1.37\times {10}^{-10}$
20	−	$2.85\times {10}^{-6}$	$1.16\times {10}^{-8}$	$7.63\times {10}^{-11}$

. The results show that at different values of N the method has superior accuracy than those addressed in references [12,15]. Figure 7 and Figure 8 depict $E\left(t\right)$ for $N=20$ and $\lambda =\frac{1}{2}$ and $\lambda =\frac{2}{3}$. Figure 9 and Figure 10 show the $MAE$ at various values of N, and $\lambda =\frac{1}{2}$ and $\lambda =\frac{2}{3}$. The results reveal good accuracy and exponential convergence of the method.

Example 5.

In this example, we address the VO-FIDE-WSK

$$D^{\delta \left(t\right)}\chi \left(t\right)=\underset{0}{\overset{t}{\int }}\frac{\chi \left(t\right)}{\sqrt{t-s}}ds+\frac{35}{128}\left(\frac{24\sqrt{\pi }{t}^{\frac{7}{2}-\delta \left(t\right)}}{\Gamma \left(\frac{9}{2}-\delta \left(t\right)\right)}-\pi t^4\right),$$

(52)

with the nonlocal condition:

$$\chi \left(0\right)+\chi \left(1\right)-3\underset{0}{\overset{1}{\int }}{s}^2\chi \left(s\right)ds=\frac{7}{13}.$$

(53)

The exact solution is a non-smooth $\chi \left(t\right)=t^{3.5}$ with $\delta \left(t\right)=\frac{t}{3}$.

The values of the MAE for Example 5 subject to the nonlocal condition (53) are listed in

Table 5. The MAE of Example 5 for various values of N, $\lambda =\frac{1}{2}$, $(\sigma ,\rho )=(0,\frac{1}{2})$, and $\delta \left(t\right)=\{\frac{t}{3},tsin\left(t\right)$}.

N	$\frac{\mathit{t}}{3}$	$\mathit{t}sin\left(\mathit{t}\right)$
2	$3.330\times {10}^{-1}$	$3.552\times {10}^{-1}$
4	$1.625\times {10}^{-2}$	$1.950\times {10}^{-2}$
6	$1.476\times {10}^{-4}$	$1.553\times {10}^{-4}$
8	$3.137\times {10}^{-15}$	$2.7113\times {10}^{-15}$

. Figure 11 and Figure 12 depict the error $E\left(t\right)$ for $N=8$, $\lambda =\frac{1}{2}$, $(\sigma ,\rho )=(0,\frac{1}{2})$, and $\delta \left(t\right)=\frac{t}{3}$ and $\delta \left(t\right)=tsin\left(t\right)$, respectively. We verify that the method yields accurate results.

6. Discussion

The examples presented in Section 5 can be summarized as follows:

Case I: Exs. 1, 2, 3 and 5 (with $N=8$).

In these examples, for $\lambda =\frac{1}{2}$, $\chi \left(t\right)$ is a polynomial in t, and N is greater than, or equal to, twice the degree of the polynomial $\chi \left(t\right)$. Therefore, the exact solution $\chi \left(t\right)$ belongs to the vector space ${\mathcal{F}}_N$, meaning that ${\chi }_N\left(t\right)=\chi \left(t\right)$, or, equivalently, $E\left(t\right)=0$. The numerical decimal errors $E\left(t\right)$ shown in Table 1, Table 2 and Table 3 and Table 5 for these cases are not due to the algorithm given in Equation (43), but from the decimal approximation of this expression. It must be mentioned that the algorithm gives the exact solution under the conditions adopted.

Case II: Exs. 4 and 5 (with $N=2,4,6$).

In these examples, we verify that $\chi \left(t\right)$ does not belong to the vector space ${\mathcal{F}}_N$. Thus, the errors $E\left(t\right)$ given in Table 4 and Table 5 (with $N=2,4,6$) are much larger than the ones of Case I.

Regarding Ex. 4, we verify that Equation (50), as well as the VO-FIDE-WSK (24), is linear in the unknown function $\chi \left(t\right)$ with the non-homogeneous term $h\left(t\right)$. The exact solution $\chi \left(t\right)=t^3+t^{\frac{4}{3}}$ is a linear combination of two power functions $t^3$ and $t^{\frac{4}{3}}$, and $h_A\left(t\right)=t^{-\delta \left(t\right)}\frac{6t^3}{\Gamma (4-\delta (t\left)\right)}$ and $h_B\left(t\right)=\frac{1}{35}{t}^{11/6}\left(22-\frac{35\sqrt{\pi }\Gamma \left(\frac{7}{3}\right)}{\Gamma \left(\frac{17}{6}\right)}\right)+t^{-\delta \left(t\right)}\frac{t^{\frac{4}{3}}\Gamma \left(\frac{7}{3}\right)}{\Gamma (\frac{7}{3}-\delta \left(t\right))}$, respectively. For $\lambda =\frac{1}{2}$ and $h\left(t\right)=h_A\left(t\right)$, we have $\chi \left(t\right)=t^3$ and, with $N\ge 6$, it yields ${\chi }_N\left(t\right)=\chi \left(t\right)$ (this is well illustrated by examples in Case I). Therefore, the function $\chi \left(t\right)=t^3$ does not contribute to the error term for the approximate solution. In this case, it turns out that $\chi \left(t\right)=t^{\frac{4}{3}}$ does not belong to ${\mathcal{F}}_N$ for any N, and the approximate solution ${\chi }_N\left(t\right)$ for $\chi \left(t\right)=t^{\frac{4}{3}}$ with $h\left(t\right)=h_B\left(t\right)$ is the main concern. For $\lambda =\frac{2}{3}$ and $h\left(t\right)=h_B\left(t\right)$, we get $\chi \left(t\right)=t^{\frac{4}{3}}$ and, with $N\ge 4$, it results in ${\chi }_N\left(t\right)=\chi \left(t\right)$. Thus, the function $\chi \left(t\right)=t^{\frac{4}{3}}$ does not contribute to the error term for the approximate solution. In this case, it turns out that $\chi \left(t\right)=t^3$ does not belong to ${\mathcal{F}}_N$ for any N, and the approximate solution ${\chi }_N\left(t\right)$ for $\chi \left(t\right)=t^3$ with $h\left(t\right)=h_A\left(t\right)$ is now the main concern.

7. Conclusions

This paper proposed a new numerical approach based on the FSJ-G-C method and R-LF operators to solve VO-FIDE-WSK subject to initial conditions. The novel algorithm reduces the solution of the original problem to a system of algebraic equations that is solved by any suitable procedure. Four numerical examples revealed that the proposed scheme is efficient and accurate when compared with alternatives reported in the literature.