Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

Abstract

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples.

1. Introduction

As approximated strategies for addressing differential equations, many numerical techniques, global or local, were listed. Local techniques provide numerical answers for specific places, whereas global methods provide solutions for the whole domain. Using finite difference techniques, the numerical approximations for differential equations [1,2,3,4] are provided at certain places. While finite element approaches [5,6,7] split the whole interval into sub-intervals and provide an approximate solution in them. Because of their ability to solve problems with infinite or finite domains [8,9], spectral techniques have recently received the most attention for dealing with various forms of differential and integral equations [10,11,12]. The major benefits of spectral approaches are their fast convergence (exponential rate) and excellent accuracy level [13,14,15,16].

Fractional calculus is a branch of calculus in which differential equations are used to represent a wide range of phenomena in mechanics, biology, engineering, economics, and physics. Fractional models can be used to closely depict a variety of phenomena. Because it is hard to provide explicit analytical solutions to space and/or time-fractional differential equations in most circumstances, developing effective numerical techniques is a critical necessity. Riesz fractional may be used to closely reflect a variety of occurrences, as shown in [17,18,19]. Only a few numerical approaches are capable of approximating such situations. The authors of [20,21,22,23] provide a number of finite difference approaches for solving the one-dimensional Riesz space-fractional issue. To address the one-dimensional Riesz space-fractional issue, finite element techniques [24,25,26,27] were applied. For both the temporal and spatial discretizations of the Riesz space fractional partial differential, Lai and Liu employed the Galerkin finite element scheme [28]. Yang et al. solved the semilinear Riesz space-fractional diffusion equations with time delay using the implicit alternating direction approach [29]. Numerous numerical approaches are utilized for the two-dimensional Riesz space-fractional issue, such as compact alternating direction implicit [30], finite volume [31], semi-alternating direction [32], Crank–Nicolson Galerkin finite element [33] and finite element [34].

In contrast to the effort put into studying finite difference and finite elements schemes for solving fractional-order differential equations in the literature, only a small amount of academic work has been invested into constructing and analyzing global spectral schemes. The spectral collocation approach is very useful, as it can estimate a broad variety of equations, including linear and nonlinear differential equations, integral, integro-differential equations, fractional differential equations, optimal control, and variational problems. For its obvious advantages, the collocation technique has been successfully used in a wide range of scientific and technical domains. The collocation approach is highly beneficial in providing very accurate result owing to its exponential rate of convergence. In contrast, for dealing with fractional problems, the collocation strategy has risen in popularity in recent decades. The truncated solutions for advection–dispersion equations with the Riesz fractional (RF-ADEs) are derived using the Legendre collocation technique based on Gauss-Lobatto and Gauss-Radau nodes. We employed the spectral collocation technique for the x and t-discretizations. For x-discretization, the SL-GL-C scheme with proper boundary management is used. While the SL-GR-C technique was used for t-discretization. This adjustment considerably improves accuracy. After that, the residuals at the SL-GR and SL-GL quadrature nodes are calculated. As a consequence, we have an algebraic system that can be solved using a suitable method. A variety of numerical problems are used to illustrate the accuracy of the new technique.

Our paper is laid out as follows. First, several interesting characteristics of Riemann–Liouville fractional derivatives (RL-FDs) are discussed. Section 3 deals with numerically solving one-dimensional RF-ADEs. Two numerical examples are examined in Section 4. Finally, remarks are offered.

2. Riemann–Liouville Fractional Derivative

Different formulations exist for fractional derivative of order $\epsilon >0$, [35]. The most frequent and commonly used expression is the Riemann–Liouville formula, which is defined as

$$\begin{array}{cc}\hfill J^{\epsilon }\phi \left(\chi \right)& =\frac{1}{\mathsf{\Gamma }\left(\epsilon \right)}{\int }_0^{\chi }{(\chi -\eta )}^{\epsilon -1}\phi \left(\eta \right)d\eta ,\epsilon >0,\chi >0,\hfill \\ \hfill J^0\phi \left(\chi \right)& =\phi \left(\chi \right).\hfill \end{array}$$

(1)

The $\epsilon (m-1<\epsilon <m)$ left- and right-sided of RL-FD on infinite domain are defined as

$$\begin{array}{cc}\hfill & {}_{-\infty }{D}_{\varsigma }^{\epsilon }\varphi (\eta ,\eta )=\frac{1}{\mathsf{\Gamma }(m-\epsilon )}\frac{{\partial }^m}{\partial {\eta }^m}{\int }_{-\infty }^{\eta }{(\eta -z)}^{m-1-\epsilon }\varphi (z,\eta )dz,\hfill \\ \hfill & {}_{\eta }{D}_{+\infty }^{\epsilon }\varphi (\eta ,\eta )=\frac{{(-1)}^m}{\mathsf{\Gamma }(m-\epsilon )}\frac{{\partial }^m}{\partial {\eta }^m}{\int }_{\eta }^{+\infty }{(z-\eta )}^{m-1-\epsilon }\varphi (z,\eta )dz.\hfill \end{array}$$

(2)

The Riesz fractional derivative is written as

$$\frac{{\partial }^{\epsilon }}{\partial {\left|x\right|}^{\epsilon }}\varphi (\eta ,\eta )=-{\left(-\mathsf{\Delta }\right)}^{\frac{\epsilon }{2}}\varphi (\eta ,\eta )=c_{\epsilon }\left[{}_{-\infty }{D}_{\eta }^{\epsilon }\varphi (\eta ,\eta )+{}_{\eta }{D}_{+\infty }^{\epsilon }\varphi (\eta ,\eta )\right],$$

(3)

where $c_{\epsilon }=-\frac{1}{2cos\left(\frac{\pi \epsilon }{2}\right)}$. The fractional Laplacian operator in (3) can formulated as

$$-{\left(-\mathsf{\Delta }\right)}^{\frac{\epsilon }{2}}\varphi (\eta ,\eta )=-{\mathcal{F}}^{-1}\left(\left|\eta \right|\mathcal{F}\left(\varphi (\eta ,\eta )\right)\right),$$

(4)

If ${\psi }_{\eta }(\mathcal{A},\eta )={\psi }_{\eta }(\mathcal{B},\eta )=0$, then the Riesz fractional derivative can be written as, see [36]

$$\frac{{\partial }^{\epsilon }}{\partial {\left|x\right|}^{\epsilon }}\varphi (\eta ,\eta )=-{\left(-\mathsf{\Delta }\right)}^{\frac{\epsilon }{2}}\varphi (\eta ,\eta )=-\frac{1}{2cos\left(\frac{\pi \epsilon }{2}\right)}\left[{}_a{D}_{\eta }^{\epsilon }\varphi (\eta ,\eta )+{}_{\eta }{D}_b^{\epsilon }\varphi (\eta ,\eta )\right].$$

(5)

For Legendre polynomial ${\mathcal{G}}_k$, get

$$\begin{array}{cc}\hfill & {}_{-1}{D}_{\eta }^{\epsilon }{\mathcal{G}}_k\left(\eta \right)=\sum _{k=0}^j\frac{{(-1)}^{k+j}\mathsf{\Gamma }(k+j+1)}{(j-k)!\mathsf{\Gamma }(k+1)2^k\mathsf{\Gamma }(k-\epsilon +1)}{(\eta +1)}^{k-\epsilon },\hfill \\ \hfill & {}_{\eta }{D}_1^{\epsilon }{\mathcal{G}}_k\left(\eta \right)=\sum _{k=0}^j\frac{{(-1)}^k\mathsf{\Gamma }(k+j+1)}{(j-k)!\mathsf{\Gamma }(k+1)2^k\mathsf{\Gamma }(k-\epsilon +1)}{(1-\eta )}^{k-\epsilon }.\hfill \end{array}$$

(6)

3. Fully Spectral Collocation Treatment

In this paper, we established a numerical approach for dealing with RF-ADEs

$$\begin{array}{cc}\hfill \frac{\partial }{\partial \eta }\mathcal{B}(\varsigma ,\eta )=& {\kappa }_{\epsilon }\frac{{\partial }^{\epsilon }}{{\partial \left|\varsigma \right|}^{\epsilon }}\mathcal{B}(\varsigma ,\eta )+{\kappa }_{\nu }\frac{{\partial }^{\nu }}{{\partial \left|\varsigma \right|}^{\nu }}\mathcal{B}(\varsigma ,\eta )+\mathsf{\Delta }(\varsigma ,\eta ),(\varsigma ,\eta )\in {\mathcal{D}}^{•}\times {\mathcal{D}}^{\diamond },\hfill \end{array}$$

(7)

where ${\mathcal{D}}^{•}\equiv [0,{\varsigma }_{end}]$, and ${\mathcal{D}}^{\diamond }\equiv [0,{\eta }_{end}]$. With

$$\begin{array}{cc}\hfill & \mathcal{B}(\varsigma ,0)={\Theta }_1\left(\varsigma \right),\varsigma \in {\mathcal{D}}^{•},\hfill \\ \hfill \mathcal{B}(0,\eta )=& {\Theta }_2\left(\eta \right),\mathcal{B}({\varsigma }_{end},\eta )={\Theta }_3\left(\eta \right),\eta \in {\mathcal{D}}^{\diamond }.\hfill \end{array}$$

(8)

The SL-GL-C and SL-GR-C techniques are used here to convert the RF-ADEs into linear algebraic systems. The abbreviated solution is written as,

$$\begin{array}{cc}\hfill {\varphi }_{\mathcal{N},\mathcal{M}}(\varsigma ,\eta )& =\sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta ),\hfill \end{array}$$

(9)

where ${\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta )={\mathcal{G}}_{{\varsigma }_{end},r_1}\left(\varsigma \right){\mathcal{G}}_{{\eta }_{end},r_2}\left(\eta \right)$ and ${\mathcal{G}}_{\mathcal{L},s}\left(\varsigma \right)$ is performed to shifted Legendre polynomials on $[0,\mathcal{L}]$.

The Riesz fractional derivative $\frac{{\partial }^{\epsilon }{\varphi }_{\mathcal{N},\mathcal{M}}(\varsigma ,\eta )}{{\partial \left|\varsigma \right|}^{\epsilon }}$ is computed as

$$\begin{array}{c}\hfill \frac{{\partial }^{\gamma }{\varphi }_{\mathcal{N},\mathcal{M}}(\varsigma ,\eta )}{{\partial \left|\varsigma \right|}^{\gamma }}=\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\varsigma }_{\gamma }}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta ),\end{array}$$

(10)

where ${\mathcal{G}}_{r_1,r_2,{\varsigma }_{\gamma }}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta )=\frac{{\partial }^{\gamma }{\mathcal{G}}_{{\varsigma }_{end},r_1}\left(\varsigma \right)}{{\partial \left|\varsigma \right|}^{\gamma }}{\mathcal{G}}_{{\eta }_{end},r_2}\left(\eta \right)$, see [37]. While, $\frac{\partial }{\partial \eta }\mathcal{B}(\varsigma ,\eta )$ is

$$\begin{array}{c}\hfill \frac{\partial }{\partial \eta }\mathcal{B}(\varsigma ,\eta )=\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\eta }_1}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta ),\end{array}$$

(11)

where ${\mathcal{G}}_{r_1,r_2,{\eta }_1}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,\eta )=\frac{\partial {\mathcal{G}}_{{\eta }_{end},r_2}\left(\eta \right)}{\partial \eta }{\mathcal{G}}_{{\varsigma }_{end},r_1}\left(\varsigma \right)$.

At certain points, the preceding derivatives are

$$\begin{array}{cc}\hfill {\left(\frac{\partial }{\partial \eta }\mathcal{B}(\varsigma ,\eta )\right)}_{\eta ={\eta }_{\mathcal{M},m}^{{\eta }_{end}}}^{\varsigma ={\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},}& =\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\eta }_1}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}}),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\left(\frac{{\partial }^{\gamma }{\varphi }_{\mathcal{N},\mathcal{M}}(\varsigma ,\eta )}{{\partial \left|\varsigma \right|}^{\gamma }}\right)}_{\eta ={\eta }_{\mathcal{M},m}^{{\eta }_{end}}}^{\varsigma ={\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},}& =\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\varsigma }_{\gamma }}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}}).\hfill \end{array}$$

(13)

The Equation (7) is constrained to be zero at the $(\mathcal{N}-1)\times \left(\mathcal{M}\right)$ points. Alternatively, the initial-boundary may be found by

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}(\varsigma ,0)=& {\Theta }_1\left(\varsigma \right),\hfill \\ \hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}(0,\eta )=& {\Theta }_2\left(\eta \right),\hfill \\ \hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},\mathcal{N}}^{{\varsigma }_{end}},\eta )=& {\Theta }_3\left(\eta \right),\hfill \end{array}$$

(14)

Therefore, adapting (7)–(14), we get

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\eta }_1}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}})=& {\kappa }_{\epsilon }\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\varsigma }_{\epsilon }}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}})+\hfill \\ & {\kappa }_{\nu }\sum _{\begin{array}{c}{r}_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2,{\varsigma }_{\nu }}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}})+\hfill \\ & \mathsf{\Delta }({\varsigma }_{\mathcal{N},n}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},m}^{{\eta }_{end}}),\hfill \end{array}$$

(15)

with $1\le n\le \mathcal{N},1\le m\le \mathcal{M}.$

Additionally,

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},k}^{{\varsigma }_{end}},0)=& {\Theta }_1\left({\varsigma }_{\mathcal{N},k}^{{\varsigma }_{end}}\right),k=1,\cdots ,\mathcal{N}-1,\hfill \\ \hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}(0,{\eta }_{\mathcal{M},l}^{{\eta }_{end}})=& {\Theta }_2\left({\eta }_{\mathcal{M},l}^{{\eta }_{end}}\right),l=0,\cdots ,\mathcal{M},\hfill \\ \hfill \sum _{\begin{array}{c}l,r_1=0,\cdots ,\mathcal{N}\\ r_2=0,\cdots ,\mathcal{M}\end{array}}{\varsigma }_{r_1,r_2}{\mathcal{G}}_{r_1,r_2}^{{\varsigma }_{end},{\eta }_{end}}({\varsigma }_{\mathcal{N},\mathcal{N}}^{{\varsigma }_{end}},{\eta }_{\mathcal{M},l}^{{\eta }_{end}})=& {\Theta }_3\left({\eta }_{\mathcal{M},l}^{{\eta }_{end}}\right),l=0,\cdots ,\mathcal{M}.\hfill \end{array}$$

(16)

A solvable algebraic system is obtained by joining Equations (15) and (16).

4. Numerical Results

In this part, we demonstrate the resilience and efficiency of the spectral collocation strategy by applying it to two test cases.

Example 1.

Taking the RF-ADEs [38]

$$\begin{array}{cc}\hfill \frac{\partial }{\partial \eta }\mathcal{B}(\varsigma ,\eta )=& \mathsf{\Gamma }(3-\epsilon )\frac{{\partial }^{\epsilon }}{{\partial \left|\varsigma \right|}^{\epsilon }}\mathcal{B}(\varsigma ,\eta )+\mathsf{\Gamma }(3-\nu )\frac{{\partial }^{\nu }}{{\partial \left|\varsigma \right|}^{\nu }}\mathcal{B}(\varsigma ,\eta )+\mathsf{\Delta }(\varsigma ,\eta ),\hfill \\ & (\varsigma ,\eta )\in [0,1]\times [0,1],\hfill \end{array}$$

(17)

$\mathsf{\Delta }(\varsigma ,\eta )$ is given where $\mathcal{B}(\varsigma ,\eta )=e^{\eta }{\varsigma }^2{(1-\varsigma )}^2.$

Table 1. MAEs for problem (17).

	Our Method
$(\mathcal{N},\mathcal{M})$	$\epsilon =1.4,\nu =0.1$	$\epsilon =1.4,\nu =0.9$	$\epsilon =1.7,\nu =0.1$	$\epsilon =1.7,\nu =0.9$
(2,2)	$5.01404\times {10}^{-2}$	$6.2237\times {10}^{-2}$	$7.80645\times {10}^{-2}$	$8.63705\times {10}^{-2}$
(4,4)	$1.23433\times {10}^{-6}$	$1.17057\times {10}^{-6}$	$1.02812\times {10}^{-6}$	$9.85684\times {10}^{-7}$
(6,6)	$2.57528\times {10}^{-9}$	$2.57725\times {10}^{-9}$	$2.63261\times {10}^{-9}$	$2.63928\times {10}^{-9}$
(8,8)	$3.86276\times {10}^{-12}$	$3.68298\times {10}^{-12}$	$3.04126\times {10}^{-12}$	$2.93278\times {10}^{-12}$
(10,10)	$1.04326\times {10}^{-14}$	$8.61811\times {10}^{-15}$	$2.50147\times {10}^{-15}$	$2.59515\times {10}^{-15}$
	Finite Difference Scheme [38]
$(\mathsf{\Delta }\varsigma ,\mathsf{\Delta }\eta )$	$\epsilon =1.4,\nu =0.1$	$\epsilon =1.4,\nu =0.9$	$\epsilon =1.7,\nu =0.1$	$\epsilon =1.7,\nu =0.9$
$(\frac{1}{80},\frac{1}{80})$	$2.53\times {10}^{-3}$	$8.35\times {10}^{-3}$	$9.63\times {10}^{-4}$	$6.18\times {10}^{-3}$
$(\frac{1}{160},\frac{1}{160})$	$1.28\times {10}^{-3}$	$4.29\times {10}^{-3}$	$4.64\times {10}^{-4}$	$3.14\times {10}^{-3}$
$(\frac{1}{320},\frac{1}{320})$	$6.43\times {10}^{-4}$	$2.17\times {10}^{-3}$	$2.21\times {10}^{-4}$	$1.58\times {10}^{-3}$
$(\frac{1}{640},\frac{1}{640})$	$3.20\times {10}^{-4}$	$1.09\times {10}^{-3}$	$1.06\times {10}^{-4}$	$7.86\times {10}^{-4}$
$(\frac{1}{1280},\frac{1}{1280})$	$1.59\times {10}^{-4}$	$5.42\times {10}^{-4}$	$4.97\times {10}^{-5}$	$3.92\times {10}^{-4}$

shows a comparison of our outcomes with those in [38]. Thus, the suggested technique outperforms the numerical results provided in [38]. It is often discovered that good estimates with a small number of nodes. The numerical solution and absolute errors of the problem (17) are shown in Figure 1 and Figure 2, respectively. We presented the $\varsigma$-direction graphs of numerical and exact solutions in Figure 3, where $\epsilon =1.4,nu=0.1$, $\mathcal{N}=\mathcal{M}=10$. The $\eta$- and $\varsigma$-graphs relating to absolute errors are depicted in Figure 4 and Figure 5, respectively. Additionally, we can see the error degradation in Figure 6.

Example 2.

Taking the RF-ADEs [38]

$$\begin{array}{cc}\hfill \frac{\partial \mathcal{B}(\varsigma ,\eta )(\varsigma ,\eta )}{\partial \eta }=& \mathsf{\Gamma }(2-\epsilon )\frac{{\partial }^{\epsilon }}{{\partial \left|\varsigma \right|}^{\epsilon }}\mathcal{B}(\varsigma ,\eta )+\mathsf{\Gamma }(2-\nu )\frac{{\partial }^{\nu }}{{\partial \left|\varsigma \right|}^{\nu }}\mathcal{B}(\varsigma ,\eta )+\mathsf{\Delta }(\varsigma ,\eta ),\hfill \\ & (\varsigma ,\eta )\in [0,1]\times [0,1],\hfill \end{array}$$

(18)

$\mathsf{\Delta }(\varsigma ,\eta )$ is supplied where $\mathcal{B}(\varsigma ,\eta )=\eta sin\left(\frac{\pi \eta }{2}\right)\varsigma (1-\varsigma ).$

Table 2. MAEs for problem (18).

	Our Method
$(\mathcal{N},\mathcal{M})$	$\epsilon =1.1,\nu =0.5$	$\epsilon =1.5,\nu =0.5$	$\epsilon =1.9,\nu =0.5$
(2,2)	$1.25883\times {10}^{-2}$	$1.25883\times {10}^{-2}$	$2.4723\times {10}^{-2}$
(6,6)	$5.36837\times {10}^{-7}$	$5.49461\times {10}^{-7}$	$1.34162\times {10}^{-6}$
(10,10)	$7.24216\times {10}^{-12}$	$3.55129\times {10}^{-12}$	$3.43689\times {10}^{-12}$
(14,14)	$7.35523\times {10}^{-16}$	$2.77556\times {10}^{-16}$	$2.88658\times {10}^{-15}$
(18,18)	$2.22045\times {10}^{-16}$	$1.38778\times {10}^{-16}$	$2.58127\times {10}^{-15}$
	Finite Difference Scheme [38]
$(\mathsf{\Delta }\varsigma ,\mathsf{\Delta }\eta )$	$\epsilon =1.1,\nu =0.5$	$\epsilon =1.5,\nu =0.5$	$\epsilon =1.9,\nu =0.5$
$(\frac{1}{40},\frac{1}{10000})$	$2.52\times {10}^{-3}$	$5.04\times {10}^{-4}$	$2.13\times {10}^{-4}$
$(\frac{1}{80},\frac{1}{10000})$	$1.27\times {10}^{-3}$	$2.50\times {10}^{-4}$	$1.07\times {10}^{-4}$
$(\frac{1}{160},\frac{1}{10000})$	$6.34\times {10}^{-4}$	$1.22\times {10}^{-4}$	$5.37\times {10}^{-5}$
$(\frac{1}{320},\frac{1}{10000})$	$3.14\times {10}^{-4}$	$5.82\times {10}^{-4}$	$2.87\times {10}^{-5}$
$(\frac{1}{640},\frac{1}{10000})$	$1.54\times {10}^{-4}$	$2.61\times {10}^{-5}$	$1.62\times {10}^{-5}$

shows a comparison of our outcomes with those in [38]. Thus, the suggested technique outperforms the numerical results provided in [38]. It is often discovered that good estimates with a small number of nodes. The numerical solution and absolute errors of the problem (18) are shown in Figure 7 and Figure 8, respectively. The $\eta$- and $\varsigma$-graphs relating to absolute errors are depicted in Figure 9 and Figure 10, respectively. Additionally, we can see the error degradation in Figure 11.

5. Conclusions

To get numerical solutions for RF-ADEs in one dimensional space, we offer an accurate and economical numerical methodology linked to the SL-GL-C and SL-GR-C methods. The approach is efficient, adaptable to a wide range of operators, and easily extendable to multi-dimensions. Based on the obtained findings, we can observe that our approach has a high degree of accuracy.