Pell Collocation Method for Solving the Nonlinear Time–Fractional Partial Integro–Differential Equation with a Weakly Singular Kernel

Abstract

This article focuses on finding the numerical solution of the nonlinear time–fractional partial integro–differential equation. For this purpose, we use the operational matrices based on Pell polynomials to approximate fractional Caputo derivative, nonlinear, and integro–differential terms; and by collocation points, we transform the problem to a system of nonlinear equations. This nonlinear system can be solved by the fsolve command in Matlab. The method’s stability and convergence have been studied. Also included are five numerical examples to demonstrate the veracity of the suggested strategy.

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

“…Table 3 shows the absolute errors at different values of α and β at N = 1. The results in the last table show that our method is more accurate when we compare our results with those obtained in table 2 at Taghipour and Aminikhah (2022b).…”

“…Also Table 6 presents the maximum absolute errors at α = 0.9 and β = 0.3. The results in Tables 5 and 6 show that our method is more accurate when we compare our results with those obtained in table 4 at Taghipour and Aminikhah (2022b). Figure 3 indicates the advantage of our method for obtaining the maximum absolute errors at small values of N .…”

“…3 on four examples. All results show that our method is applicable and effective when we compare it with results in Taghipour and Aminikhah (2022b). Example 1 (Taghipour and Aminikhah 2022b;Guo et al 2020) Consider the following equation…”

“…Tables 1 and 2 show the comparison of the absolute errors between our method at N = 12 with the method developed in Taghipour and Aminikhah (2022b) at different values of α and β. In addition, Fig.…”

“…For example, the authors in Guo et al (2020) proposed a finite difference method for solving this problem. In Taghipour and Aminikhah (2022b), the nonlinear TFPIDE is solved by using Pell collocation method.…”