A numerical study on the non‐smooth solutions of the nonlinear weakly singular fractional Volterra integro‐differential equations

Abstract

The solutions of weakly singular fractional Volterra integro-differential equations involving the Caputo derivative typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. In this paper, we design an algorithm to prevail on this non-smooth behavior of solutions of the nonlinear fractional Volterra integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated.The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

An efficient numerical method based on ultraspherical polynomials for linear weakly singular fractional Volterra integro‐differential equations

“…The weakly singular fractional integro‐differential equations appear in the mathematical modeling of many physical problems such as polymer physics and biophysics[1], heat conduction problems [3], and radiative equilibrium [4]. In recent years, several numerical methods have been presented to numerical solution of the integral equations, especially the fractional equations (see, e.g., earlier studies [5–35]).…”