Spectral Collocation Method for a Class of Integro-Differential Equations with Erdélyi-Kober Fractional Operator

Abstract

The Erd\u00e9lyi-Kober fractional operators have found a large number of applications in many disciplines such as porous media and viscoelasticity. The purpose of this paper is to express the fractional integro-differential equations with Erd\u00e9lyi-Kober derivative in terms of a class of nonlinear weakly singular integral equations of mixed type in order to analyze their numerical solvability. The resulting mixed type Volterra equations will have kernels containing both an end point and diagonal singularity, with solutions that their derivatives typically are unbounded. Applications of such problems are described to reformulate the fractional integro-differential equations with\u00a0 Erd\u00e9lyi-Kober derivative in terms of a particular class of cordial weakly singular integral equations of mixed type. The existence and uniqueness results of solutions under some verifiable conditions on the kernels and nonlinear functions are discussed. The corresponding nonlinear weakly singular equation can be solved numerically in terms of the implicitly linear collocation method. The error analysis of the method is also discussed and the feasibility of the introduced strategy is illustrated by some numerical experiments. The reformulation proposed here might be used to develop a computational method to solve fractional integro-differential equations.