Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

Abstract

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time  t as well as how the absolute error in each example is calculated.