An approximate solution of multi-term fractional telegraph equation with quadratic B-spline basis functions

Abstract

This paper introduces the Galerkin Method for the approximate solution ofMulti-term Fractional Telegraph Equations (MFTE). The Galerkin Method (GM)is one of the most popular techniques for the solution of Partial Differential Equations (PDEs), and it uses the idea of mapping a solution onto a set of basis functions and then seeking the residual error through minimization. In GM, the weight and basis functions are the same, and as such, the basis functions are selected appropriately to satisfy the given conditions imposed. In this paper, the quadratic B-spline functions are adopted as shape and test functions for resolving the approximate solution of MFTE. Here, the Caputo fractional derivative takes care of the fractional part, and the Gauss-Mamadu-Njoseh quadrature scheme handles numerical integration. Numerical illustrations are examined for single-term and two-term MFTE, respectively, with numerical evidence measured using$L_2$ and $L_{\infty }$ error norms. Consequently, the resulting numerical evidence, as presented in Tables and Figures, shows the accuracy and reliability of the method. Also, the study examined and presented relevant theorems of convergence and error analyses of method. MAPLE 18 was used for all computational frameworks in this research.