A Second-Order Difference Scheme for Solving a Class of Fractional Differential Equations

Abstract

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.