Product integration techniques for fractional integro-differential equations

Abstract

This article presents an application of approximate product integration (API) to find the numerical solution of fractional order Volterra integro-differential equation based on Caputo non-integer derivative of order (Formula presented.), where (Formula presented.). Also, the idea is extended to a class of fractional order Volterra integro-differential equation with a weakly singular kernel. For this purpose, two numerical schemes are established by utilizing the concept of the API method, and L1 and L1-2 formulae. We applied L1 and L1-2 discretization to approximate the Caputo non-integer derivative. At the same time, Taylor's series expansion of an unknown function is taken into consideration when approximating the Volterra part in the considered mathematical model using the API method. Combination of API method with L1 and L1-2 formula provided the order of convergence (Formula presented.) and (Formula presented.) for Scheme-I and Scheme-II, respectively. The derived techniques reduced the proposed model to a set of algebraic equations that can be resolved using well-known numerical algorithms. Furthermore, the unconditional stability, convergence, and numerical stability of the formulated schemes have been rigorously investigated. Finally, we conducted some numerical experiments to validate our theoretical findings and guarantee the accuracy and efficiency of the recommended schemes. The comparison between the numerical outcomes obtained by proposed schemes and existing numerical techniques has also been provided through tables and graphs. © 2024 John Wiley & Sons Ltd.