Product integration techniques for generalized fractional integro-differential equations

Abstract

In this article, two numerical schemes are proposed and analyzed for solving the general Volterra integro-differential equation of fractional order based on the Caputo non-integer derivative of order ν∈(0,1). The schemes involve L1 and L2 approximations of the general Caputo derivative and the use of the approximate product integration formula to handle the Volterra part. Specifically, Taylor’s series expansion of an unknown function is taken into consideration when approximating the Volterra part using the approximate product integration method. Additionally, the techniques are extended to encompass a class of general fractional-order Volterra integro-differential equations with a weakly singular kernel. These methods transform the proposed model into a system of algebraic equations which is solvable by established numerical algorithms. The article rigorously investigates the unconditional stability, convergence, and numerical stability of the Scheme-I. It demonstrates that combining the approximate product integration method with the L1 approximation yields a convergence rate of (2-ν) for Scheme-I. Finally, numerical simulations are conducted to validate the theoretical results, ensuring the efficiency and accuracy of the proposed schemes. The obtained numerical results for various test problems are compared and presented through tables and graphs. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.