An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay

Abstract

This paper introduces an iterative-based numerical scheme for solving nonlinear fractional-order Volterra integro-differential equations involving delay. Additionally, we provide sufficient conditions for the existence and uniqueness of the solution. The composite trapezoidal rule is applied to approximate the integral involved in the equation, followed by discretizing the Caputo fractional derivative operator of arbitrary order $$\alpha \in (0,1)$$ by using the classical L1 scheme. Further, the Daftardar-Gejji and Jafari method is employed to solve the implicit algebraic equation. The convergence analysis and error bounds of the proposed scheme are presented. It is shown that the approximate solution converges to the exact solution with order $$( 2-\alpha ).$$ We illustrate the efficacy and applicability of the proposed method through a couple of examples.