Numerical Scheme for Fractional Volterra Integro-Differential Equations: Handling Modified Atangana-Baleanu Operators

Abstract

This study presents a comparative numerical investigation of Volterra integrodifferential equations (VIDEs) incorporating the Modified Atangana-Baleanu fractional derivative in the Caputo sense. Fractional-order VIDEs play a vital role in modeling biological, physical, and engineering phenomena. The non-local nature and complexity of the fractional derivative make it difficult to solve them analytically. Therefore, efficient numerical methods have to be used to solve these types of problems. We develop a numerical technique based on Laplace transform (LT). First, we transform the governing equation into a simple algebraic equation via the LT. Then, we solve the transformed equation algebraically and numerically inverted. Two best numerical methods are employed for inverting the LT: the Gauss-Hermite quadrature; and the improved Talbot's method. The inversion methods are validated via numerical experiments and numerical results are compared against the exact solutions. Our findings prove the convergence of the two inversion methods. The numerical results demonstrate the adaptability of the suggsted methods, making them promising tools for fractional VIDEs.