Theoretical and Computational Analysis of Delay Volterra Integro-Differential Equations via LaplaceTransform and Numerical Inversion (European Journal of Pure and Applied Mathematics, 2025)
Delay integro-differential equations (DIDEs) represent a significant class of integro-differential equations where state evolution depends on its past history. This paper presents a numerical approach for delay integro-differential equations (DIDEs), utilizing the Laplace transform (LT) and its inversion as the core methodology. The proposed technique begins by transforming the given equation into an algebraic equation in the Laplace domain using the LT. The resulting transformed equation is subsequently solved for the unknown function within the Laplace domain. Finally, two inversion methods the Gauss-Hermite quadrature method and the Weeks methods are used to invert the solution back to the time domain. Additionally, the existence and uniqueness of the solution are rigorously analyzed using the functional analysis. To demonstrate the effectiveness of the methods several examples from the literature are provided. The results obtained using the two techniques are compared and analyzed through tables and figures, highlighting their accuracy and computational efficiency.
Keywords: Delay integro-differential equation; Laplace transform; Existence; Uniqueness; Gauss-Hermite quadrature method; Weeks method