MODIFIED LAPLACE-VARIATIONAL ITERATION METHOD FOR SOLVING LINEAR AND NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

This study presents a modified Laplace-variational iteration method (MLVIM) designed to solve linear and nonlinear Volterra integro-differential equations (VIDEs) with specified initial conditions. The MLVIM is a hybrid approach that integrates the strengths of the Laplace transform and the variation iteration method (VIM), effectively enhancing the overall solution process by improving both the efficiency and convergence rate. Specifically, the method refines the correction functional and optimizes the handling of the integral term, which directly leads to a reduction in the number of iterations needed and decreases the associated computational complexity. To demonstrate the effectiveness of MLVIM, the study applies it to two illustrative examples involving both linear and nonlinear VIDEs, with initial conditions. The results are then compared to those obtained using the Adomian decomposition method (ADM) and the fourth-order Runge-Kutta (RK4) algorithm. The findings show that MLVIM consistently exhibits a faster convergence rate and higher accuracy compared to both ADM and RK4 in all the examples presented. The MLVIM can be applied to a broad range of linear and nonlinear VIDEs. This makes it a valuable tool with potential applications in various scientific and engineering fields, where integro-differential equations frequently arise in modeling complex systems and processes.