Exact Solutions of Nonlinear Convolution-Type Differential Equations Using the Modified-Laplace Variational Iteration Method

Abstract

This paper presents a novel application of the Modified-Laplace Variational Iteration Scheme (MLVIS) for solving nonlinear convolution differential equations. Convolution differential equations are integral to various fields, including engineering, physics, and applied mathematics, due to their ability to model systems with memory and hereditary properties. Traditional methods often struggle with the nonlinearity and complexity inherent in these equations. The approach combines the robustness of the Laplace transform with the flexibility of variational iteration methods, resulting in a powerful tool for tackling these challenges. This study begins by deriving the modified Laplace variational iteration method, emphasizing its theoretical foundation and practical implementation. The derivation process involves transforming the original nonlinear convolution differential equation into the Laplace domain, simplifying the convolution terms into algebraic forms. This transformation facilitates the application of variational principles, which are used to construct approximate solutions iteratively. The modified method enhances accuracy and convergence compared to standard techniques. The effectiveness of the MLVIM is validated through illustrative examples, demonstrating its capability to produce exact solutions for a range of nonlinear convolution differential equations. The results highlight the method's efficiency, precision, and potential for broader applications. The findings suggest that the MLVIM is a robust and reliable approach for solving complex nonlinear convolution differential equations.