Exact Solutions of Nonlinear Delay Voltera Integro-Differential Equations Using Modified Homotopy Perturbation Method

Abstract

This paper presents a robust and efficient approach for solving delay Volterra integro-differential equations (DVIDEs), which model systems with memory and delay effects commonly encountered in fields such as biology, control theory, and epidemiology. Due to their complexity, these equations require accurate and efficient solution methods. The proposed method modifies the traditional homotopy perturbation method (HPM) to form a new version (MHPM), integrating it with the Laplace transformation and Pad ́e approximants. The incorporation of the Laplace transformation simplifies the problem by converting integro-differential equations into algebraic equations, streamlining the solution process. To further enhance the accuracy and convergence of the solution series, Pad ́e approximants are employed, enabling the method to overcome the limitations of standard perturbation techniques. This hybrid approach effectively combines the strengths of homotopy perturbation, Laplace transformation, and Pad ́e approximants, yielding highly accurate solutions that closely approximate the exact ones for various nonlinear DVIDEs. Numerical experiments and illustrative examples confirm the method’s efficiency and superior accuracy, even for equations with complex delay terms. The results highlight the potential of this combined approach as a powerful analytical tool for solving nonlinear delay integro-differential equations of the Volterra type in scientific and engineering applications.