IMPROVEMENT IN VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD (Journal of Mathematics Mechanics and Computer Science, 2025)
The Adomian Decomposition Method (ADM) is widely recognized as a powerful and versatile semi-analytical tool designed to solve a broad range of problems, including linear and nonlinear differential equations, as well as integral equations. This method has been extensively applied across various scientific and engineering disciplines due to its simplicity and efficiency in generating accurate approximate solutions. In this note, we introduce an enhanced and refined scheme based on the ADM framework to obtain approximate solutions for Volterra-Fredholm integro-differential equations (IDEs) with specified initial conditions. Our proposed scheme not only simplifies the computational process but also ensures improved accuracy and convergence. Additionally, we rigorously prove the uniqueness of the solutions to the Volterra-Fredholm IDEs by leveraging the mathematical foundation of Banach’s Fixed Point Theorem, providing theoretical validity to our approach. To validate the effectiveness of the enhanced scheme, we apply it to a diverse set of linear and nonlinear Volterra-Fredholm IDEs with initial conditions. The numerical results obtained are systematically compared with those from existing methods reported in the literature. Our findings reveal that the proposed approach demonstrates remarkable accuracy, efficiency, and reliability in solving complex IDEs. Consequently, this method represents a significant advancement in the field of integro-differential equations.
Keywords: Adomian decomposition method; Volterra-Fredholm Integrodifferential equation; Approximate solution; Uniqueness solution; Adomian poliynomials; Banach’s fixed point theorem