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All Journal YASIN: Jurnal Pendidikan dan Sosial Budaya Asian Journal of Science, Technology, Engineering, and Art
A., Jeremiah
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A Simplified Hybrid Analytical Method for Solving Integer and Fractional-Order Differential Equations without Adomian Polynomials or Lagrange Multipliers O., Okai J.; M., Cornelius; I., Abdulmalik; A., Jeremiah; M., Nasir U.; U., Hafsat Y.; O., Abichele
Asian Journal of Science, Technology, Engineering, and Art Vol 3 No 3 (2025): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v3i3.5719

Abstract

In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and nonlinear first-order initial value problems (IVPs), including those of fractional order. The principal aim of this approach is to overcome the computational challenges typically encountered in each individual method—namely, the complexity of generating Adomian polynomials in ADM and the requirement for Lagrange multipliers in VIM. By synthesizing the strengths of both methods, the hybrid scheme constructs analytical series solutions without necessitating linearization, Adomian polynomials, or the explicit formulation of Lagrange multipliers. This significantly streamlines the solution process while preserving accuracy and generality. The validity and computational efficiency of the proposed method are substantiated through a series of illustrative examples, encompassing both integer-order and fractional differential equations. The results demonstrate that the hybrid approach not only simplifies implementation but also yields precise and rapidly converging solutions, making it a robust alternative for tackling a broad spectrum of initial value problems in mathematical modeling and applied sciences.
A Hybrid Approach of the Variational Iteration Method and Adomian Decomposition Method for Solving Fractional Integro-Differential Equations O, Okai J.; Adamu, M. S.; M., Cornelius; I., Abdulmalik; A., Jeremiah; M., Nasir U.; U., Hafsat Y.; O., Abichele; Araga, Hassan
YASIN Vol 5 No 4 (2025): AGUSTUS
Publisher : Lembaga Yasin AlSys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/yasin.v5i4.5720

Abstract

In this study, we propose a hybrid analytical technique that integrates the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve both linear and nonlinear integro-differential equations of integer and fractional orders. This approach extends and refines the Odibat Decomposition Method (ODM) by addressing key limitations inherent in ADM and VIM—specifically, the reliance on linearization, Adomian polynomials, and Lagrange multipliers. By circumventing these computational complexities, the proposed method enables the direct and efficient construction of series solutions with improved convergence properties. The hybrid scheme is designed for broader applicability and enhanced computational simplicity, making it a powerful tool for analyzing complex integro-differential systems. Its effectiveness and robustness are demonstrated through a range of illustrative examples, confirming the method’s capability to provide accurate analytical approximations with minimal computational overhead.
Co-Authors Adamu, M. S. Araga, Hassan I., Abdulmalik M., Cornelius M., Nasir U. O, Okai J. O., Abichele O., Okai J. U., Hafsat Y.