A Simplified Hybrid Analytical Method for Solving Integer and Fractional-Order Differential Equations without Adomian Polynomials or Lagrange Multipliers
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940-957
Digital Object Identifier:
10.58578/ajstea.v3i3.5719
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Authors:
Okai J. O.1*, Cornelius M.2, Abdulmalik I.3, Jeremiah A.4, Nasir U. M.5, Hafsat Y. U.6, Abichele O.7 
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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.
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References
Bhalekar, S., and Daftardar-gejji, V. (2012). Solving Fractional-Order Logistic Equation Using a
New Iterative Method. International Journal of Differential Equations Volume 2012, Article ID 975829, 12 pages. 2012. https://doi.org/10.1155/2012/975829
Hemeda, A. A. (2018). A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations. International Journal of Computational Methods, 15(1), 1–15. https://doi.org/10.1142/S0219876218500160
Rashidi, M.M., Rabiei, F. S., and Abbasbandy, S. (2020). A Review : Differential Transform Method. Int. Journal. of Applied Mechanics and Engineering, 25(2), 122 129.https://doi.org/10.2478/ijame-2020-0024
Momani, S., and Odibat, Z. (2007). Homotopy perturbation method for nonlinear partial differential equations of fractional order. Physics Letters, Section A: General, Atomic and Solid State Physics, 365(5–6), 345–350. https://doi.org/10.1016/j.physleta.2007.01.046
Odibat, Z. (2019). An optimized decomposition method for nonlinear ordinary and partial differential equations. Physica A .https://doi.org/10.1016/j.physa.2019.123323
Tate, S., and Dinde, H. T. (2019). A New Modification of Adomian Decomposition Method for Nonlinear Fractional-Order Volterra Integro-Differential Equations. World Journal of Modelling and Simulation 15(1), 33–41.
Wazwaz, A. M. (2009). The variational iteration method for analytic treatment for linear and nonlinear ODEs. Applied Mathematics and Computation, 212(1), 120–134. https://doi.org/10.1016/j.amc.2009.02.003
Ziane, D., and Cherif, M. H. (2018). Variational iteration transform method for fractional differential equations. Journal of Interdisciplinary Mathematics, 21(1), 185–199. https://doi.org/10.1080/09720502.2015.1103001