An Enhanced Variational Iteration Method for Solving Ordinary and Partial Differential Equations
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470-489
Digital Object Identifier:
10.58578/mikailalsys.v3i2.5317
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Authors:
Araga Hassan1*, M. Y. Adamu2, A. G. Madaki3, Yohanna Nehemiah4, Michael Cornelius5, U. M. Nasir6 
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Abstract
The Variational Iteration Method (VIM) has proven to be a powerful technique for solving both ordinary and partial differential equations. However, its reliance on Lagrange multipliers for each type of equation has posed significant limitations, complicating its application and reducing its efficiency. This study introduces a Modified Variational Iteration Method (MVIM) that eliminates the need for Lagrange multipliers, addressing these challenges. The MVIM reformulates the correctional functional, simplifying the solution process and enhancing computational efficiency. The method is applied to both linear and nonlinear ordinary and partial differential equations, demonstrating its ability to provide accurate and fast-converging solutions. Numerical examples show that the MVIM outperforms traditional VIM in terms of computational time and convergence speed, and compares favourably with other methods such as the Adomian Decomposition Method (ADM) and New Iteration Method (NIM). The results highlight the potential of MVIM as a versatile and efficient tool for solving complex differential equations in a variety of scientific and engineering applications.
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References
Abdou, M. A., Soliman, A. A., & Aty, M. A. A. (2020). On a discussion of Volterra – Fredholm integral equation with discontinuous kernel. Journal of the Egytian Mathematical Society, 28(11), 1–10.
Al-Mdallal, Q. A., Al-Khazali, H. A., & Abbas, S. M. (2016). Modified variational iteration method for solving nonlinear equations. Journal of Computational and Applied Mathematics, 290, 89-101.
Goswami, P., & Alqahtani, R. T. (2016). Solutions of fractional differential equations by Sumudu transform and variational iteration method. J. Nonlinear Sci. Appl, 9(2016), 1944–1951.
Hamoud, A. A., & Ghadle, K. P. (2016). On The Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations by Variational Iteration Method. June 2017.
He, J. (2007). Variational iteration method — Some recent results and new interpretations. 207, 3–17. https://doi.org/10.1016/j.cam.2006.07.009
He, J. H., & Wu, X. H. (2007). Variational iteration method: New development and applications. Computers and Mathematics with Applications, 54(7–8), 881–894. https://doi.org/10.1016/j.camwa.2006.12.083
Mamun, A. Al, & Tao, W. (2019). Solution of Volterra ’ s Integro-Differential Equations by Using Variational Iteration Method. May, 1–9. https://doi.org/10.20944/preprints201905.0164.v1
Mechee, M. S., Ramahi, A. M. Al, & Kadum, R. M. (2016). Applications of Variational Iteration Method for Solving A Class of Volterra Integral Equations. 9.
Porshokouhi, M. G., & Ghanbari, B. (2011). Variational Iteration Method for Solving Volterra and Fredholm Integral Equations of the Second Kind. 2(1), 143–148.
Saleh, M. H. (2016). Variational Iteration Method for Solving Two Dimensional Volterra - Fredholm Nonlinear Integral Equations. 152(3), 29–33.
Scheiber, E., & Brasov, U. T. (2019). On the Variational Iteration Method for the Nonlinear Volterra Integral Equation. AMS Researchgate, 1(July), 1–8.
Wu, G. (2015). Challenge in the variational iteration method – A new approach to identification of the Lagrange multipliers. Journal of King Saud University - Science, 25(2), 175–178. https://doi.org/10.1016/j.jksus.2012.12.002
Yildirim, E. (2013). An overview of the variational iteration method and its applications. Mathematical Methods in the Applied Sciences, 36(11), 1439-1450.
Zongo, G., So, O., & Barro, G. (2024). Variational Iteration Method on Linear and Nonlinear Schrödinger’S Equations. International Journal of Numerical Methods and Applications, 24(2), 181–192. https://doi.org/10.17654/0975045224012