Hybrid Integral Transform Techniques for the Solution of Third-Order Nonlinear Ordinary Differential Equations
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Page Numbers:
192-200
Digital Object Identifier:
10.58578/mjaei.v3i2.9236
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Authors:
Umar Mujahid Aliyu1*, David Opeoluwa Oyewola2, Joel John Taura3, Salisu Lukunti4, Hassan Muhammad5, Abubakar Yahya Adamu6, Abdulhalim Isah Ibrahim7, Mubarak Muhammad8, Imafidor Hassan Ibrahim9, Mohammed Abubakar Kolo10, Isah Adamu11, Wallen Juliet Piapna'an12, Mustapha Mohammed Mansur13, Ibrahim Abubakar Adamu14, Mohammed Yusuf Marafa15, Abdulwasiu Umar16, Sulaiman Ahmad17, Nura Hashim18 
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Abstract
Third-order nonlinear ordinary differential equations frequently arise in the mathematical modeling of complex engineering and physical phenomena; however, exact analytical solutions remain difficult to obtain because of strong nonlinearities and higher-order derivative effects. Classical integral transform techniques, including the Laplace and Fourier transforms, are widely used for solving differential equations but often have limitations when extended to nonlinear systems. Although modern integral transforms such as the Sumudu, Mahgoub, and Elzaki transforms offer computational advantages, their applicability is generally restricted to linear models. This study introduces a hybrid analytical approach that integrates the Mahgoub transform with the Variational Iteration Method (VIM) to solve third-order nonlinear ordinary differential equations more effectively. The proposed method converts the governing equation into the transform domain and applies an iterative correction functional to address nonlinear terms without linearization or discretization. The resulting solutions are expressed in rapidly convergent series form. Numerical validation demonstrates strong agreement with exact solutions, confirming the efficiency, accuracy, and stability of the hybrid Mahgoub–VIM approach. The study concludes that this hybrid semi-analytical method provides a reliable framework for solving higher-order nonlinear differential equations in applied mathematics and engineering analysis. These findings contribute to the development of transform-based analytical methods by extending the applicability of the Mahgoub transform to nonlinear differential equation models through variational iteration.
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References
Ahmed, A., Khan, Y., & Wu, Q. (2023). Hybrid integral transform methods for nonlinear fractional differential equations. Applied Mathematics and Computation, 432, Article 127345.
Alzaki, A., & Jassim, H. (2024). A hybrid Mahgoub transform approach for nonlinear evolution equations. Chaos, Solitons & Fractals, 178, Article 114302.
Audu, K. J., Ogwuche, M. O., Akande, S., & Amuda, Y. Y. (2025). Advancements in solving higher-order ordinary differential equations via the variational iterative method. Akdeniz University Journal of Science and Engineering, 1(1), 36–46.
Ganie, A. H., Singh, J., & Kumar, D. (2024). Integral transform techniques for fractional differential equations: A review. Fractional Calculus and Applied Analysis, 27(1), 1–32.
Hussein, A., & Ziane, D. (2024). Nonlocal operators and memory effects in fractional models. Journal of Computational Physics, 498, Article 112685.
Mahgoub, M. A. (2016). The Mahgoub transforms and its applications. British Journal of Mathematics & Computer Science, 17(2), 1–13.
Mikail, R. (2023). Variational iteration-based hybrid methods for nonlinear fractional systems. Nonlinear Dynamics, 112, 2341–2358.
Onuoha, C. (2023). Comparative analysis of modern integral transforms in applied mathematics. International Journal of Applied Mathematics, 36(4), 589–605.
Waqas, M. (2022). Applications of Sumudu transform in fractional calculus. Results in Physics, 39, Article 105760.
Watugala, G. K. (1993). Sumudu transform: A new integral transform to solve differential equations and control engineering problems. International Journal of Mathematical Education in Science and Technology, 24(1), 35–43. https://doi.org/10.1080/0020739930240105