A Comparative Study of Matched Asymptotic Expansion and Homotopy Perturbation Method for the Reaction-Convection-Diffusion Boundary Value Problem with Mathematica-Based Validation
Keywords:
Homotopy Perturbation Method, Asymptotic Expansions, Singular Perturbations, Boundary Value Problems, Reaction-Convection-Diffusion, MathematicaAbstract
In this paper, we have given comparative error analysis of Matched Asymptotic Expansion (MAE) and Homotopy Perturbation Method (HPM) for singularly perturbed boundary value problems. Here we have considered two Reaction-Convection-Diffusion problems, firstly homogenous Reaction-Convection-Diffusion problem without force term and then Reaction-Convection-Diffusion with oscillating force term for applying MAE and HPM. First, we used Mathematica for computing exact solution and then for computing errors by both the methods with different values of and also have given comparison between errors by plotting graphs with the help of Mathematica. Also, we could determine the best method for solving the singularly perturbed boundary value problem as per the plots and error tables. Even with plot function in Mathematica, we could see boundary layer formation of solution for matching the initial conditions.
Downloads
References
[1] Manoj Kumar, Parul. Methods for solving singular perturbation problems arising in science and engineering. Mathematical and Computer Modelling, 54(1-2), 556-575, 2011.
[2] Ravendra Singh, and Anoop Kumar. A Comparative Study of Homotopy Perturbation Method with Differential Transformation Method to Solve a Reaction Diffusion Equation. Indian Journal of Science and Technology, 10(30), 1-9, 2017.
[3] S. Gh. Hosseini, S. M. Hosseini, M. Heydari, M. Amini. The analytical solution of singularly perturbed boundary value problems. Journal of mathematics and computer science, 10, 7-22, 2014.
[4] Mubashir Qayyum, Khadim Hussain. Homotopy Perturbation Laplace Method for Boundary Value Problems. International Journal of Emerging Multidisciplinaries: Mathematics, 03(01), 1-6, 2024.
[5] Mubashir Qayyum, Imbsat Oscar. Least Square Homotopy Perturbation Method for Ordinary Differential Equations. Journal of Mathematics, 1-16,2021.
[6] Nidal Anakira, Sameer Bawaneh, Areen Al-khateeb, Ala Amourah, Tala Sasa. Modified Optimal Homotopy Asymptotic Method for Singular Two-Point Boundary Value Problems. European Journal of Pure and Applied Mathematics, 18(4), 1-11, 2025.
[7] Shaher Momani, Zaid Odibat. Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Computers & Mathematics with Applications. 54(7-8), 910-919, 2007.
[8] Tapas Roy & Dilip K. Maiti. An optimal and modified homotopy perturbation method for strongly nonlinear differential equations. Nonlinear Dynamics, 111, 15215–15231, 2023.
[9] Malak Hijazi, Mahmood Shareef Ajeel, Kamel Al-Khaled, Hala Al-Khalid. Perturbation Methods for Solving Non-Linear Ordinary Differential Equations. International Journal of Applied Mathematics, 54(10), 2070-2082, 2024.
[10] Azhar Shareef Islubi, Muayyad Mahmood Khalil. Using Optimal Homotopy Asymptotic Method for Solving Differential Equations. Central Asian Journal of Mathematical Theory and Computer Sciences, 5(3), 329-338, 2024.
[11] Hasan Almefleh. Approximating the Solution of Cell-cell Adhesion Model via the Homotopy Perturbation Method. International Journal of Mathematics and Computer Science, 19(2), 509-519, 2024.
[12] Igor V. Andrianov, Jan Awrejcewicz. Asymptotic Methods for Engineers. Taylor & Francis Group, 2024.
[13] A. Lagerstrom. Matched Asymptotic Expansions: Ideas and Techniques. Springer Science + Business Media, LLC, 2013.
[14] W Eckhaus. Matched asymptotic expansions and singular perturbations. North-Holland Publishing Company-Amsterdam, 2011.
[15] AW Bush. Perturbation methods for engineers and scientists. Taylor & Francis Group, 2018.
[16] H Li, MK Alam, H Li. Asymptotic analysis on viscoelastic rheological fluid on gravity driven drainage flow through a vertical heated tube. Journal of Taibah University for Science. 20(1), 1-17, 2026.
[17] SP Sathiyapriya, C Jana, N Ivkovic. Stability Analysis of Fuzzy Generalized Abel Integral Equations Using Homotopy Perturbation Method. European Journal of Pure and Applied Mathematics. 19(1), 1-13, 2026.
[18] KF Fazea, NR Seewn, MT Yasser. A New Technique for Solving Nonlinear Partial Differential Equations Using the Homotopy Perturbation Method. Journal of Education for Pure Science, 15(2), 60-68, 2025.
[19] N Anakira, A Amourah, A Almalki, A Alsoboh, T Sasa. Exact solutions of nonlinear delay Volterra Integro-differential equations using modified homotopy perturbation method. European Journal of Pure and Applied Mathematics. 18(3), 1-16,2025.
[20] Ji-Huan He, Yusry O. El-Dib. Homotopy perturbation method with three expansions. Journal of Mathematical Chemistry, 59, 1139-1150, 2021.
