Authors:
Nishtha,Sidharth Monga,Vansh Garg,Yogesh, Himanshu,DOI NO:
https://doi.org/10.26782/jmcms.spl.12/2025.08.00010Keywords:
Homotopy Perturbation method,Partial differential equations,Abstract
In this paper, we present an effective semi-analytical method for solving non-linear partial differential equations that arise in various scientific and engineering fields. The Homotopy Perturbation Method (HPM) combines the concepts of homotopy analysis and perturbation theory to obtain approximate solutions for diverse partial differential equations. Several numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.Refference:
I. Abdel-Aty, A. H., M. Khater, R. A. Attia, and H. Eleuch. “Exact Traveling and Nano-Soliton Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells.” Mathematics, vol. 8, no. 6, 2020, article 697.
II. Allahviranloo, Tofigh, Atefeh Armand, and Saeed Pirmohammadi. “Variational Homotopy Perturbation Method: An Efficient Scheme for Solving Partial Differential Equations in Fluid Mechanics.” Journal of Mathematics and Computer Science, vol. 9, no. 4, 2014, pp. 362–69.
III. Biazar, J., K. Hosseini, and P. Gholamin. “Homotopy Perturbation Method for Solving KdV and Sawada–Kotera Equations.” Journal of Applied Mathematics, vol. 6, 2009, pp. 23–29.
IV. Daga, A., and V. Pradhan. “A Novel Approach for Solving Burger’s Equation.” Applications & Applied Mathematics, vol. 9, 2014, pp. 541–52.
V. Foursov, M. V., and M. M. Maza. “On Computer-Assisted Classification of Coupled Integrable Equations.” Journal of Symbolic Computation, vol. 33, 2002, pp. 647–60.
VI. He, J. H. “Comparison of Homotopy Perturbation Method and Homotopy Analysis Method.” Applied Mathematics and Computation, vol. 156, 2004, pp. 527–39.
VII. Khater, M. M., B. Ghanbari, K. S. Nisar, and D. Kumar. “Novel Exact Solutions of the Fractional Bogoyavlensky–Konopelchenko Equation Involving the Atangana-Baleanu-Riemann Derivative.” Alexandria Engineering Journal, vol. 59, 2020, pp. 2957–67.
VIII. Liao, S. Beyond Perturbation: Introduction to Homotopy Analysis Method. CRC Press, 2003.
IX. Liang, S., and D. J. Jeffrey. “Comparison of Homotopy Analysis Method and Homotopy Perturbation Method Through an Evolution Equation.” Department of Mathematics, University of Western Ontario, 2009, pp. 1–12.
X. Maini, P. K., D. L. S. McElwain, and D. I. Leavesley. “Traveling Wave Model to Interpret a Wound Healing Cell Migration Assay for Human Peritoneal Mesothelial Cells.” Tissue Engineering, vol. 10, 2004, pp. 475–82.
XI. “Travelling Waves in a Wound Healing Assay.” Applied Mathematics Letters, vol. 17, 2004, pp. 575–80.
XII. Matinfar, M., M. Mahdavi, and Z. Raeisy. “The Implementation of Variational Homotopy Perturbation Method for Fisher’s Equation.” International Journal of Nonlinear Science, vol. 9, no. 2, 2010, pp. 188–94.
XIII. Park, C., M. M. Khater, A. H. Abdel-Aty, R. A. Attia, H. Rezazadeh, A. Zidan, and A. B. Mohamed. “Dynamical Analysis of the Nonlinear Complex Fractional Emerging Telecommunication Model with Higher–Order Dispersive Cubic–Quintic.” Alexandria Engineering Journal, vol. 59, 2020, pp. 1425–33.
XIV. Qin, H., M. Khater, and R. A. Attia. “Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids.” Mathematical Problems in Engineering, 2020, article 3498796.
XV. Sherratt, J. A., and J. D. Murray. “Models of Epidermal Wound Healing.” Proceedings of the Royal Society B, vol. 241, 1990, pp. 29–36.
XVI. Simpson, M. J., K. K. Treloar, B. J. Binder, P. Haridas, K. J. Manton, D. I. Leavesley, D. L. S. McElwain, and R. E. Baker. “Quantifying the Roles of Cell Motility and Cell Proliferation in a Circular Barrier Assay.” Journal of the Royal Society Interface, vol. 10, 2013, article 20130007.
XVII. Xue, C., D. Lu, M. M. Khater, A. H. Abdel-Aty, W. Alharbi, and R. A. Attia. “On Explicit Wave Solutions of the Fractional Nonlinear DSW System via the Modified Khater Method.” Fractals, vol. 28, no. 2, 2020, article 2040034.
XVIII. Yue, C., D. Lu, M. M. Khater, A. H. Abdel-Aty, W. Alharbi, and R. A. Attia. “On Explicit Wave Solutions of the Fractional Nonlinear DSW System via the Modified Khater Method.” Fractals, vol. 28, no. 2, 2020, article 2040034.