An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).
| Published in | American Journal of Applied Mathematics (Volume 8, Issue 6) |
| DOI | 10.11648/j.ajam.20200806.16 |
| Page(s) | 334-343 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2020. Published by Science Publishing Group |
Homotopy Perturbation Method, Burger’s Equation, Nonlinear Partial Differential Equations, Approximate Solutions, Adomian Decomposition Method
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APA Style
Amanat Ali Khan, Musammet Tahmina Akter. (2020). Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method. American Journal of Applied Mathematics, 8(6), 334-343. https://doi.org/10.11648/j.ajam.20200806.16
ACS Style
Amanat Ali Khan; Musammet Tahmina Akter. Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method. Am. J. Appl. Math. 2020, 8(6), 334-343. doi: 10.11648/j.ajam.20200806.16
AMA Style
Amanat Ali Khan, Musammet Tahmina Akter. Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method. Am J Appl Math. 2020;8(6):334-343. doi: 10.11648/j.ajam.20200806.16
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Download@article{10.11648/j.ajam.20200806.16,
author = {Amanat Ali Khan and Musammet Tahmina Akter},
title = {Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method},
journal = {American Journal of Applied Mathematics},
volume = {8},
number = {6},
pages = {334-343},
doi = {10.11648/j.ajam.20200806.16},
url = {https://doi.org/10.11648/j.ajam.20200806.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.16},
abstract = {An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).},
year = {2020}
}
TY - JOUR T1 - Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method AU - Amanat Ali Khan AU - Musammet Tahmina Akter Y1 - 2020/12/11 PY - 2020 N1 - https://doi.org/10.11648/j.ajam.20200806.16 DO - 10.11648/j.ajam.20200806.16 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 334 EP - 343 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20200806.16 AB - An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM). VL - 8 IS - 6 ER -
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