Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method
American Journal of Applied Mathematics
Volume 8, Issue 6, December 2020, Pages: 334-343
Received: Nov. 12, 2020;
Accepted: Nov. 26, 2020;
Published: Dec. 11, 2020
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Amanat Ali Khan, Department of Mathematics, Cuet College, Chattogram, Bangladesh
Musammet Tahmina Akter, Department of Mathematics, Chittagong University of Engineering &Technology, Chattogram, Bangladesh
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).
Amanat Ali Khan,
Musammet Tahmina Akter,
Solving Highly Nonlinear Partial Differential Equations Using Homotopy Perturbation Method, American Journal of Applied Mathematics.
Vol. 8, No. 6,
2020, pp. 334-343.
M. Esmaeilpour, D. D. Ganji, Application of He’s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate, Physics Letters A, 372, 33-38, Iran, (2007).
J. H. He, Application of homotopy perturbation method to nonlinear wave equations; Chaos, Solitons and Fractals, 26 (3), 695-700, (2005).
G. Adomian. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston, (1994).
M. Wazwaz. Adomian decomposition method for a reliable treatment of the Bratu-type equations. Applied Mathematics and Computation, 166: 652–663, (2005).
D. D. Ganji, H. Tari, M. Bakhshi Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations, International Journal of Computational and Applied Mathematics, 54, 1018- 1027, (2007).
Mario Basto, Viriato Semiao, Francisco L. Calheiros. Numerical study of modified Amomian’s method applied to Burger’s Equation. Journal of Computational and Applied Mathematics, 206, 927-949, (2007).
Doğan Kaya, The use of Adomian decomposition method for solving a speciﬁc nonlinear partial differential equations, Bulletin of the Belgian Math. Soc., Research Gate, 9, 343-349, (2002).
Fatheah Ahmad Alhendi, Aisha Abdullah Alderremy, Numerical Solutions of Three- Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods, Journal of Applied Mathematics and Physics, 4, 2011-2030, Saudi Arabia, (2016).
J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Comput. Methods Appl. Mech. Engrg., vol. 167, pp. 57–68, (1998).
J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35 (1), 37-43, (2000).
J. H. He, Comparsion of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, 156, 527-539, (2004).
J. Biazar, H. Ghazvini, Convergence of the homotopy perturbation method for partial differential equations, Non Linear Anal. RWA, (2008).
M. Tahmina Akter, A. S. M. Moinuddin and M. A. Mansur Chowdhury, Semi-Analytical Approach to Solve Non-Linear Differential Equations and Their Graphical Representations, International Journal of Applied Mathematics and Statistical Science, Vol. 3, Issue 1, 35-56, India, (2014).
M. Tahmina Akter and M. A. Mansur Chowdhury, Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations, IOSR Journal of Mathematics, Vol. 12, Issue 5, Ver. VI, PP 59-69, (2016), India.
J. Biazar, M. Eslami, H. Ghazvini, Homotopy perturbation method for systems of partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation.
O. Abdul Aziz, I. Hashim, S. Momani, Application of homotopy perturbation method to fractional IVPs, J. Comput. Appl. Math., 216 (2), pp. 574-584, (2008).
Shaher Momani, Zaid Odibat, Homotopy perturbation method for nonlinear partia differential equations of fractional order, Physics Letters A, 365, 345-350, (2007).
J. H. He. Homotopy perturbation technique. Computer methods in Applied Mechanics and Engineering, vol. 178, pp. 257-262, (1999).
J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, pp. 73-79, (2003).
J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151, 287-290, (2004).
J. H. He, Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A 374 (4-6), 228-230, (2005).
J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (1-2), 87-88, (2006).
Desai, K. R. and Pradhan, V. H., Solution of Burger’s Equation and Coupled Burger’s Equations by Homotopy Perturbation Method, International Journal of Engineering, Re- search and Applications, 2, 2033-2040, (2012).
Y. Cherruault and G. Adomian. Decomposition methods: a new proof of convergence. Math. Comput. Modelling, 8, 103–106, (1993).
G. Adomian, A new approach to the heat equationan application of the Decomposition method, J. Math. Anal. Appl., 113, 202-209, (1986).
A. M. Wazwaz, A Reliable Modification of Adomian Decomposition method, Appl. Math. Comp., 102, 77-86 (1999).
J.-L. Li, “Adomian's decomposition method and homotopy perturbation method in solving nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 168–173, 2009.
T. Öziş and A. Yıldırım, “Comparison between Adomian's method and He's homotopy perturbation method,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1216–1224, 2008.
L. Cveticanin, “Homotopy-perturbation method for pure nonlinear differential equation,” Chaos Solitons & Fractals, vol. 30, no. 5, pp. 1221-1230, (2006).