Application of the Homotopy Perturbation Method to Nonlinear Partial Differential Equations
Keywords:
Homotopy Perturbation Method, Heat equation, Linear and Nonlinear PDE.Abstract
In this paper, we investigate the efficiency of the Homotopy Perturbation Method (HPM) in solving both linear and nonlinear partial differential equations (PDEs). The method is applied to a series of classical problems such as the heat equation, nonlinear Schrödinger equation, telegraph equation, and reaction–diffusion models. In all cases, HPM provides solutions that either coincide with the exact analytical results or yield rapidly convergent series approximations. Moreover, several open problems and future directions are discussed, including convergence analysis, stiff and singular problems, fractional PDEs, high-dimensional systems, and real-world applications.
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References
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, 1994.
J. Biazar and H. Ghazvini, Homotopy perturbation method for solving systems of Volterra integral equations, Chaos, Solitons Fractals, 39(3), (2004), 1253–1258.
S. M. El-Sayed, Application of homotopy perturbation method to the linear and nonlinear heat equations, Appl. Math. Comput., 190(1), (2007), 365–373.
D. D. Ganji and A. Sadighi, Application of He’s homotopy-perturbation method to nonlinear coupled equations, Int. J. Nonlinear Sci. Numer. Simul., 8(3), (2007), 435-443.
I. Hashim, Application of homotopy perturbation method for fractional diffusion-wave equations, Phys. Lett. A, 374(34), (2010), 3474-3479.
J. H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A, 350(1–2), (2006), 87–88.
J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178(3–4), (1999), 257–262.
J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35(1), (2000), 37–43.
J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26(3), (2005), 695-700.
H. Jafari and H. Tajadodi, Application of homotopy analysis method for solving fractional KdV–Burgers equation, Comput. Methods Appl. Mech. Eng., 16(3), (2011), 1138-1149.
S. A. Khuri, A new approach to solving linear and nonlinear PDEs via the homotopy analysis method, Appl. Math. Comput., 218(6), (2012), 2763-2775.
S. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD Thesis, Shanghai Jiao Tong University, 1992.
S. Momani and Z. Odibat, Analytical approach to fractional differential equations by modified homotopy perturbation method, Appl. Math. Comput., 177(2), (2006), 488–494.
M. A. Noor and S. T. Mohyud-Din, Variational iteration method for solving higher-order nonlinear PDEs, Int. J. Nonlinear Sci. Numer. Simul., 11(1), (2010), 7-12.
Z. Odibat and S. Momani, Modified homotopy perturbation method: application to fractional differential equations, Chaos Solitons Fractals, 36(1), (2008), 167-174.
M. Rafei, D.D. Ganji, H. Daniali, and H. Pashaei, The variational iteration method and homotopy perturbation method for solving Burgers’ equation, Appl. Math. Comput., 190(2), (2007), 437–445.
A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111(1), (2000), 53-69.
A. Yildirim, A homotopy perturbation method for nonlinear evolution equations with variable coefficients, Appl. Math. Comput., 219(6), (2013), 2888-2895.
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