On The Simulation of Partial Differential Equations Using the Hybrid of Fourier Transform and Homotopy Perturbation Method

Document Type : Research Article


1 Associate Professor, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

2 Researcher, Tehran university alumnus in mechanical engineering, Tehran, Iran

3 M.Sc. Student, Department of Physics, Helsinki University, Helsinki, Finland


In the present work, a hybrid of Fourier transform and homotopy perturbation method is developed for solving the non-homogeneous partial differential equations with variable coefficients. The Fourier transform is employed with combination of homotopy perturbation method (HPM), the so called Fourier transform homotopy perturbation method (FTHPM) to solve the partial differential equations. The closed form solutions obtained from the series solution of recursive sequence forms are obtained. We show that the solutions to the non-homogeneous partial differential equations are valid for the entire range of problem domain. However the validity of the solutions using the previous semi-analytical methods in the entire range of problem domain fails to exist. This is the deficiency of the previous HPMs caused by unsatisfied boundary conditions that is overcome by the new method, the Fourier transform homotopy perturbation method. Moreover, it is shown that solutions approach very rapidly to the exact solutions of the partial differential equations. The effectiveness of the new method for three non-homogenous differential equations with variable coefficients is shown schematically. The very rapid approach to the exact solutions is also shown schematically.


[1] J. H. He, “Non-perturbative Methods for Strongly Nonlinear Problems,” dissertation.de-Verlag im Internet GmbH, Berlin, 2006.
[2] J. H. He, “Some asymptotic methods for strongly nonlinear equations,” Int. J. Mod. Phys., vol. B 20, no. 10, pp. 1141-1199, 2006.
[3] J. H. He, “Homotopy perturbation method for solving boundary value problems,” Phys. Lett., vol. A 350, no. 1–2, pp. 87-88, 2006.
[4] J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos Solitons Fractals, vol. 26, no. 3, pp. 695-700, 2005.
[5] J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Comput. Math. Appl. Mech. Eng., vol. 167, pp. 57-68, 1998.
[6] J. H. He, “Approximate solution of nonlinear differential equations with convolution product nonlinearities,” Comput. Math. Appl. Mech. Eng., vol. 167, pp. 69-73, 1998.
[7] J. H. He, “Variational iteration method – a kind of non-linear analytical technique: some examples,” Int. J. Non-Linear Mech., vol. 34, pp. 699-708, 1999.
[8] J. H. He, “Homotopy perturbation technique,” J. Comput. Math. Appl. Mech. Eng., vol. 178, pp. 257-262, 1999.
[9] J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-
linear problems,” Int. J. Non-Linear Mech., vol. 35, pp. 37-43, 2000.
[10] J. H. He, X.H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos Solitons Fractals, vol. 29, no. 1, pp. 108-113, 2006.
[11] J. H. He, “Periodic solutions and bifurcations of delay-differential equations,” Phys. Lett., vol. A 347, no. 4–6, pp. 228-230, 2005.
[12] J. H. He, “Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals,” vol. 26, no. 3, pp. 827-833, 2005.
[13] J. H. He, “Homotopy perturbation method for bifurcation of nonlinear problemsint,” J. Nonlinear Sci. Numer. Simul., vol. 6, no. 2, pp. 207-208, 2005.
[14] B. Jang, “Two-point boundary value problems by extended Adomian decomposition method,” J. of Comput. and Appl. Math. vol. 219, pp. 253–262, 2008.
[15] M. Madani, M. Fathizadeh, Y. Khan and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,” J. Mathematical and Computer Modelling, vol. 53 no. 9-10, pp. 1937-1945, 2011.
[16] S. Q. Wang, J. H. He, “Nonlinear oscillator with discontinuity by parameter-expansion methodChaos Solitons Fractals,” vol. 35, no. 4, pp. 688-691, 2008.
[17] M. Fathizadeh, F. Rashidi, “Boundary layer convective heat transfer with pressure gradient using Homotopy Perturbation Method (HPM) over a flat
plate, Chaos Solitons Fractals,” I. S. J. of Thermal Scince, vol. 42, no. 4, pp. 2413-2419, 2009.
[18] M. Omidvar, A. Barari, M. Momeni, D.D. Ganji, Geomech. Geoeng. New class of solutions for water infiltration problems in unsaturated soils,” Int. J. Geo-mech. and Geoeng., vol. 5, no. 2, pp. 127-135, 2010.
[19] Magdy A. El-Tawil, Noha A. Al-Mulla, “Using Homotopy WHEP technique for solving a stochastic nonlinear diffusion equation,” Math. Comput. Modelling, vol. 51, no. 9-10, pp. 1277-1284, 2010.
[20] H. E. Qarnia, Application of homotopy perturbation method to non-homogeneous parabolic
[21] partial and non linear differential equations, World J. Modelling Simul., vol. 5, no. 3, pp. 225-231, 2009.
[22] Md. S. H. Chowdhury, I.Hasim, “Solution of time-dependent emden-fowler type equation by homotopy perturbation method,” Phy. Lett. Vol. 365, no. 3, pp. 305-313, 2007.
[23] S. Kumar, O. P. Singh, S. Dixit, “Generalized Abel inversion using Homotopy perturbation method,” Applied Mathematics, vol. 2, no. 2, pp. 254-257, 2011.
[24] Y. Liu, Z. Li, Y. Zhang, “Homotopy perturbation method to fractional biological population equation,” Fractional Differential Equation, vol. 1, no. 1, pp. 117–124, 2011.
[25] Md. S.H. Chowdhury, “A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations,” J.applied Sci., vol. 11, no. 8, pp. 1416-1420, 2011.
[26] M. Akbarzade, J. Langari, “Application of homotopy-perturbation method and variational iteration method to three dimensional diffusion problem,” Int. Journal of Math. Analysis, vol. 5, no. 18, pp. 871 – 880, 2011.
[27] B. Ganjavi, H. Mohammadi, D. D. Ganji and , A. Barari Am., “Homotopy perturbation method and variational iteration method for solving Zakharov-Kuznetsov equation,” J. Applied Sci., vol. 5, no. 7, pp. 811-817, 2008.
[28] S. Z. Rida, A. A. M. Arafa, “Exact solutions of fractional-order biological population model,” Commun. Theor. Phys., vol. 52, pp. 992–996, 2009.
[29] E. Shakeri, M. Dehghan, “Numerical solutions of a biological population model using He’s variational iteration method,” Comput. and Maths, with Appl., vol. 54, pp. 1197-1207, 2007.
[30] P. Y. Tsai, C. K. Chen, “Free vibration of the nonlinear pendulum using hybrid Laplace Adomian
decomposition method,” Int. J. Numer. Meth. Biomed. Engng., vol. 27, no. 2, pp. 262-272, 2011.
[31] M. Y. Ongun, “The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+T cells,” Mathematical and Computer Modeling, vol. 53, no. 5-6, pp. 597-603, 2011.
[32] A. M. Wazwaz, M. S. Mehanna, “The Combined Laplace-Adomian method for handling singular integral equation of heat transfer,” International Journal of Nonlinear Science, vol. 10, no. 2, pp. 248-252, 2010.
[33] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2009.
[34] S. Nourazar, A. Nazari-Golshan, A. Yildrim and M. Nourazar, “On the hybrid of Fourier transform and Adomian decomposition method for the solution of nonlinear Cauchy problems of the reaction-diffusion equation,” Z. Naturforsch, vol. 67a, pp. 355-362, 2012.