Analytical solution of Gas Flow Through a Micro-Nano Porous Media by Homotopy Perturbation method

Authors: Jamal Amani Rad, Kourosh Parand

Abstract:

In this paper, we have applied the homotopy perturbation method (HPM) for obtaining the analytical solution of unsteady flow of gas through a porous medium and we have also compared the findings of this research with some other analytical results. Results showed a very good agreement between results of HPM and the numerical solutions of the problem rather than other analytical solutions which have previously been applied. The results of homotopy perturbation method are of high accuracy and the method is very effective and succinct.

Keywords: Unsteady gas equation, Homotopy perturbation method(HPM), Porous medium, Nonlinear ODE

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1062416

References:

[1] I. Hashim, M. S. M. Noorani, M. R. S. Hadidi, Solving the generalized Burgers-Huxley equation using the Adomian decomposition method, Mathematical and Computer Modelling, vol. 43, pp. 1404-1411, 2006.
[2] M. Tatari, M. Dehghan, M. Razzaghi, Application of the Adomian decomposition method for the Fokker-Planck equation, Mathematical and Computer Modelling, vol. 45, pp. 639-650, 2007.
[3] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, vol. 178, pp. 257-262, 1999.
[4] F. Shakeri and M. Dehghan, Numerical solution of the Klein-Gordon equation via He-s variational iteration method, Nonlinear Dynamics, vol. 51, pp. 89-97, 2008.
[5] J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, vol. 30, pp. 700-708, 2006.
[6] K. Parand, J. A. Rad, Some solitary wave solutions of generalized Pochhammer-Chree equation via Exp-function method, International Journal of Computational and Mathematical Sciences, vol. 3, pp. 142-147, 2010.
[7] K. Parand, J. A. Rad, Exp-function method for some nonlinear PDE-s and a nonlinear ODE-s, Journal of King Saud University (Science), (2010), doi:10.1016/j.jksus.2010.08.004.
[8] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, vol. 167, pp. 57-68, 1998.
[9] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulations, vol. 35, pp. 37-43, 2000.
[10] J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation, vol. 135, pp. 73-79, 2003.
[11] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, vol. 156, pp. 527-539, 2004.
[12] S. Abbasbandy, Application of He-s homotopy perturbation method to functional integral equations, Chaos, Solitons and Fractals, vol. 31, pp. 1243-1247, 2007.
[13] S. Abbasbandy, Application of He-s homotopy perturbation method for Laplace transform, Chaos, Solitons and Fractals, vol. 30, pp. 1206-1212, 2006.
[14] A.M. Siddiqui, R. Mahmood and Q.K. Ghori, Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos, Solitons and Fractals, vol. 38, pp. 506-515, 2008.
[15] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A, vol. 352, pp. 404-410, 2006.
[16] L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals, vol. 30, pp. 1221-1230, 2006.
[17] P. D. Ariel,T. Hayat, S. Asghar, Homotopy perturbation method and axisymmetric flow over a stretching sheet, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, pp. 399-406, 2006.
[18] D.D. Ganji, A. Sadighi, Application of He-s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, pp. 411-418, 2006.
[19] J. Biazar, H. Ghazvini, Exact solutions for non-linear Schro¨odinger equations by He-s homotopy perturbation method, Physics Letters A, vol. 366, pp. 79-84, 2007.
[20] D.D. Ganji, The application of He-s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A, vol. 355, pp. 337-341, 2006.
[21] Z. Odibat, S. Momani, Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos, Solitons and Fractals, vol. 36, pp. 167-174, 2008.
[22] J.H. He, Asymptotology by homotopy perturbation method, Applied Mathematics and Computation, vol. 156, pp. 591-596, 2004.
[23] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos, Solitons and Fractals, vol. 35, pp. 140-147, 2008.
[24] L. Cveticanin, Homotopy perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals, vol. 30, pp. 1221-1230, 2006.
[25] J. Biazar, M. Eslami, H. Ghazvini, Homotopy perturbation method for systems of partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 413-418, 2007.
[26] J. Biazar, H. Ghazvini, He-s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, vol. 39, pp. 770-777, 2009.
[27] J. Biazar, H. Ghazvini, Numerical solution for special non-linear Fredholm integral equation by HPM, Applied Mathematics and Computation, vol. 195, pp. 681-687, 2008.
[28] O.K. Ghori, M. Ahmed, A.M. Siddiqui, Application of homotopy perturbation method to squeezing flow of a Newtonian fluid, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 179- 184, 2007.
[29] M.A. Rana, A.M. Siddiqui, Q.K. Ghori, Application of He-s homotopy perturbation method to Sumudu transform, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 185-190, 2007.
[30] H. Tari, D.D. Ganji, M. Rostamian, Approximate solutions of K (2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 203- 210, 2007.
[31] A. Ghorbani, J. Saberi-Nadjafi, He-s homotopy perturbation method for calculating Adomian polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 229-232, 2007.
[32] T. Ozis, A. vildirim, Traveling wave solution of Korteweg-de vries equation using He-s Homotopy Perturbation Method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, pp. 239-242, 2007.
[33] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, vol. 151, pp. 287-292, 2004.
[34] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, vol. 26, pp. 695-700, 2005.
[35] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals, vol. 26, pp. 827-833, 2005.
[36] M. Rafei, DD. Ganji, Explicit solutions of Helmholtz equation and fifthorder KdV equation using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, pp. 321- 328, 2006.
[37] M. Ghasemi, M. Tavassoli Kajanic, E. Babolian Numerical solution of the nonlinear Voltrra-Fredholm integral equations by using homotopy perturbation method, Applied Mathematics and Computation, vol. 188, pp. 446-449, 2007.
[38] M. Dehghan, J. Manafian, The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method, Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences, vol. 64, pp. 420-430,2009.
[39] M. Dehghan, F. Shakeri, Use of he-s homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, Journal of Porous Media, vol. 11, pp. 765-778,2008.
[40] R.E. Kidder, Unsteady flow of gas through a semi-infinite porous medium, J. Appl. Mech., vol. 24, pp. 329-332,1957.
[41] T.v. NA , Computational Methods in Engineering Boundary Value Problems, Academic Press, New vork, 1979.
[42] R.P. Agarwal, D. O-Regan, Infinite Interval Problems Modeling the Flow of a Gas Through a Semi-Infinite Porous Medium, Studies in Applied Mathematics, vol. 108, pp. 245-257, 2002.
[43] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, J.W. Edwards, Ann Arbor, 1946.
[44] H.T. Davis, Introduction to Nonlinear Di.erential and Integral Equations, Dover Publications, New vork, 1962.
[45] R.C. Roberts, Unsteady fow of gas through a porous medium, Proceedings of the First US National Congress of Applied Mechanics, Ann Arbor, MI, pp. 773-776, 1952.
[46] A.M. Wazwaz, The modified decomposition method applied to unsteady flow of gas through a porous medium, Applied Mathematics and Computation, vol. 118, pp. 123-132, 2001.
[47] K. Parand, A. Taghavi, H. Fani, Lagrangian method for solving unsteady gas equation, International Journal of Computational and Mathematical Sciences, vol. 3, pp. 40-44, 2009.
[48] K. Parand, M. Shahini, A. Taghavi, Generalized Lagurre Polynomials and Rational Chebyshev Collocation Method for solving unsteady gas equation, Int. J. Contemp. Math. Sciences, Vol. 4, pp. 1005-1011, 2009.
[49] Muhammad Aslam Noor, Syed Tauseef Mohyud-Din,, Variational iteration method for unsteady flow of gas through a porous medium using He-s polynomials and Pade approximants, Computers and Mathematics with Applications, vol. 58, pp. 2182-2189, 2009.
[50] A.H. Nayfeh, Problems in perturbation, New vork, John Wiley, 1985.