Lagrangian Method for Solving Unsteady Gas Equation

Authors: Amir Taghavi, kourosh Parand, Hosein Fani

Abstract:

In this paper we propose, a Lagrangian method to solve unsteady gas equation which is a nonlinear ordinary differential equation on semi-infnite interval. This approach is based on Modified generalized Laguerre functions. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare this work with some other numerical results. The findings show that the present solution is highly accurate.

Keywords: Unsteady gas equation, Generalized Laguerre functions, Lagrangian method, Nonlinear ODE.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070687

References:

[1] B. Y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., vol. 86, pp. 635-654, 2000.
[2] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., vol. 38, pp. 1113-1133, 2000.
[3] Y. Maday, B. Pernaud-Thomas and H. Vandeven, Reappraisal of Laguerre type spectral methods, La Recherche Aerospatiale, vol. 6, pp. 13-35, 1985.
[4] H. I.Siyyam, Laguerre tau methods for solving higher order ordinary differential equations, J. Comput. Anal. Appl., vol. 3, pp. 173-182, 2001.
[5] B. Y. Guo, Jacobi spectral approximation and its applications to differential equations on the half line, J. Comput. Math., vol. 18, pp. 95-112, 2000.
[6] J. P.Boyd, Chebyshev and Fourier Spectral methods, Second Edition, 2rd ed., New York:Dover, 2000.
[7] J. P. Boyd, The optimization of convergence for Chebyshev polynomial methods in an unbounded domain, J. Comput. Phys., vol. 45, pp. 43, 1982.
[8] B. Y.Guo, J. Shen, Z. Q.Wang, A rational approximation and its applications to differential equations on the half line, J. Sci. Comput., vol. 15, pp. 117-147, 2000.
[9] J. P. Boyd, Pseudospectral methods on a semi-infinite interval with application to the Hydrogen atom: a comparison of the mapped Fouriersine method with Laguerre series and rational Chebyshev expansions, J. Comput. Phys., vol. 70, pp. 63-88, 1987.
[10] K. Parand, M. Razzaghi, Rational Legendre approximation for solving some physical problems on semi-infinite intervals, Phys. Scripta, vol. 69, pp. 353-357, 2004.
[11] K. Parand, M. Razzaghi, Rational Chebyshev tau method for solving higher-order ordinary differential equations, Appl. Math. Comput., vol. 149, pp. 73-80, 2004.
[12] K. Parand, M. Razzaghi, Rational Chebyshev tau method for solving Volterra-s population model, Int. J. Comput. Math., vol. 81, pp. 893- 900, 2004.
[13] R.E. Kidder, Unsteady flow of gas through a semi-infinite porous medium, IJ. Appl. Mech., vol. 24, pp. 329-332, 1957.
[14] T.Y. NA, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York, 1979.
[15] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, J. Edwards, Michigan:Ann Arbor, 1946.
[16] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, New York:Dover Publications, 1962.
[17] R.C. Roberts, Unsteady flow of gas through a porous medium, in: Proceedings of the First US National Congress of Applied Mechanics, Michigan:Ann Arbor, 1952.
[18] A.M. Wazwaz, The modified decomposition method applied to unsteady flow of gas through a porous medium, Appl. Math. Comput. vol. 118, pp. 123-132, 2001.
[19] B. Y. Guo, J. Shen, C. L. Xu, Generalized Laguerre approximation and its applications to exterior problems, J. Comp. Math., vol., vol. 53, pp. 63, 2005.
[20] O. Coulaud , D. Funaro, O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Method. Appl. M., vol. 80, pp. 451-458, 1990.
[21] S. S. Bayin, Mathemathical Methods in science And Engineering, New York:John Wiley & Sons, 2006.
[22] G. Gasper, K. Stempak and W. Trembels, Fractional integration for Laguerre expansions, J. Math. Appl. Anal., vol. 67, pp. 6775, 1995.
[23] H. Taseli, On the exact solution of the Schrodinger equation with a quartic anharmonicity, Int. J .Quantom .Chem. vol. 63, pp. 63-71, 1996.
[24] D. Funaro, Polynomial Approximation of Differential Equations, Berlin:Springer-Verlag, 1992.