Collocation method applied to unsteady flow of gas through a porous medium
In this article, we study a two point boundary value problem of non linear differential equation on a semi infinitedomain that describes the unsteady flow of gas through a porous medium. Under special transform, we convert thisproblem to boundary value problem in compactly supported domain [0,1]. An algorithm provided for obtainingsolution by Legendre wavelet collocation method. This method is effectively used to determine y (t) and its initialslope at the origin. The convergence and stability analysis is provided. The results thus obtained are compared withthe those obtained from modified decomposition method , Variational iterational method , rational Chebyshevfunctions method (RCM)  and radial basis function (RBF) collocation method . It has been observed thatthe proposed method provide better results with lesser computational complexity.
Keywords: Convergence and stability analysis, Legendre Wavelets, Legendre wavelet collocation method, Kidder's equation.
H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962.
R. P. Agarval, D. Regan, Infinite interval problems for differential, diifference and integral Equations, Springer Science-Business Media, 2001.
R. P. Agarval, D. Regan, Non-linear boundary value problems on the semi infinite interal, all upper and lower solution approach, MATHHMATIKA, 49 (2002), 129-140.
R.E. Kidder, Unsteady flow of gas through a semi-infinite porous medium, J. Appl. Mech. 27(1957) 329332.
A. M. Wazwaz, The modified decomposition method applied to unsteady flow of gas through a porous medium, Appl. Math. Comput. 118 (2001) 123-132.
M. A. Noor, S. T. 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 58 (2009) 2182-2189.
K. Parand, M. Shahini and A. Taghavi, Generalized Laguerre Polynomials and Rational Chebyshev Collocation Method for Solving Unsteady Gas Equation, Int. J. Contemp. Math. Sciences, 4 (2009) 1005- 1011.
A. Taghavi, K.Parand, and H. Fani, Lagrangian method for solving unsteady gas equation, World Academy of Science, Engineering and Technology35 (2009) 1016-1020.
K. Parand, A. Taghavi, M. Shahini, Comparison between Rational Chebyshev and Modified generalized La-guerre Functions Psuedospectral method for solving LaneEmden and unsteady gas equation, ACTA PHYSICA POLONICA B, 40 (2009) 1749-1763.
S. Kazem, J. A. Rad, K. Parand, M. Shaban, and H. Saberi, The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation, International Journal of Computer Mathematics, 89 (2012) 22402258.
C. Lanczos, Trigonometric interpolation of empirical and analytical functions, Journal of Mathematics and Physics 17 (1938) 123-129.
C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, Berlin Springer 1988.
A. Finlayson, L.E. Scriven, The method of weighted residuals: a review, Applied Mechanics Reviews. 19 (1966) 735-748.
B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge, Cambridge University Press 1996.
E. Babolian, M.M. Hosseini, A modified spectral method for numerical solution of ordinary differential equations with non-analytic solution, Applied Mathematics and Computation. 132 (2002) 341-351.
F. Mohammadi, M.M. Hosseini and S. T. Mohyud-Din, Legendre wavelet galerkin method for solving ordinary differential equations with non-analytic solution, Int. J. of Sys. 42 (2011) 579-585.
M. Razzaghi, S. Yousefi, Legendre wavelets operational matrix of integration, International Journal of Systems Sci. 32 (2001) 495-502.