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Research Papers

# An Accurate Numerical Method for Solving Unsteady Isothermal Flow of a Gas Through a Semi-Infinite Porous Medium

[+] Author and Article Information
Kourosh Parand

Department of Computer Sciences,
Shahid Beheshti University,
Tehran 19697-64166, Iran
e-mail: k_parand@sbu.ac.ir

Mehdi Delkhosh

Department of Computer Sciences,
Shahid Beheshti University,
Tehran 19697-64166, Iran
e-mail: mehdidelkhosh@yahoo.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 8, 2017; final manuscript received July 1, 2017; published online October 9, 2017. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 13(1), 011007 (Oct 09, 2017) (9 pages) Paper No: CND-17-1021; doi: 10.1115/1.4037225 History: Received January 08, 2017; Revised July 01, 2017

## Abstract

The Kidder equation, $y″(x)+2xy′(x)/1−βy(x)=0, x∈[0,∞), β∈[0,1]$ with $y(0)=1$, and $y(∞)=0$, is a second-order nonlinear two-point boundary value ordinary differential equation (ODE) on the semi-infinite domain, with a boundary condition in the infinite that describes the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium and has widely used in the chemical industries. In this paper, a hybrid numerical method is introduced for solving this equation. First, by using the method of quasi-linearization, the equation is converted to a sequence of linear ODEs. Then these linear ODEs are solved by using the rational Legendre functions (RLFs) collocation method. By using 200 collocation points, we obtain a very good approximation solution and the value of the initial slope $y′(0)=−1.19179064971942173412282860380015936403$ for $β=0.50$, highly accurate to 38 decimal places. The convergence of numerical results is shown by decreasing the residual errors when the number of collocation points increases.

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## Figures

Fig. 1

Graphs of the logarithm of coefficients |ai| for various values of L, m = 200 and sixth iteration to calculate an approximation of the optimal value of L

Fig. 3

Graph of the logarithm of coefficients |ai| with m = 200 and n = 6 for showing the convergence of the method

Fig. 2

Graph of residual errors RESnm with m=100,125,150,175,200, n = 6 and β=0.50 for showing the convergence of the method

## Errata

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