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

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.