Numerical solution of non-linear partial differential equation for porous media using operational matrices

Abstract

Nonlinear partial differential equations (NPDEs) play a significant role in modeling of many complex physical problems and engineering processes. Development of reliable and efficient techniques to solve such types of NPDEs is the need of the hour. In this article, numerical solutions of a class of NPDEs subject to initial and boundary conditions are derived. In the proposed approach, shifted Chebyshev polynomials are considered to approximate the solutions together with shifted Chebyshev derivative operational matrix and spectral collocation method. The benefit of this method is that it converts such problems in the systems of algebraic equations which can be solved easily. To show the efficiency, high accuracy and reliability of the proposed approach, a comparison between the numerical results of some illustrative examples and their existing analytical results from the literature is reported. There is high consistency between the approximate solutions and their exact solutions to a higher order of accuracy. The error analysis for each case exhibited through graphs and tables confirms the super-linear convergence rate of the proposed method.