A Newton-Modified Weighted Arithmetic Mean Solution of Nonlinear Porous Medium Type Equations

Elayaraja Aruchunan,Andang Sunarto,Jackel Vui Lung Chew,Jumat Sulaiman,Mohana Sundaram Muthuvalu

doi:10.3390/sym13081511

Elayaraja Aruchunan, Andang Sunarto + Show 3 more

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https://doi.org/10.3390/sym13081511

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Abstract

The mathematical theory behind the porous medium type equation is well developed and produces many applications to the real world. The research and development of the fractional nonlinear porous medium models also progressed significantly in recent years. An efficient numerical method to solve porous medium models needs to be investigated so that the symmetry of the designed method can be extended to future fractional porous medium models. This paper contributes a new numerical method called Newton-Modified Weighted Arithmetic Mean (Newton-MOWAM). The solution of the porous medium type equation is approximated by using a finite difference method. Then, the Newton method is applied as a linearization approach to solving the system of nonlinear equations. As the system to be solved is large, high computational complexity is regulated by the MOWAM iterative method. Newton-MOWAM is formulated technically based on the matrix structure of the system. Some initial-boundary value problems with a different type of nonlinear diffusion term are presented. As a result, the Newton-MOWAM showed a significant improvement in the computation efficiency compared to the developed standard Weighted Arithmetic Mean iterative method. The analysis of efficiency, measured by the reduced number of iterations and computation time, is reported along with the convergence analysis.

Highlights

Introduction published maps and institutional affilThe porous medium type equation is one of the classes of nonlinear parabolic evolution equation
In order to evaluate the efficacy of the Newton-MOWAM iterative method, two examples of one-dimensional porous medium type initial-boundary value problems are selected with different levels of difficulties
This paper presented the formulation of a new numerical method called NewtonMOWAM for solving several porous medium type equations

Summary

Introduction

The porous medium type equation is one of the classes of nonlinear parabolic evolution equation. This class of partial differential equations appears in the description and modeling of natural physical phenomena such as gas diffusion, fluid flow, and heat propagation. The authors of [2] developed a mathematical model to describe the flow of a mixture of ideal gases in a highly porous electrode for fuel cell engineering. In their developed model, a porous medium type equation is used to simulate the evolution of the gas mixture

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