Numerical method for solving a nonlinear inverse diffusion equation

Abstract

Numerical Method is a way in solving a model of a problem mathematically and predicts the behaviour of the problem. That is why this method is very important for both natural and manmade process. Some basic theory of the Numerical Method has been applied in our daily life such as chemistry, physics, engineering, biology and others. The purpose of this project is to develop a numerical method for solving a one-dimensional inverse problem. In order to solve the equation problem, some assumptions such as the existence and uniqueness of the inverse problem, are taken into consideration where auxiliary problem and Schauder Fixed-point theorem were take place in order to prove it. Furthermore, a Numerical Algorithm such as Fully implicit Finite-different method and least square minimization method for solving a nonlinear inverse problem is proposed. At first, Taylor’s series Expansion is employed to linearize the nonlinear terms and then the finite-different method is used to discretize the problem. The present approach is to rearrange the system of linear differential equation into matrix form and then estimate the unknown diffusion coefficient via Least-square minimization method. Computer programming namely Maple 13 will be used as an additional method to improve the accuracy between the exact solution of the problem and the result from comparing the numerical method with a exact solution. Lastly, the graphing of the curve based from the result obtained will be done by using Maple 13. Throughout the project, we can conclude that all three objectives have been achieved.