Numerical Solution of MHD Equations By Spectral Quasilinearization Method

Abstract

Differential equations play an important role in modeling many natural and engineering processes such as ecology, aerospace engineering, fluid engineering etc. There are several methods which help us in finding the analytical solution of the differential equations in simple situations. During the mathematical modeling processe, as we get closer towards the real life situations, the system of differential equations become more and more complex and finding an analytical solution becomes almost impossible, such as in the case of Navier-Stokes' equations governing the fluid flow of the most realistic fluid. In such situations we are left with the option of using a numerical technique to obtain an approximate solution of the system.There are several methods which can help us in solving non-linear ordinary differential equations such as finite difference method, finite element method, shooting technique coupled with some classical method etc. During the recent past spectral methods were widely used for solving non-linear ordinary differential equations.

The aim of the present thesis is to investigate the effectiveness of the spectral method based quasilinearization technique known as spectral quasilinearization method (SQLM) in solving similar ordinary non-linear differential equations arising in the magnetohydrodynamics (MHD) flow along a stretching cylinder.

Chapter 1 is introductory which presents a brief overview of differential equations, their classifications and a short account of the history of spectral methods.

In chapter 2 the numerical solution of the problem considered by Mukhopadhyay \cite{Mukhopadhyay2013}using spectral quasilinearization method (SQLM) is presented. The main idea of the study presented in chapter 2 is to compare the results obtained by SQLM with those obtained by Mukhopadhyay \cite{Mukhopadhyay2013}. It is observed that the results of Mukhopadhyay \cite{Mukhopadhyay2013} are not as accurate as they claim to be.

Chapter 3 is an extension of the research study carried out by Mukhopadhyay \cite{Mukhopadhyay2013} considering the effects of induced magnetic field. The main objective of present work is to examine the effectiveness of the SQLM in solving the non linear ordinary differential equations governing the model. It is concluded that the spectral quasilinearization method is an effective technique to deal with such type of fluid flow problems.