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Title of Thesis

Higher Order Techniques for Heat Equation Subject to Non-local Specifications

Author(s)

Muhammad Aziz ur Rehman

Institute/University/Department Details
Department of Mathematics / GC University, Lahore
Session
2009
Subject
Mathematics
Number of Pages
130
Keywords (Extracted from title, table of contents and abstract of thesis)
Higher, Order,Numerical, Techniques, Heat, Equation, Non-local, Specifications, homogeneous, integral

Abstract
Higher-order numerical techniques are developed for the solution of
(i) homogeneous heat equation ut = uxx
and
(ii) inhomogeneous heat equation ut = uxx + s(x; t) subject to initial condition u(x; 0) = f(x); 0 < x < 1, boundary condition u(0; t) = g(t)0 < t T and with non-local boundary condition(s)
(i) Rb 0 u(x; t)dx = M(t) 0 < t T; 0 < b < 1
(ii) u(0; t) = R1 0 (x; t)u(x; t)dx + g1(t); 0 < t T and
(iii) u(1; t) = R1 0 (x; t)u(x; t)dx + g2(t); 0 < t T
as appropriate.
The integral conditions are approximated using Simpson's 1/3 rule while the space derivatives are approximated by higher-order finite difference approximations. Then method of lines, semidiscritization approach, is used to transform the model partial differential equations into systems of first-order linear ordinary differential equations whose solutions satisfy recurrence relations involving matrix exponential functions. The methods are higher-order accurate in space and time and do not require the use of complex arithmetic. Parallel algorithms are also developed and implemented on several problems from literature and are found to be highly accurate. Solutions of these problems are compared with the exact solutions and the solutions obtained by alternative techniques where available.

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423 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

i
54 KB
2

1

INTRODUCTION

1.1 Introduction

1.2 Literature Survey

1.3 Motivation and Objectives

1.4 Applications

1
102 KB
3 2 THIRD-ORDER TECHNIQUES FOR THE SOLUTION OF HEAT EQUATIONS

2.1 Homogeneous Heat Equation Subject to a Non-local Boundary Condition

2.2 Inhomogeneous Heat Equation Subject to Non-local Boundary Conditions

2.3 Numerical Experiments

9
182 KB
4 3 FOURTH-ORDER TECHNIQUES FOR THE SOLUTION OF HEAT EQUATIONS

3.1 Homogeneous Heat Equation Subject to a Non-local Boundary Condition

3.2 Inhomogeneous Heat Equation Subject to Non-local Boundary Conditions

3.3 Numerical Experiments

36
192 KB
5 4 FIFTH-ORDER TECHNIQUES FOR THE SOLUTION OF HEAT EQUATIONS

4.1 Homogeneous Heat Equation Subject to a Non-local Boundary Condition

4.2 Inhomogeneous Heat Equation with Non-local Boundary Condition

4.3 Inhomogeneous Heat Equation with Non-local Boundary Conditions

4.4 Numerical Experiments

66
260 KB
6 5 RESULT AND DISCUSSION 114
54 KB
7 6 BIBLIOGRAPHY

117


81 KB