KENYATTA UNIVERSITY
SMA 433
PARTIAL DIFFERENTIAL EQUATIONS II
Iyaya Wanjala
Department of Mathematics
MORE NOTES AND SERVICES>> WWW.ROWHESS.COM
, FOREWORD
This monograph is the outgrowth of my lecture notes for a one semester course of partial
differential equations taught at the University of Nairobi from 1993 to 1996, Maseno
University (1996 to 1997), Catholic University of Eastern Africa and Kenyatta University
from 1998 to date.
The aim of this monograph is to present a clear and well organized treatment of the concept
behind the development of mathematics. Many practical problems have been illustrated
displaying a wide variety of solution techniques.
The three types of equations that we have discussed in this manuscript are namely hyperbolic,
elliptic and parabolic equations. In a study purely of partial differential equations, it would be
natural to take each equation separately and examine its properties, emphasizing the
essentially different pattern of behavior of solutions of the above named equations.
Our point of view is on techniques rather than on the intrinsic properties which characterize
the boundary value problems and thus we are going to group together particular mathematical
techniques rather than patterns of behavior of solutions.
I owe a debt of gratitude to many mathematicians from whom, as lecturers, colleagues,
authors or correspondents, I have drawn my knowledge of this subject. I especially cherish
the personal contacts I have had with many of them and in particular I would like to thank the
Late M.S. Alala and Professor B.W. Ogana whom I have in the past had the privilege of
calling colleagues and from both whom I have learned much. I am also grateful to the
students of the Kenyatta University who have always posed thrilling questions during
lectures.
ii
, TABLE OF CONTENTS
Foreword (iii)
Chapter 1
1.0 Introduction (1)
Exercise (6)
Chapter 2
2.0 Classification of Second Order Partial Differential Equations (6)
Summary (10)
2.1 Differential Equations of the Characteristics (10)
Exercise (15)
Chapter 3
3.0 The Method of Separation of Variables (17)
Exercise (28)
Chapter 4
4.0 Transform Methods (31)
4.1 Laplace Transformation (31)
Exercise
4.2 Fourier Series (35)
4.3 Fourier Transform (36)
4.4 Convolution Theory for Fourier Transform (40)
Exercise (44)
Chapter 5
5.0 Riemann’s Methods (45)
5.1 Hyperbolic Equations (45)
5.2 Riemann’s Method and the Riemann-Green Function (45)
5.3 The Euler-Poisson-Darboux Equation (55)
Exercise (58)
Chapter 6
6.0 Non-Linear Second Order Partial Differential Equations (60)
6.1 Monge’s Methods (60)
iii
, CHAPTER ONE
1.0 INTRODUCTION
Ordinary differential equations involve derivatives of one of more dependent variables with
respect to a single independent variable. By using ordinary differential equation to solve
applied problems, we are in effect greatly simplifying the mathematical model governing the
physical situation. In practice, the solution function of a physical problem depends upon the
space variable in addition to time t . For example, the temperature in a given body depends
upon the point of measurement which may be defined by the Cartesian coordinates x, y , z
and the time of measurement t . In such cases, a realistic approach must take into
consideration the fact that the dependent variable depends not only on t , but also on one or
more space variables. Whenever more than a single independent variable must be taken into
consideration, the formulation of such problems leads to partial differential equation (pde)
rather than ordinary differential equation (ode).
In this course, the first five chapters, boundary value problems for linear, second order, partial
differential equations will be discussed. As a result of the work done on such equations since
their existence and importance were first recognized, great progress has been made in the
development of mathematics. The study of these equations inextricably linked with what is
still regarded, even in an age in which more and more abstraction is demanded of and sought
by mathematicians as true classical analysis. More such equations are of fundamental
importance in mathematical description of a wide range of physical phenomena extending
from electro-magnetic theory to heat conduction and from gravitational force to sound waves.
The basic equations which will be considered are set out below. In all cases the letter u is
used to denote the dependent variable x, y , z being the independent variables and t the time
variable.
(i) Laplace’s equation 2u 0 ,
1
The diffusion equation u 0,
2
(ii)
k t
1 2u
(iii) The wave equation 2u 0.
c 2 t 2
The Laplacian may operate in one, two or three space variables in each equation and the
appropriate dimension will have to be made clear in individual cases.
iv
SMA 433
PARTIAL DIFFERENTIAL EQUATIONS II
Iyaya Wanjala
Department of Mathematics
MORE NOTES AND SERVICES>> WWW.ROWHESS.COM
, FOREWORD
This monograph is the outgrowth of my lecture notes for a one semester course of partial
differential equations taught at the University of Nairobi from 1993 to 1996, Maseno
University (1996 to 1997), Catholic University of Eastern Africa and Kenyatta University
from 1998 to date.
The aim of this monograph is to present a clear and well organized treatment of the concept
behind the development of mathematics. Many practical problems have been illustrated
displaying a wide variety of solution techniques.
The three types of equations that we have discussed in this manuscript are namely hyperbolic,
elliptic and parabolic equations. In a study purely of partial differential equations, it would be
natural to take each equation separately and examine its properties, emphasizing the
essentially different pattern of behavior of solutions of the above named equations.
Our point of view is on techniques rather than on the intrinsic properties which characterize
the boundary value problems and thus we are going to group together particular mathematical
techniques rather than patterns of behavior of solutions.
I owe a debt of gratitude to many mathematicians from whom, as lecturers, colleagues,
authors or correspondents, I have drawn my knowledge of this subject. I especially cherish
the personal contacts I have had with many of them and in particular I would like to thank the
Late M.S. Alala and Professor B.W. Ogana whom I have in the past had the privilege of
calling colleagues and from both whom I have learned much. I am also grateful to the
students of the Kenyatta University who have always posed thrilling questions during
lectures.
ii
, TABLE OF CONTENTS
Foreword (iii)
Chapter 1
1.0 Introduction (1)
Exercise (6)
Chapter 2
2.0 Classification of Second Order Partial Differential Equations (6)
Summary (10)
2.1 Differential Equations of the Characteristics (10)
Exercise (15)
Chapter 3
3.0 The Method of Separation of Variables (17)
Exercise (28)
Chapter 4
4.0 Transform Methods (31)
4.1 Laplace Transformation (31)
Exercise
4.2 Fourier Series (35)
4.3 Fourier Transform (36)
4.4 Convolution Theory for Fourier Transform (40)
Exercise (44)
Chapter 5
5.0 Riemann’s Methods (45)
5.1 Hyperbolic Equations (45)
5.2 Riemann’s Method and the Riemann-Green Function (45)
5.3 The Euler-Poisson-Darboux Equation (55)
Exercise (58)
Chapter 6
6.0 Non-Linear Second Order Partial Differential Equations (60)
6.1 Monge’s Methods (60)
iii
, CHAPTER ONE
1.0 INTRODUCTION
Ordinary differential equations involve derivatives of one of more dependent variables with
respect to a single independent variable. By using ordinary differential equation to solve
applied problems, we are in effect greatly simplifying the mathematical model governing the
physical situation. In practice, the solution function of a physical problem depends upon the
space variable in addition to time t . For example, the temperature in a given body depends
upon the point of measurement which may be defined by the Cartesian coordinates x, y , z
and the time of measurement t . In such cases, a realistic approach must take into
consideration the fact that the dependent variable depends not only on t , but also on one or
more space variables. Whenever more than a single independent variable must be taken into
consideration, the formulation of such problems leads to partial differential equation (pde)
rather than ordinary differential equation (ode).
In this course, the first five chapters, boundary value problems for linear, second order, partial
differential equations will be discussed. As a result of the work done on such equations since
their existence and importance were first recognized, great progress has been made in the
development of mathematics. The study of these equations inextricably linked with what is
still regarded, even in an age in which more and more abstraction is demanded of and sought
by mathematicians as true classical analysis. More such equations are of fundamental
importance in mathematical description of a wide range of physical phenomena extending
from electro-magnetic theory to heat conduction and from gravitational force to sound waves.
The basic equations which will be considered are set out below. In all cases the letter u is
used to denote the dependent variable x, y , z being the independent variables and t the time
variable.
(i) Laplace’s equation 2u 0 ,
1
The diffusion equation u 0,
2
(ii)
k t
1 2u
(iii) The wave equation 2u 0.
c 2 t 2
The Laplacian may operate in one, two or three space variables in each equation and the
appropriate dimension will have to be made clear in individual cases.
iv
