DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter VIII
Suggested Readings
Chapter I
This material is covered in almost every text on functional analysis. We
mention specifically references [22], [25], [47].
Chapter II
Our definition of distribution in Section 1 is inadequate for many purposes. Fo
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter VI
Second Order Evolution
Equations
1
Introduction
We shall find wellposed problems for evolution equations which contain the
second order time derivative of the solution. These arise, for example, when
we attempt to use the techniques of the pre
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter V
Implicit Evolution Equations
1
Introduction
We shall be concerned with evolution equations in which the timederivative
of the solution is not given explicitly. This occurs, for example, in problems
containing the pseudoparabolic equation
t u(x,
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Hilbert Space Methods
for
Partial Differential Equations
R. E. Showalter
Electronic Journal of Differential Equations
Monograph 01, 1994.
i
Preface
This book is an outgrowth of a course which we have given almost periodically over the last eight years. It
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Advectiondiffusion equation
The conservation equation and flux constitutive equation are
cp + j = F (x) ,
(0.1)
j = ap + b p .
where c = c(x), a = a(x) and b = b(x).
Gravitydriven Fluid Flow. Let p denote pressure of a slightly compressible fluid in the
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter II
Distributions and Sobolev
Spaces
1
Distributions
1.1
We shall begin with some elementary results concerning the approximation
of functions by very smooth functions. For each > 0, let C0 (Rn ) be
given with the properties
0
,
supp( ) cfw_x Rn :
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter VII
Optimization and
Approximation Topics
1
Dirichlets Principle
When we considered elliptic boundary value problems in Chapter III we
found it useful to pose them in a weak form. For example, the Dirichlet
problem
)
n u(x) = F (x) ,
xG,
(1.1)
u(s
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
The Periodic BoundaryValue Problem
Denote the unit cube in IRN by Q (0, 1)N . Let a() L (Q) be
uniformly positive: a(x) a0 > 0, x Q.
Consider the periodic boundaryvalue problem
a(x)u(x) = F (x), x Q,
uxj =0 = uxj =1 and
u
u
(a
)xj =0 = (a
)x =1 on Q
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter IV
First Order Evolution
Equations
1
Introduction
We consider first an initialboundary value problem for the equation of heat
conduction. That is, we seek a function u : [0, ] [0, ] R which satisfies
the partial differential equation
ut = uxx ,
0
DIFFERENTIAL AND INTEGRAL EQUATIONS OF MATHEMATICAL PHYSICS
MTH 623

Spring 2014
Chapter III
Boundary Value Problems
1
Introduction
We shall recall two classical boundary value problems and show that an
appropriate generalized or abstract formulation of each of these is a wellposed problem. This provides a weak global solution to each