Environmental Engineering Reference
In-Depth Information
Chapter 1
Introduction
In this chapter we discuss basic concepts and definitions of partial differential
equations, general methods for developing equations of mathematical physics, the
progress in heat-conduction theory and three types of heat-conduction equations.
Also discussed are supplementary conditions and problems for determining solu-
tions.
1.1 Partial Differential Equations
1.1.1 Partial Differential Equations and Their Orders
An
ordinary differential equation
(
ODE
for short) is a differential equation that
contains one or more derivatives of the dependent variable (in addition to the de-
pendent variable and the independent variable). The order of an ordinary differen-
tial equation is the order of the highest-ordered derivative appearing in the equation.
Similarly, a partial differential equation and its order can be defined for the case of
several independent variables.
Definition 1.
A differential equation that contains the dependent variable, more than
one independent variables and one or more partial derivatives of the dependent vari-
able is called a
partial differential equation
,or
PDE
for short. The order of the
highest-ordered partial derivative appearing in the equation is called the
order
of
the PDE.
Independent and dependent variables may not appear in a PDE explicitly for
some special cases. A PDE must contain, however, at least one partial derivative of
the dependent variable. For an unknown function
u
(
x
,
y
)
of two independent vari-
ables,
u
x
=
1
,
u
xy
=
0
and
u
xxy
+
2
u
x
=
0
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