Environmental Engineering Reference
In-Depth Information
The general solution of an ODE of
n
-th order is a family of functions depending
on
n
independent arbitrary constants. In the case of a PDE, the general solution
contains arbitrary functions. To illustrate this, consider the equation
x
2
y
u
xy
=
,
u
=
u
(
x
,
y
)
.
If we integrate this equation with respect to
y
, we obtain
1
2
x
2
y
2
u
x
(
x
,
y
)=
+
f
(
x
)
.
A second integration with respect to
x
yields
1
6
x
3
y
2
u
=
+
ϕ
1
(
x
)+
ϕ
2
(
y
)
,
where
are arbitrary functions.
Suppose that
u
is a function of three variables,
x
,
y
and
z
. Then for the equation
u
yy
=
ϕ
1
(
x
)
and
ϕ
2
(
y
)
2, we find the general solution
y
2
u
(
x
,
y
,
z
)=
+
y
ϕ
1
(
x
,
z
)+
ϕ
2
(
x
,
z
)
,
where
2
are arbitrary functions of two variables
x
and
z
.
We recall that in the case of ordinary differential equations, the first task is to find
its general solution, and then a particular solution is determined by finding the val-
ues of arbitrary constants from the prescribed conditions. But, for partial differential
equations, selecting a particular solution satisfying supplementary conditions from
the general solution of a PDE may be as difficult as, or even more difficult than, the
problem of finding the general solution itself. This is so because the general solu-
tion of a partial differential equation involves arbitrary functions; the specialization
of such a solution to the particular form which satisfies supplementary conditions
requires the determination of these arbitrary functions, rather than merely the deter-
mination of constants.
For linear homogeneous ODE of order
n
, a linear combination of
n
linearly-
independent solutions is a general solution. Unfortunately, this is not true for PDE
in general. This is due to the fact that the solution space of every homogeneous
linear partial differential equation is infinite dimensional. For example, the partial
differential equations
ϕ
1
and
ϕ
u
x
−
u
y
=
0ad
u
=
u
(
x
,
y
)
can be transformed into the equation 2
u
η
=
0 by the transformation of variables
ξ
=
x
+
y
,
η
=
x
−
y
.
The general solution is
(
,
)=
(
+
)
,
u
x
y
f
x
y
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