Environmental Engineering Reference
In-Depth Information
where
f
(
x
+
y
)
is an arbitrary function. Thus, we see that each of the functions
n
e
n
(
x
+
y
)
,
(
x
+
y
)
,
sin
n
(
x
+
y
)
,
cos
n
(
x
+
y
)
,
n
=
1
,
2
,
3
, ···
is a solution. The fact that a simple equation such as
u
x
−
0 yields infinitely
many solutions is an indication of an added difficulty which must be overcome in
the study of partial differential equations. Thus, we generally prefer to directly de-
termine the particular solution of a PDE satisfying prescribed supplementary condi-
tions.
u
y
=
1.1.4 Classification of Linear Second-Order Equations
The classification of partial differential equations is suggested by the classification
of quadratic equations in analytic geometry. The equation
Ax
2
Cy
2
+
2
Bxy
+
+
Dx
+
Ey
+
F
=
0
B
2
is elliptic, parabolic, or hyperbolic accordingly as
Δ
=
−
AC
is negative, zero, or
positive.
Consider a second-order linear equation in the dependent variable
u
and the in-
dependent variables
x
and
y
,
a
11
u
xx
+
2
a
12
u
xy
+
a
22
u
yy
+
b
1
u
x
+
b
2
u
y
+
cu
+
f
=
0
(1.2)
where the coefficients are functions of
x
and
y
. The equation is said to be elliptic,
parabolic, or hyperbolic at a point
(
x
0
,
y
0
)
accordingly as
a
12
(
Δ
=
x
0
,
y
0
)
−
a
11
(
x
0
,
y
0
)
a
22
(
x
0
,
y
0
)
is negative, zero, or positive. If this is true at all points in a domain
D
, the equation
is said to be elliptic, parabolic, or hyperbolic in
D
.
It should be remarked here that a given PDE may be of a different type in a dif-
ferent domain. For example,
Tricomi equation
yu
xx
+
u
yy
=
0
y
.
To generalize the classification to the case of more than two independent vari-
ables, consider the quadratic form of the equation (1.2)
is hyperbolic for
y
<
0, parabolic for
y
=
0, and elliptic for
y
>
0, since
Δ
=
−
2
∑
1
2
A
(
λ
)=
a
11
λ
+
2
a
12
λ
1
λ
2
+
a
22
λ
=
a
ij
λ
i
λ
j
i
,
j
=
1
whose matrix is
a
11
a
12
a
21
a
22
,
a
12
=
a
21
.
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