Environmental Engineering Reference
In-Depth Information
Definition 2.
A PDE is said to be
linear
if it is linear in the unknown function and
all its derivatives. An equation which is not linear is called a
nonlinear
equation.
A nonlinear equation is said to be
quasi-linear
if it is linear in all highest-ordered
derivatives of the unknown function.
For example, the above-mentioned three typical equations of mathematical
physics and the equations
2
u
2
u
∂
u
)
∂
u
∂
∂
u
x
=
∂
x
+
a
(
x
,
y
x
=
f
(
x
,
y
)
,
y
=
2
y
−
x
,
y
+
u
(
x
,
y
)
∂
∂
∂
x
∂
∂
∂
x
∂
are linear, whereas
∂
2
∂
2
u
u
∂
u
)
∂
u
u
2
+
=
0
and
x
+
a
(
x
,
y
y
=
∂
x
∂
y
∂
∂
are nonlinear. Equations
2
u
2
u
2
u
2
u
∂
u
x
∂
x
2
+
∂
u
y
∂
u
∂
x
2
+
∂
u
x
∂
u
2
y
2
+
=
0
and
y
2
=
0
∂
∂
∂
∂
∂
∂
∂
are quasi-linear.
The most general second-order linear PDE in
n
independent variables has the
form
n
∑
n
i
=
1
b
i
u
x
i
+
cu
=
f
a
ij
u
x
i
x
j
+
(1.1)
i
,
j
=
1
where we assume
u
x
i
x
j
=
a
ji
without loss of generality. We also as-
sume that
a
ij
,
b
i
,
c
and
f
are known functions of the
n
independent variables
x
i
.
If all coefficients
a
ij
,
b
i
and
c
are constants, the equation is called a
PDE with con-
stant coefficients
; otherwise it is a
PDE with variable coefficients
. As for ordinary
differential equations, we can classify linear PDE into homogeneous and nonhomo-
geneous equations.
u
x
j
x
i
and
a
ij
=
Definition 3.
The
free term
in a PDE is the term that contains no unknown function
and its partial derivatives. If the free term is identically zero, a linear equation is
called a
homogeneous PDE
; otherwise it is called a
nonhomogeneous PDE
.
Equation (1.1) is
homogeneous
if
f
0 ; otherwise it is
nonhomogeneous
.Note
that the definition of homogeneity is only for linear PDE.
≡
1.1.3 Solutions of Partial Differential Equations
Definition 4.
A function
u
is called a
classical solution
of the PDE, a
solution
for
short, if it has continuous partial derivatives of all orders appearing in a PDE and
satisfies the equation.
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