Environmental Engineering Reference
In-Depth Information
Remark 2
.
Principle of superposition of linear equations
. Linear equations have
several remarkable properties that are very useful in finding their solutions. We dis-
cuss these properties here by using second-order partial differential equations as the
n
ij
=
1
a
ij
2
n
i
=
1
b
i
∂
∂
example. Let
L
be a linear operator
L
=
x
j
+
x
i
+
c
.Themost
∂
x
i
∂
∂
general homogeneous or nonhomogeneous second-order linear partial differential
equation in
n
independent variables [Eq. (1.1)] may respectively be written in the
form
Lu
=
0
and
Lu
=
f
.
The following properties can be readily shown.
Property 1.
A linear combination of two solutions of a homogeneous equation is
also a solution of the equation. That is
L
(
+
)=
=
=
c
1
u
1
c
2
u
2
0, if
Lu
1
0and
Lu
2
0.
Here
c
1
and
c
2
are arbitrary constants.
Property 2.
Let a sequence of functions
{
u
i
}
,
i
=
1
,
2
, ···
be solutions of a ho-
∞
i
=
1
c
i
u
i
,
c
i
(
i
=
1
,
2
, ···
)=
constants be uni-
formly convergent and twice differentiable with respect to the independent variables
x
1
=
=
mogeneous equation
Lu
0, and
u
,
x
2
, ··· ,
x
n
term by term in a domain. Then
u
is also a solution of the equation,
L
∞
i
=
1
c
i
u
i
that is
Lu
=
=
0.
u
2
is a solution of a nonhomogeneous equation if
u
1
and
u
2
are
the solutions of the homogeneous and the nonhomogeneous equations, respectively.
Therefore, we have
Lu
Property 3.
u
=
u
1
+
f
.
These properties are collectively called the
principle of superposition
,whichis
important in finding solutions of linear equations. It should be remarked that the
supplementary conditions must be linear as well in order to apply this principle.
=
L
(
u
1
+
u
2
)=
f
,if
Lu
1
=
0and
Lu
2
=
1.2 Three Basic Equations of Mathematical Physics
1.2.1 Physical Laws and Equations of Mathematical Physics
In physics and engineering, we normally describe or characterize the state of a phys-
ical system and the variation of system states by physical variables that are functions
of time and position in space. Physical laws govern and provide the fundamental re-
lations among basic physical variables. Equations of mathematical physics come
from and are the quantitative representation of physical laws. They are different
from general partial differential equations.
The fundamental relations established by the physical laws can be local or global
in space. They can also be for an instant or a period in time. The equations of math-
Search WWH ::
Custom Search