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Then we find that the unique smooth solution of this problem is given by the formula
u
.x; t /
D
2:3e
9
2
t
sin.3x/
C
10e
36
2
t
sin.6x/:
Indeed, the reader should verify by hand that this function satisfies the three
equations (
8.66
), (
8.67
), and (
8.68
).
Example 8.4.
Let us determine a formula for the solution of the following problem:
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0 for t>0;
u
.x; 0/
D
20 sin. x/
C
8 sin.3 x/
C
sin.67 x/
C
1002 sin.10
4
x/
for x
2
.0; 1/:
This is easily accomplished by setting
for k
ยค
1; 3; 67; 10
4
;
c
k
D
0
D
20; c
3
D
8; c
67
D
1; c
10
4
D
1002;
c
1
in formula (8.65). The reader should verify that the solution of this problem is
given by
u
.x; t /
D
20e
2
t
sin.x/
C
8e
9
2
t
sin.3x/
C
e
.67/
2
t
sin.67x/
C
1002e
.10
4
/
2
t
sin.10
4
x/
8.2.5
Initial Conditions Given by a Sum of Sine Functions
The technique used in the last two examples is generalized in a straightforward
manner to handle cases where the initial condition consists of any finite number of
sine modes. This can be expressed in mathematical terms as follows.
Let S be any finite set of positive integers and consider an initial condition of the
form
f.x/
D
X
k
2
S
c
k
sin.kx/;
where
f
c
k
g
k
2
S
are arbitrary given constants. In this case our model problem takes the form
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0
for t>0;
u
.x; 0/
D
X
c
k
sin.kx/
for x
2
.0; 1/;