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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/;
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