Fourier methods in one dimension
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Using the method of separation of variables, solve the partial differential equation
u subscript(tt)+2(pi)u subscript(t)-u subscript(xx)=-3sin(3(pi)x)
for 0 less than or equal to x less than or equal to 1
with boundary conditions u(0,t)=u(1,t)=0
and initial conditions u(x,0)=u subscript (t)(x,0)=0
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Solution Summary
The expert examines Fourier methods in one dimension. The method of separation of variables is examined.
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Expansion of a function into a series of orthogonal eigenfunctions.
We define a basis of orthogonal functions with respect to a weight function over the interval as:
(1.1)
Where A is some constant.
We can expand the function into a series of the orthogonal functions:
(1.2)
The question is to find the set of expansion coefficients
For this we multiply both sides by :
(1.3)
Since the summation is over n and not m, we can bring the term into the sum:
(1.4)
Now we integrate both sides:
(1.5)
Integration of a sum is the same as sum of integrals, and since are constants:
(1.6)
According to the orthogonal relations (1.1) we see that the only term that is not zero on the right hand side sum is the term where and we are left with:
(1.7)
And this completes the expansion (1.2).
It is important to note that the expansion is unique. That is, there is only one set of coefficients that forms the expansion.
We prove this by contradiction.
Set
(1.8)
And for at least one n we have
Then:
(1.9)
Or:
(1.10)
Define then:
(1.11)
But according to (1.7) this means that for all n we have , and therefore for all n we have
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