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Separation of variables, sine and cosine expansion.

Full equations are shown in the attached file.

1. Let f(x) = x(1-x), 0 < x < 1

Find:
a. cosine series expansion of f(x).
b. sine series expansion of f(x).
Sketch the extended function for series in (a) and (b).

2. Solve the wave equation: [attached] with boundary conditions: u(0,t) = u(1,t) = 0 and initial condition: [attached]. You may use the results from number 1.

3. Find the eigenvalue λ and Eigen function [see the attachment for the function] with boundary conditions [attached].

4. By the method of separation of variables and by Eigen function expansion, solve the initial boundary value problem: [attached] with boundary conditions: [attached] and initial conditions: [attached].
You may use the results from number 3.

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1.
We know that a function defined on the interval can be expanded into Fourier series:
(1.1)
Where
(1.2)
If is an even function, then is an odd function.
If is an odd function, then is an odd function.
When we integrate an odd function over a symmetric interval, the result is identically 0.
Therefore the Fourier coefficients in these cases are:
(1.3)

Furthermore, sue to the symmetry of the even integrands we can write:
(1.4)
Any function can be written as a sum of an odd and even functions:
(1.5)
Where the even part is:
(1.6)
While the odd part is:
(1.7)
If the function is defined on the interval we can expand it into an even function on the interval :
(1.8)
We can expand it into an odd function on the interval :
(1.9)
Then we can expand the function into either a cosine series (even expansion) or a sine series (odd expansion).
In our case L=1 and
(1.10)
It is easy to see that
(1.11)
And the even expansion is:
(1.12)
It looks like:

Since this is the even expansion we get only a cosine series where:
(1.13)
And:
(1.14)
We use the auxiliary integrals (integrating by parts twice):
(1.15)

And:

(1.16)
Thus we get:

...

Solution Summary

The expert examines separation of variables for sine and cosine expansions.

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