# 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.