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    Separation of variables, sine and cosine expansion.

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

    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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    We know that a function defined on the interval can be expanded into Fourier series:
    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:

    Furthermore, sue to the symmetry of the even integrands we can write:
    Any function can be written as a sum of an odd and even functions:
    Where the even part is:
    While the odd part is:
    If the function is defined on the interval we can expand it into an even function on the interval :
    We can expand it into an odd function on the interval :
    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
    It is easy to see that
    And the even expansion is:
    It looks like:

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


    Thus we get:


    Solution Summary

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