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    The 1D heat equation

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    Solve the heat equation

    u_t = ku_xx

    for 0<x<L

    With boundary conditions u(0,t)=u(L,t)=0

    Solve for the initial value conditions:

    a. u(x,0) = sin(5*Pi*x/L)
    b. u(x,0) = x
    c. For part b, plot the solution at t=0, 0.1, 1

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    The heat equation is:

    The boundary conditions are:

    And the initial condition is:

    We start with setting:
    That is, and are single-variable independent functions. Thus:
    Plugging (1.5) and (1.6) into (1.1) we get:

    Dividing both sides by and we get a separated equation:
    Now, the left hand side is independent of t while the right hand side is independent of x. This can be true if and only if the two sides of the equation are equal to the same constant.

    Solution Summary

    The 9 pages solution includes full derivations explanations of the method of separation of variables and Fourier series expansion.