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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

    © BrainMass Inc. brainmass.com October 9, 2019, 11:01 pm ad1c9bdddf
    https://brainmass.com/math/numerical-analysis/1d-heat-equation-241523

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    The heat equation is:

    (1.1)
    The boundary conditions are:

    (1.2)
    And the initial condition is:

    (1.3)
    We start with setting:
    (1.4)
    That is, and are single-variable independent functions. Thus:
    (1.5)
    And:
    (1.6)
    Plugging (1.5) and (1.6) into (1.1) we get:
    (1.7)

    Dividing both sides by and we get a separated equation:
    (1.8)
    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.
    Thus:
    ...

    Solution Summary

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

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