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    Heat equation on a rectangular domain

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    2-1 a, b

    Consider the heat equation for a rectangular region, 0 < x < a, 0 < y < b, t > 0

    ut = k(uxx + uyy) , 0 < x < a, 0 < y < b, t > 0

    subject to the initial conditions: u(x,y) = f(x,y)

    a) ux (0, y, t) = 0, ux (a, y, t) = 0, 0 < y < b, t > 0
    uy (x, 0, t) = 0, uy (x, b, t) = 0, 0 < x < a, t > 0

    b) u (0, y, t) = 0, ux (a, y, t) = 0, 0 < y < b, t > 0
    uy (x, 0, t) = 0, uy (x, b, t) = 0, 0 < x < a, t > 0

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    https://brainmass.com/math/fourier-analysis/heat-equation-rectangular-domain-626271

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    Hi C
    Here it is.
    The difference between part (a) and (b) is only in the x-dependent eigenfunction, so most of the analysis of part (a) is valid to part (b).

    Part (a)
    The equation is:
    (1.1)
    With Neumann boundary conditions:
    (1.2)
    And
    (1.3)
    While the initial condition is:
    (1.4)
    We use separation of variables:
    (1.5)
    The boundary conditions become:

    (1.6)
    And:

    (1.7)
    Substituting (1.5) into (1.1) we get:

    (1.8)
    The left hand side is a function of t while the right hand sides is s function of x and y.
    Since it must be true for any both sides must be equal a constant:
    (1.9)
    Furthermore,

    (1.10)
    The left hand side of (1.10) is a function of x while the right hand side is a function of y, therefore both sides must be equal a constant:
    (1.11)

    And by the same token we can write:

    (1.12)
    so again the equation is separated and both sides equal a different constant:
    (1.13)
    And from (1.9) we see that :

    (1.14)
    So we start with equation (1.11):
    (1.15)
    with the boundary conditions
    (1.16)
    We distinguish between three cases:
    Case 1:
    The equation is and its solution is
    (1.17)

    Applying the boundary conditions:

    We get the trivial solution
    Case ...

    Solution Summary

    The 17-pages solution shows in great detail how to apply the method of separation of variables to the heat equation on a rectangle with mixed boundary conditions, and how to obtain the expansion coefficients using Fourier analysis.

    $2.49

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