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    2D Heat equation with mixed boundary conditions

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    Solve:

    ut = k1 uxx + k2 uyy

    on a rectangle (0<x<L, 0<y<H) Subject to

    u(0 , y, t) = 0 uy = (x, 0, t) = 0
    u(x, y, 0) = f(x,y)
    u(L, y, t) = 0 uy = (x, H, t) = 0

    Left and Right sides are kept at zero temp top and bottom are insulated

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    https://brainmass.com/math/fourier-analysis/heat-equation-mixed-boundary-conditions-625953

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    The equation is:
    (1.1)
    With 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
    ...

    Solution Summary

    The 12-pages solution contains detailed explanation how to solve the homogeneous heat equation on a 2D rectangle with mixed boundary conditions.

    $2.49

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