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Math Help - non-homogeneous PDE

  1. #1
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    non-homogeneous PDE

    Can someone help me solve this non-homogeneous PDE (Poisson's equation)?

    uxx + uyy = sin(x)sin(2y), 0 < x < pi, 0 < y < pi
    with the conditions:
    u(x,0) = sin(3x), u(x,pi) = 0, u(0,y) = 0, and u(pi,y) = sin(y)
    Any help is greatly appreciated.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    Can someone help me solve this non-homogeneous PDE (Poisson's equation)?

    uxx + uyy = sin(x)sin(2y), 0 < x < pi, 0 < y < pi
    with the conditions:
    u(x,0) = sin(3x), u(x,pi) = 0, u(0,y) = 0, and u(pi,y) = sin(y)
    Any help is greatly appreciated.
    Let v(x,y) = -\tfrac{1}{5}\sin (x) \sin (2y) then v_{xx} + v_{yy} = \sin(x)\sin(2y).

    We will try to find a function w(x,y) so that u=w+v would solve the equation you want to ask. This would require (w+v)_{xx} + (w+v)_{yy} = \sin(x)\sin(2y) \implies w_{xx}+w_{yy} = 0. We also want u(x,0) = \sin(3x) therefore w(x,0)+v(x,0) = \sin(3x) \implies w(x,0) = \sin(3x). We also want u(x,\pi) = 0 therefore w(x,\pi) + v(x,\pi) = 0 \implies w(x,\pi) = 0. We also want u(0,y) = 0 therefore w(0,y) + v(0,y) = 0 \implies w(0,y) = 0. We also want u(\pi,y) = \sin(y) therefore w(\pi,y) + v(\pi,y) = \sin (y) \implies w(\pi,y) = \sin (y).

    Now we can find the solution. Solve w_{xx}+w_{yy}=0 subject to the boundary value problem w(x,0) = \sin(3x),w(x,\pi)=0,w(0,y)=0,w(\pi,y)=\sin(y). This is a standard exercise with seperation of variables. One you get the solution w(x,y) then the solution to the original problem would be w(x,y) - \tfrac{1}{5}\sin (x)\sin (2y).
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  3. #3
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    Well, the PDE you give at the end has two nonhomogeneous conditions. So how do you solve it?
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  4. #4
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    Quote Originally Posted by PvtBillPilgrim View Post
    Well, the PDE you give at the end has two nonhomogeneous conditions. So how do you solve it?
    Solve the two problems

    w_{xx} + w_{yy} = 0,
    w(x,0) = \sin(3x),w(x,\pi)=0,w(0,y)=0,w(\pi,y)=0

    and

    w_{xx} + w_{yy} = 0,
    w(x,0) = 0,w(x,\pi)=0,w(0,y)=0,w(\pi,y)=\sin(y)

    and then add the two solutions.
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