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Math Help - Verify 2D Laplace Solution

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    Verify 2D Laplace Solution

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    Re: Verify 2D Laplace Solution

    What is your question?
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    Re: Verify 2D Laplace Solution

    In part A I am not sure how to verify that the analytic solution is of the form described? Additionally I have done Laplace's transforms before but i have no idea how to do part B as well. Thanks for all your help!
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    Re: Verify 2D Laplace Solution

    Quote Originally Posted by helpthanks View Post
    In part A I am not sure how to verify that the analytic solution is of the form described?
    To verify the solution, just apply the Laplace operator to it and see that it gives 0 indeed. Note that your solution is separable u(x,y) = X(x)Y(y), which should ease the procedure.
    Quote Originally Posted by helpthanks View Post
    Additionally I have done Laplace's transforms before but i have no idea how to do part B as well.
    No need for Laplace transforms. This is the Laplace equation.
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    Re: Verify 2D Laplace Solution

    Would you mind doing or starting this problem for me, i have quite a few like this i have to do but am having trouble going through the whole process of one problem. Thanks again!
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    Re: Verify 2D Laplace Solution

    Attached is my work so far.
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    Re: Verify 2D Laplace Solution

    You fixed a and b such that the boundary conditions are satisfied. Good. But you need to check that the solution satisfies the Laplace equation, i.e. \nabla^2 u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0.
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    Re: Verify 2D Laplace Solution

    The first page is where i verified that it is equal to zero and the second page I solved for A & B. What should i do differently on the first page that i did not do to show it is equal to zero?
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    Senior Member vincisonfire's Avatar
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    Re: Verify 2D Laplace Solution

    On the first page you checked that the Dirichlet boundary conditions u(x,0) = 0 and u(x,L_y ) = 0. This shows that 2 boundary conditions are satisfied. You need to compute the Laplacian of the function to show that the equation is true everywhere in the domain.
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