Laplace's equation on a rectangle with mixed b.c.s

  1. 1. The problem statement, all variables and given/known data

    Solve Laplace's equation on the rectangle 0< x< L, 0< y< H with the boundary conditions du/dx(0, y) = 0, du/dx(L, y)=y, du/dy(x, 0)=0, U(x, H)=x.

    2. Relevant equations



    3. The attempt at a solution

    I would be able to solve it by separation of variables if the last boundary condition were du/dx(x,H)=x. How can I apply the last boundary condition? Does principle of superposition work?
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,550
    Staff Emeritus
    Science Advisor

    You need a function that has derivative 0 at y= 0 and value x at y= H. The first condition makes me think of a quadratic. Look at [itex]v(x, y)= (x/H^2)y^2[/itex]. And then change your function to w(x,y)= u(x,y)- v(x,y).
     
  4. Then how can I go further with separation of variables?
     
  5. By the way, Can I assume U=X(x)Y(y) and say Y(H)=1 and Xx(L)=1 ?
     
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