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Solve Laplace equation on rectangle domain

  • #1
Dor
10
0

Homework Statement


I'm having issues with a Laplace problem. actually, I have two different boundary problems which I dont know how to solve analytically.
I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic.
It's just Laplace's equation on the square [0,a]x[0,b] with a mixed boundary.

Homework Equations



  1. the first one is:

Uxx+Uyy=0
Ux(0,y)= 0
Ux(a,y)= f(y) (some function)
U(x,0)= 0
Uy(x,b)= v (some constant)

  1. the second is:

Uxx+Uyy=0
Ux(0,y)= 0
U(a,y)= g(y) (some function)
U(x,0)= 0
Uy(x,b)= v (some constant)

The Attempt at a Solution



I tried to do ordinary separable solution but I don't really know how to do this in such problems.
 

Answers and Replies

  • #3
Dor
10
0
I know the variable separation method.
But I'm not sure how to do it when in one side of the rectangle there is a Dirichlet boundary U(x,b)=const and in the other one I have Neumann boundary Uy(x,0)=0
 
  • #4
pasmith
Homework Helper
1,735
410
Laplace's equation is linear. You can always write the solution as a sum of solutions satisfying simpler boundary conditions. For example in the second problem you can set [itex]U = U_1 + U_2[/itex] where [itex]U_1[/itex] satisfies [tex]
U_x(0,y) = 0 \\ U(a,y) = f(y) \\ U(x,0) = 0 \\ U_y(x,b) = 0 [/tex] and [itex]U_2 [/itex] satisfies [tex]
U_x(0,y) = 0 \\ U(a,y) = 0 \\ U(x,0) = 0 \\ U_y(x,b) = v[/tex]
 
  • #5
Dor
10
0
I'm not sure how to solve it when I have one side with Dirichlet boundary and the other side with Neumann boundary.
U ( x , 0 ) = 0
U y ( x , b ) = 0
or
U ( x , 0 ) = 0
U y ( x , b ) = v
 
  • #6
pasmith
Homework Helper
1,735
410
We're looking for [itex]U = \sum_n X_n(x)Y_n(y)[/itex]

For [itex]U(x,0) = U_y(x,b) = 0[/itex] you need [itex]Y_n(0) = 0[/itex] and [itex]Y_n'(b) = 0[/itex]. Therefore set [itex]Y_n(y) = \sin k_ny[/itex] so [itex]Y_n(0) = 0[/itex] is satisfied and [itex]k_n[/itex] is determined by [itex]Y_n'(b) = k_n\cos k_nb = 0[/itex].

For [itex]U(x,0) = 0[/itex] and [itex]U_y(x,b) = v[/itex], you should regard [itex]v[/itex] as a function of [itex]x[/itex] to be expanded as a fourier series; accordingly [itex]Y_n[/itex] will be an exponential function with [itex]Y_n(0) = 0[/itex] and (for convenience) [itex]Y_n'(b) = 1[/itex]. The combination of exponentials which satisfies these is [itex]Y_n(y) = \sinh(k_ny)/\sinh(k_nb)[/itex] where [itex]k_n[/itex] is determined by the boundary conditions on [itex]X_n[/itex].
 

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