How to solve this BVP

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  • #1
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Summary:
How to determine the boundary conditions to this elliptic BVP given an analytical solution
For this BVP in ##(0,1)^2##,
$$
-u_{xx} - u_{yy} = 0
$$
subject to some boundary data it is said the analytical solution is ##u(x,y) = \theta##. I've thought about this for awhile I can't seem to figure out how to determine the boundary conditions for this BVP. Moreover, ##\theta## is illustrated in figure attached. Some comments would be greatly appreciated.
 

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Answers and Replies

  • #2
pasmith
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In polar coordinates, [tex]x = r \cos \theta \qquad y = r \sin \theta.[/tex] Since you're in the first quadrant both [itex]x[/itex] and [itex]y[/itex] are positive, so you don't lose any information by dividing: [tex]
\frac yx = \tan \theta.[/tex]
 
  • #3
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Thank you for the reply. However, given that I am also trying to also solve this numerically, I was curious what BCs I would need to satisfy the equations? This is my main concern.
 
  • #4
Office_Shredder
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The question is given that ##u=\theta## is the solution to the differential equation given unknown boundary conditions, what must those unknown boundary conditions be?

It's stupidly obvious actually, the boundary condition must be ##u=\theta## on the boundary. I feel like I must not understand the question right.
 
  • #5
pasmith
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Substitute into the given equation: [itex]u = 0[/itex] on [itex]y = 0[/itex], [itex]u = \pi/2[/itex] on [itex]x = 0[/itex], [itex]u = \arctan(y)[/itex] on [itex]x = 1[/itex], and [itex]u = \arctan(x^{-1})[/itex] on [itex]y = 1[/itex].
 
  • #6
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Transform the differential equation (Laplace's equation) to cylindrical polar coordinates and see what it says.
 

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