Solving BVP with Analytical Solution in (0,1)^2

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In summary, the conversation discusses a boundary value problem in the region (0,1)^2 with the analytical solution u(x,y) = theta. The speaker is struggling to determine the boundary conditions for this problem and asks for clarification. The reply suggests using polar coordinates to determine the boundary conditions, and the speaker realizes that the boundary condition must be u = theta on the boundary. The final comment suggests transforming the differential equation to cylindrical polar coordinates.
  • #1
ty1998
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TL;DR 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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  • #2
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
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
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.
 
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  • #5
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
Transform the differential equation (Laplace's equation) to cylindrical polar coordinates and see what it says.
 

1. What is a BVP?

A BVP stands for Boundary Value Problem. It is a type of mathematical problem where the solution is sought in a specific domain with known values at the boundaries.

2. What does it mean to solve a BVP with analytical solution?

Solving a BVP with analytical solution means finding the exact mathematical expression for the solution, rather than using numerical methods to approximate it.

3. Why is it important to solve BVPs with analytical solution in (0,1)^2?

The domain (0,1)^2 is a common region in many physical and engineering problems. By solving BVPs with analytical solution in this domain, we can obtain precise and general solutions that can be applied to a wide range of problems.

4. What are the steps involved in solving a BVP with analytical solution in (0,1)^2?

The steps may vary depending on the specific problem, but generally, they involve setting up the differential equation, applying boundary conditions, solving the equation using techniques such as separation of variables or Green's function, and then verifying the solution.

5. What are the advantages of using analytical solutions for BVPs in (0,1)^2?

Some advantages of using analytical solutions for BVPs in (0,1)^2 include the ability to obtain precise solutions, insight into the behavior of the system, and the potential for generalization to other problems. Additionally, analytical solutions can often be obtained more efficiently than numerical methods, which can be computationally intensive.

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