Solve Laplace equation on rectangle domain

In summary, the conversation discusses a problem with a Laplace equation on a square domain with mixed boundary conditions. The individual attempts at solving the problem are described, including using variable separation and expanding solutions as a sum of simpler solutions. The conversation ultimately suggests using a combination of sine and hyperbolic sine functions to satisfy the given boundary conditions.
  • #1
Dor
12
0

Homework Statement


I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't 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.
 
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  • #3
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
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
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
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].
 

1. What is a Laplace equation?

The Laplace equation is a mathematical equation that describes the relationship between the second order derivatives of a function and its values at different points. It is commonly used in physics and engineering to model the distribution of potential or heat within a domain.

2. What is a rectangle domain?

A rectangle domain is a two-dimensional region that is defined by four sides, where each side is parallel to the x or y-axis. It is commonly used in mathematics and physics to represent a bounded area of study.

3. How do you solve a Laplace equation on a rectangle domain?

To solve a Laplace equation on a rectangle domain, you can use different methods such as separation of variables, Fourier series, or numerical methods. The specific method chosen will depend on the specific characteristics of the problem and the desired level of accuracy.

4. What are the boundary conditions for a rectangle domain?

The boundary conditions for a rectangle domain are the values of the function at the boundary of the domain. These values can be specified as either Dirichlet boundary conditions (prescribed values) or Neumann boundary conditions (prescribed derivatives). It is important to have well-defined boundary conditions in order to obtain a unique solution to the Laplace equation on a rectangle domain.

5. What are some real-world applications of solving a Laplace equation on a rectangle domain?

The Laplace equation on a rectangle domain has many applications in physics and engineering, such as modeling the flow of heat or electricity within a rectangular conductor, analyzing the distribution of potential in a capacitor, and predicting the behavior of fluids in a rectangular container. It is also used in image processing, where it can be used to smooth out noisy images.

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