Why Does Solving Laplace's Equation in a Square Yield an Infinite Sum?

[cgmeytanperos]

Homework Statement

i need to solve the laplace equation in square with length side 1 i tried to solve by superposition and i got infinite sum enen thouth i know that the answer should be finite

Homework Equations

1.ψ(x=0,0≤y≤1)=0
2.ψ(y=0,0≤x≤1)=0
3.ψ(x=1,0≤y≤1)=10sin(∏*y)+3x
4.ψ(y=1,0≤x≤1)=9sin(2∏*y)+3x

The Attempt at a Solution

from 1 and to 2 have (by superposition):
ψ n=Asin(n∏x)sinh(n∏y) or ψ n=Asin(n∏y)sinh(n∏x)
and after multiply by sin(n∏x)
A n=2∫((9sin(2∏x)+3x)*sin(n∏x))/sinh(n∏) (the integral from 0 to 1)
the problem is that i heve the infinite sum from ∫3x*sin(n∏x)
thank you very much!

 

[BvU]

Hello cg, and welcome to PI.
Are you sure your relevant equations are relevant equations ? They look like boundary conditions to me.
In that case, filling in x=1 changes 10sin(∏*y)+3x to 10sin(∏*y)+3 and 9sin(2∏*y)+3x changes into 3x.
And then the relevant eqations still have to be found out ...

 

[cgmeytanperos]

hi BvU and thanks.
you were right this is boundry conditions and they are also incorrect (i am sorry)
3.ψ(x=1,0≤y≤1)=10sin(∏*y)+3y
4.ψ(y=1,0≤x≤1)=9sin(2∏*x)+3x

 

[BvU]

And then the relevant eqations still have to be found out ... What do you have available ?

 

[paranormal]

It seems like you have made a good attempt at solving the Laplace equation in a square with length side 1. However, as you have mentioned, the solution you obtained is an infinite sum, which is not the expected result. This could be due to a mistake in your calculations or in the application of the superposition principle.

One possible way to solve the Laplace equation in this case is by using separation of variables. This involves assuming that the solution can be written as a product of two functions, one depending only on x and the other only on y. By plugging this assumption into the Laplace equation and solving for each function separately, you can obtain a finite solution that satisfies the boundary conditions given in the problem.

Alternatively, you could also try using a different method such as the method of images or the method of Green's functions. These methods may provide a more direct solution to the problem without the need for an infinite sum.

In any case, it is important to carefully check your calculations and make sure that you are correctly applying the boundary conditions and the Laplace equation. If you are still having trouble, it may be helpful to consult with your instructor or a colleague for further guidance. Good luck!