Solve PDE Uxx+Uyy=-2 with Boundary Conditions

chwala
Gold Member
Messages
2,825
Reaction score
413
Homework Statement
solve the inhomogenous pde below by considering the boundary conditions given
Relevant Equations
separation of variables.
1614169978889.png
ial

this is the question.
 
Physics news on Phys.org
its very long since i solved such questions during my post graduate class, i am trying to see if i can remember...
i attempted to solve the homogenous part, i just want to know if i am on the right track...
1614170283272.png

am i thinking right or i am missing something...from literature i can see that this does not qualify to be solved by the approach i have used of trying to solve homogenous pde...breaking them into ode. Since we have ##-2## on the rhs then it becomes poisson equation?
 
Last edited:
As with any linear problem the answer is "take something which satisfies the RHS and add something from the kernel".

So here: Find a function of (x,y) which satisfies \nabla^2 f = -2 and add solutions of Laplace's equation to satisfy the boundary condition. This is easier if your particular solution satisfies as much of the boundary condition as possible. Here I think you have less work if your function vanishes on x = 0 and x = 1.

Here you have u(x,0) = u(x,1) = u(1,-x) = -\sinh \pi \sin(\pi x).

Don't forget that you can use \cosh (kx) and \sinh(kx) in place of e^{\pm kx} and that \sinh(k(1 - x)) is a linear combination of these.
 
I should be able to look at this during december...i have had too much work on my desk... and at same time need to familiarise with literature on inhomegenous pde's...
 
* Am on this now...allow me to deal with it slowly...am now conversant with pde's. This is Laplace equation...steady-state equation (Equilibrium solution).

We shall solve this by separation of variables, we shall seek solutions of the form:

##U(x,y) = X(x) Y(y)##

For the Homogenous part we shall have;

##X^{''}(x) Y(y) + X(x) Y^{''} (y) = 0##

it follows that,

##\dfrac{X^{''}(x)}{X(x)} = -\dfrac{Y^{''}(y)}{Y(y)} =±μ^2##

case 1; using eigenvalue ##[λ=μ^2]##

##X^{''} -μ^2X=0##
##Y^{''}+μ^2Y=0##

##X^{''} -μ^2X=0##

it follows that

##m^2-μ^2=0## is our characteristic equation, solving we get;

##m^2=μ^2##

##⇒m=\sqrt{μ^2}=μ## our solution would therefore be of the form;

##X=e^{μx}=A \cosh μx +B \sinh μx##

##X=A \cosh μx +B \sinh μx##

...will continue later.
 
Last edited:
Solution
<br /> u(x,y) = x(1-x) - \sum_{n=1}^\infty Y_n(y) a_n \sin (n \pi x) + \sinh(\pi x)\sin(\pi y) <br /> - ( \sinh (\pi y) + \sinh (\pi(1 - y)) )\sin (\pi x) where <br /> Y_n(y) = \frac{\sinh(n\pi y) + \sinh(n\pi(1-y))}{\sinh (n \pi)}<br /> is chosen such that Y_n(0) = Y_n(1) = 1 and <br /> \sum_{n=1}^\infty a_n \sin (n \pi x) = x(1-x).
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top