Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given the PDE, $\Delta u=x$ in the region $x^2+y^2<1$. And $\frac{\partial u}{\partial r}=y$ on $x^2+y^2=1$. I am supposed to find all solutions.

The only machinery I know for finding solutions on bounded domains is separation of variables. But I only know how to do separation of variables for homogeneous problems. So I found the solution to $\Delta u=0$ is $u(x,y)=y$. But it is unclear how to extend this to inhomogeneous case. (It amounts to solving $\Delta u=x$ with $\frac{\partial u}{\partial r}=0$)

Maybe there some transform that turns $\Delta u=x$ into Laplace's equation? Thank-you.

share|improve this question
    
Procedure is the same – find general solution of homogeneous problem, particular solution of inhomogeneous problem, and then add them up to get general solution of inhomogeneous problem. –  Kaster Mar 5 '13 at 2:47

2 Answers 2

One approach goes as follows. First, you make the boundary condition homogeneous. This can be done by introducing $v=u-g$, where $g$ is some function that satisfies $\partial_r g = y$ at the boundary of the disk. The function $v$ must then satisfy $\Delta v = x-\Delta g$. Now you use separation of variables. The idea is to expand the right hand side $f(x,y)=x-\Delta g$ in terms of the eigenfunctions of the Neumann Laplacian on the disk. So let $\phi_n$ be those eigenfunctions, and let $$ f = \sum_n b_n\phi_n. $$ We look for the solution in the form $$ v = \sum_n a_n\phi_n. $$ Then the equation is $$ \Delta v = \sum_n a_n\Delta \phi_n = \sum_n a_n\lambda_n\phi_n=\sum_n b_n\phi_n, $$ and so $$ a_n=b_n/\lambda_n, $$ that is $$ v = \sum_n \frac{b_n}{\lambda_n} \phi_n. $$ Note that we need $b_0=0$ because $\lambda_0=0$ for the Neumann Laplacian, and by a related reason the solution $v$ is unique up to a constant.

share|improve this answer
    
I think we can use a green's function approach. The Neumann green representation is $$u(x) = -\int_U G(x,y)\Delta u(y) + \int_{\partial U} G(x,y)\frac{\partial u}{\partial \nu}$$ But I am still trying to think of the neumann green's function on disk. For your solution, how can I calculate the eigenfunctions of the neumann laplacian? When I try seperation of variables on $\Delta u =\lambda u$, I get a 2nd order ODE with non-constant coefficients, so I don't know how to solve. Thank-you –  StuartHa Mar 6 '13 at 0:44
    
@Stuart: The eigenfunctions are epxressed in terms of the Bessel functions. –  timur Mar 6 '13 at 2:14
    
Thank-you. But I am looking for a more explicit solution. I should have said this, sorry. –  StuartHa Mar 13 '13 at 20:06
up vote 1 down vote accepted

I'll start by decomposing problem as follows, \begin{cases} \Delta u=x\\ \partial u/\partial r=0 \end{cases}

\begin{cases} \Delta v=0\\ \partial v/\partial r=y \end{cases}

The second problem is easy. You can do it with seperation of variables or just observe that the solution is $v=y$. For the first problem we can start by converting to polar coordinates,

$$u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^{2}}u_{\theta\theta}=r\cos\theta$$

Now it is natural to guess the solution is like $r^{3}\cos\theta$. Plugging this in yields that $\frac{1}{8}r^{3}\cos\theta$ works, and then you need to fix the BC so we have $u=\frac{1}{8}r^{3}\cos\theta-\frac{3}{8}r\cos\theta$. Adding $u$ and $v$ yields the solution,

$$\frac{x}{8}\left(x^{2}+y^{2}-3\right)+y + C$$

Now since I added the constant $C$, we have all solutions. This is because solution to Poisson with Neumann data is unique up to a constant. (Easy energy method to show this)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.