Solving Odd Elliptic Equation with Greens Functions

[hunt_mat]

I have the following elliptic equation that I must solve:
$$a\frac{\partial^{2}\phi}{\partial y^{2}}+\frac{\partial^{2}\phi}{\partial u^{2}}+b\frac{\partial\phi}{\partial u}=-ce^{-y^{2}}$$
Where a,b and c are constants. Along with the conditions:
$$\phi (y,0)=\frac{\partial\phi}{\partial u}(y,0)=0,\quad\lim_{y\rightarrow\infty}\phi (y,u)=\lim_{y\rightarrow\infty}\frac{\partial\phi}{\partial y}(y,u)=0$$
I have tried the usual approach using Laplace transforms but I got into a horrible mess with looking at the particular solution which involved horrible things like complimentary error functions and the like. As well as that mess, computing the inverse transforms were waay waay to hard as I had to look at $$\sqrt{\alpha s^{2}+\beta s}$$ where s was my Laplace transform variable and I had no idea what to do regarding them so I dropped that method.

My current idea is to use Greens functions but I don't know a great deal about them and was after some advice.

Regards

Mat

 

[fzero]

Using Green functions would be a good way to solve this. The homogeneous equation is separable and has exponential solutions. After imposing the boundary conditions, I believe you'll need to use Gram-Schmidt to construct an orthonormal basis for the solution space. The Green function can then be obtained as a series expansion in terms of these orthonormal basis. Finally the solutions to the inhomogeneous equation can be obtained in the usual manner by integrating the forcing function.

 

[hunt_mat]

I have tried looking at the problem from the point of view of Laplace transform and as you can see I haven't gotten very far at all. I have no idea how to go about inverting the Laplace transform. The Homogeneous solution has sin and cos solutions and I feel that they will inverse transform to zero which doesn't help me much.

It looks as if I will have to learn some Green's functions. Can you recommend anything?

Mat

 

[fzero]

Any EM or QM text that discusses scattering theory will contain a discussion of Green functions, as will any decent math methods book. I would look at Arfken or Jackson myself, but you may prefer something else.

 

[hunt_mat]

Did you have a look at the Laplace transform thing I did? Did it look promising?

Mat

 

[fzero]

Did you have a look at the Laplace transform thing I did? Did it look promising?

Mat

I hadn't looked too closely, but did now. It might be fine if you want to do the integrals numerically. Of course, there's no guarantee that the Green function method will give a result that can be expressed in a nice form either. But it seems like a good exercise to learn the method regardless.

 

[hunt_mat]

There is the boundary conditions to consider. We need to compute the constants. Can we do this numerically?

 

[fzero]

There is the boundary conditions to consider. We need to compute the constants. Can we do this numerically?

I've never studied numerical methods in much detail, but I'd expect you can turn it into a root-finding problem which is well-suited to numerical methods.