Toronto Math Forum
APM3462019 => APM346Lectures & Home Assignments => Home Assignment 6 => Topic started by: Sebastian Lech on March 12, 2019, 01:00:45 PM

I was wondering if anyone could let me know how to move forward on problem one:
Consider Dirichlet problem:
$$\begin{equation} u_{xx}+u_{yy}=0\end{equation}, \infty<x<\infty, y>0$$
$$\begin{equation} u_{y=0}=f(x)\end{equation}$$
We need to make a Fourier Transform by x and leave the solution in the form of a Fourier Integral.
What I did first was make the Fourier transform:
$$\begin{equation} \hat{u}_{yy}\xi^2\hat{u}=0 \end{equation}$$
$$\begin{equation} \hat{u}_{y=0}=\hat{f}(\xi) \end{equation}$$
Which has general solution:
$$\begin{equation} \hat{u}(\xi, y)=A(\xi)e^{\xiy}+B(\xi)e^{\xiy}\end{equation}$$
and using equation (4):
$$\begin{equation}\hat{u}(\xi,0)=A(\xi)+B(\xi)=\hat{f}(\xi) \end{equation}$$
Which I am now stuck on, how do we solve for $A, B$ ?

There was an explanation why one of the solutions in halfplane should be rejected (does not satisfy condition at infinity)

So for this question why we don’t use cos and sin for the general solution?

What condition is given at infinity? I don't see any condition given in the question for the behaviour of $u$ at infinity.

I think we need to make assumption that the Fourier transformation u will be 0 as y goes to infinity,that’s what my TA did in tutorial

I think we need to make assumption that the Fourier transformation u will be 0 as y goes to infinity,that’s what my TA did in tutorial
Or, at least, does not grow exponentially as $k\to \infty$

Here is my solution for solving A.