# Toronto Math Forum

## APM346--2019 => APM346--Lectures & Home Assignments => Home Assignment 6 => Topic started by: Sebastian Lech on March 12, 2019, 01:00:45 PM

Title: Laplace Fourier Transform S5.3.P Q1
Post 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:
$$$$u_{xx}+u_{yy}=0$$, -\infty<x<\infty, y>0$$
$$$$u|_{y=0}=f(x)$$$$
We need to make a Fourier Transform by x and leave the solution in the form of a Fourier Integral.
What I did first was make the Fourier transform:
$$$$\hat{u}_{yy}-\xi^2\hat{u}=0$$$$
$$$$\hat{u}|_{y=0}=\hat{f}(\xi)$$$$

Which has general solution:

$$$$\hat{u}(\xi, y)=A(\xi)e^{-|\xi|y}+B(\xi)e^{|\xi|y}$$$$
and using equation (4):
$$$$\hat{u}(\xi,0)=A(\xi)+B(\xi)=\hat{f}(\xi)$$$$
Which I am now stuck on, how do we solve for $A, B$ ?

Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: Victor Ivrii on March 12, 2019, 06:25:23 PM
There was an explanation why one of the solutions in half-plane should be rejected (does not satisfy condition at infinity)
Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: Zengyue Lin on March 16, 2019, 10:22:00 PM
So for this question why we don’t use cos and sin for the general solution?
Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: MikeMorris on March 17, 2019, 12:27:26 PM
What condition is given at infinity? I don't see any condition given in the question for the behaviour of $u$ at infinity.
Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: Zengyue Lin on March 17, 2019, 01:59:15 PM
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
Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: Victor Ivrii on March 19, 2019, 02:14:40 PM
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 $|k|\to \infty$
Title: Re: Laplace Fourier Transform S5.3.P Q1
Post by: shuxian on March 31, 2019, 05:19:13 PM
Here is my solution for solving A.