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Home Assignment 6 / Laplace Fourier Transform S5.3.P Q1
« 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:
$$\begin{equation} \hat{u}(\xi, y)=A(\xi)e^{-|\xi|y}+B(\xi)e^{|\xi|y}\end{equation}$$
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 $A, B$ ?
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:
$$\begin{equation} \hat{u}(\xi, y)=A(\xi)e^{-|\xi|y}+B(\xi)e^{|\xi|y}\end{equation}$$
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 $A, B$ ?