Toronto Math Forum
APM3462022S => APM346Lectures & Home Assignments => Chapter 5 => Topic started by: Zicheng Ding on March 13, 2022, 02:32:37 PM

For this problem we have the following equations: $u_{xx} + u_{yy} = 0 \space \space (\infty < x < \infty , y > 0)$ and $u_{y=0} = f(x)$.
After doing the Fourier Transformation, the equations become $\hat{u}_{yy}  k^2\hat{u} = 0$ and $\hat{u}(k,0) = \hat{f}(k)$.
In my understanding, we should get a general form as $\hat{u}(k,y) = A(k)e^{ky} + B(k)e^{ky}$ and drop the first term if $k>0$ since as $y \rightarrow \infty$ the term $e^{ky} \rightarrow \infty$, and if $y<0$ we drop the second one.
On the answer provided by Prof. Kennedy it said $\hat{u}(k,y) = \hat{f}(k)e^{ky}$, so I am a little confused since $\hat{u}(k,y) = \hat{f}(k)e^{ky}$ makes more sense to me.

You are right, it was a misprint