So I have the two following exercises :
1) Solve the following differential equation : $$\frac{\partial^2}{\partial x^2}u(x,y)+\frac{\partial^2}{\partial y^2}u(x,y)=0$$
where $x\in \mathbb{R}$, $y > 0$
knowing that $u(x,0)=h(x)$ where $h: \mathbb{R}\rightarrow\mathbb{C}$ is an absolute, integrable, continuous function.
Hint: use the Fourier transform and the convolution theorem. Keep in mind for the solution of the system that $u(\cdot,y)$ should be bounded (german :beschränkt).
2) Let $\alpha>0$. Solve
$$\frac{\partial}{\partial t}u(t,x)=\alpha^2\frac{\partial^2}{\partial x^2}u(t,x)$$
$t>0$, $x\in \mathbb{R}$
knowing that $u(x,0)=h(x)$ where $h: \mathbb{R}\rightarrow\mathbb{C}$ is an absolute, integrable, continuous function.
My attempt : For the first one, I tried to transform both sides. So I wrote $$(-2\pi if_1)^2U(f_1,f_2)+(-2\pi if_2)^2U(f_1,f_2)=0$$
($f$ is the frequency)
Now, I know about the convolution theorem : $$F(f\cdot g) = F(f)*F(g)$$ (where $*$ denotes the convolution) and I have an initial condition, but I don't really see how to proceed. I also looked on the internet, but I didn't really find a lot of help.
For the second one, I suppose the method to proceed is similar.
Thanks for your help !