Find the temperature distribution $T(x,y)$ in the upper half-plane, given that the temperature along the $x$-axis is at:
$$T(x,0)=T_0, \quad x<-1$$
$$T(x,0)=T_1, \quad x>1$$
And $$\frac{\partial T}{\partial y}(x,0)=0, \quad |x|<1.$$
Assume the temperature follows $$\nabla ^2 T=0.$$
I have only dealt with Dirichlet and Neumann boundary conditions previously. I was unable to find any helpful materials on the internet. I would be grateful for any reference or hints on how to generate a solution. I know that for the Dirichlet conditions one took the Fourier transform of the equation, and thus produced a solution, and in the case of Neumann conditions one expressed the function in terms of the derivative which also satisfied Laplace's.
The only idea I can come up with is to solve for the function $T$ on the segment $|x|<1, y=0$, since we're given the derivative. So there, $T(x,y)$ is really just a function of $x$, and preferably continuous, so on the left its value will be approaching $T_0$ and on the right, $T_1$. But I don't see how to find the rest of the distribution on this segment. I bet one has to use the fact that it satisfies Laplace's equation in the region, but I'm unsure how to proceed. I can't just write that $T(x,y)=f(x)$ on the segment, and $f''(x)=0$, then solve that with $f(-1)=T_0$, $f(1)=T_1$, right?
Best Answer
First subtracting from $T(x,t)$ a constant $(T_1+T_0)/2$ the problem can be reduced to the case of odd initial data $T_0=-T_1$. The solution will be odd with respect to $x$ too.
Namely, putting $\tilde T(x,t)=T(x,t)-(T_0+T_1)/2\ $ we have $\tilde T(x,0)=\tilde T_1=(T_1-T_0)/2\,$ for $x>1$, $\tilde T(x,0)=-\tilde T_1\,$ for $x<-1$ and $\frac{\partial \tilde T(x,0)}{\partial y}=0\,$ for $x\in(-1,1)\,$.
Function $\frac{\partial^2 T(x,0)}{\partial x\partial y}=0$ on $(-\infty,-1)\cup(-1,1)\cup(1,\infty)$. Hence $\psi(x)=\frac{\partial T(x,0)}{\partial y}$ is piecewise constant and odd: $$ \psi(x)=\begin{cases} -a, & x<-1, \\ 0, & -1<x<1, \\ a, & x>1. \end{cases} $$ So the problem in question reduces to the Neumann problem: $\Delta u=0$, $y>0$, and $\frac{\partial T}{\partial y}\Big|_{y=0}=\psi$. The parameter $a$ to be defined from the condition $T\big|_{y=0}=T_1$ then $x>1$.