I have been asked to solve the following problem:
$$\left\{ \begin{array}{c} \Delta u =0 \;\;\; (x,y)\in(0,1)\times(0,1) \\ u(0,y)=1 \;,\; u(1,y)=0 \;,\; u(x,0)=0 \;,\; u(x,1)=0 \;,\; x,y\in(0,1) \end{array} \right. $$
They ask me to compute the value $u(1/2,1/2)$
When solving the equation, I get a series from I don't know to compute the convergence on the value that they ask me to solve the equation.
I have seen on other questions on the forum that this value can be obtained without solving the equation (using the rotational invariance of Laplace equation), by as the mean of the values of the function in the boundaries. So that in this case this would be $\frac{0+0+0+1}{4}=1/4$.
However I am not able to reach this point. I would be grateful if anybody could help
Thank you very much