I need to solve Laplace equation in the domain D= 0 < x,y < pi
Neumann boundary conditions are given:
2. The attempt at a solution
first, we check that the integral of directional derivative of u on the edge of D is zero. This should be the necessary condition for the existence of a solution to the problem. the condition is satisfied (the integral is zero indeed), but why is this condition sufficient and not only necessary?
Anyway, assuming a solution does exist, I propose a solution of the form u=X(x)Y(t), and solving the appropriate SL system I find that the solution sould be of the form:
but the solution should have the form
where does the By come from? and does Cosh[n(y-pi)] equal Sinhny? It doesn't make sense because for example if we have y=pi , Cosh[n(y-pi)] = 1, while Sinh(ny) gives an infinite number of answers, depending on n.
Thanks in advance...:)