- 58

- 0

**1. Homework Statement**

I need to solve Laplace equation in the domain D= 0 < x,y < pi

Neumann boundary conditions are given:

du/dx(0,y)=du/dx(pi,y)=0

du/dy(x,pi)=x^2-pi^2/3+1

du/dy(x,0)=1

**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:

u(x,y)= A+sigma{Cos(nx)*[Cosh(ny)+Sinh(ny)]}

but the solution should have the form

u(x,y)= A+By+sigma{Cos(nx)*[Coshny+Cosh[n(y-pi)]}

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...:)