1. ## PDE in circle

i have a PDE in polar coordinates with 0 < theta < pi/2 and a < r < b
use laplace's equation for the initial conditions u(r,0) = 0, u(r, pi/2) = 0, and u(r,0) = 0 and the boundary condition u(b,theta) = theta

does anyone know how to solve this PDE? i know how to do the fourier series at the end, but how do i get u(r,theta)?

I do not think this is a correctly stated Dirichlet problem. I am assuming you want to solve $u_{xx}+u_{yy} = 0$ on the region depicted on the left. For each point $(x,y)$ in that region there exists a smooth bijection with $(r,\theta)$ so that $x=r\cos\theta \text{ and }y=r\sin \theta$. So if we define $U(r,\theta) = u(x,y)$ then we need to solve $U_{rr}+\tfrac{1}{r}U_r + \tfrac{1}{r^2}U_{\theta \theta}= 0$ on the region on the right. However, the Dirichlet problem only tells us the boundary value and not the initial values. The boundary values that you providied do not completely define the boundary and so I do not think your problem is fairly stated to be solved.