11.5.2.31 Fourier method

Problem: Solve the Poisson equation \( u_{x x}+u_{y y}=1 \) in the ring \( r \leq \sqrt{x^{2}+y^{2}} \leq R \). The values of the function \( u(x, y) \) on the inner and outer boundaries of the ring are equal relatively to \( a, b \).