Solve the Laplace equation ∂²u/∂ x² + ∂ 2u/∂ y² = 0 in the region 0 x 1, 0 y 1 subject to the boundary conditions u(0, y) = 0, u(x, 0) = 0, u(1, y) = 1, u(x, 1) = 1 by separation methods.
The answer given :
u = x + 2π Σ[infinity]ₙ₌₁ sin ηπχ/n sinh ηπ
× {sinh nπy + (-1)ⁿ sinh nπ(1 – y)}