Boundary and initial value problems
General · Mathematics
Study notes
Q: For u_t = u_xx on (0,π), state Dirichlet IBVP with u(x,0) = x. PDE: u_t = u_xx, 0<x<π, t>0. BCs (Dirichlet): u(0,t) = 0, u(π,t) = 0. IC: u(x,0) = x. Solve: u = ΣBₙe^(-n²t)sin(nx), Bₙ = (2/π)∫₀^πx sin(nx)dx = 2(-1)^(n+1)/n. (Well-posed: exists, unique, stable!)