Heat equation and separation of variables
General · Mathematics
Study notes
Q: Solve u_t = u_xx, u(0,t) = u(π,t) = 0, u(x,0) = sin x. Separate: X''+λX = 0, T'+λT = 0. BCs: λₙ = n², Xₙ = sin(nx). Tₙ = e^(-n²t). u = ΣBₙe^(-n²t)sin(nx). Initial sin x: B₁ = 1, rest 0! u = e^(-t)sin x! (Heat decays: high modes faster!)