Partial differential equations
Engineering Mathematics · Engineering
Study notes
Solve heat equation du/dt = d2u/dx2 on 0<x<pi, u(0,t)=u(pi,t)=0, u(x,0)=sin x. Separate u=X(x)T(t): T'/T = X''/X = -L. BCs give Xn=sin(nx), Ln=n2. Un=e^(-n2t). So u = sum Bn sin(nx)e^(-n2t). IC: sin x = sum Bn sin(nx) gives B1=1. Answer: u(x,t) = e^(-t) sin x.