Let f (ac) = if 0 bk(t) sin((2k + 1)x), where bk(t) = k=0

C if 0 < C T/ 2 Let f(c) C T if T/ 2 C < T Then the solution to the heat equation = ut with Dirichlet boundary conditions u(0, t) = 0, (0 < t < 00) u(T, t) = 0, (0 < t < 00) u(c, 0) = f(c), (0 < c < T) is u(c, t) bk(t) sin((2k + l)c), where bk(t)