Answer Happy

7 (Heat equation) Consider the heat equation apa Kaxa at' (8) where K is a positive constant, 2 € 0,a] and is real-valued. For t > 0, we demand Dirichlet boundary conditions v(t,0) = o at x = 0 (where to is a constant) and von Neumann boundary conditions (t, a) = 0 at r=a. The initial condition is (0,x) = 0. (a) As a preparation, we introduce the functions gx : (0,a] → R defined by 9x(x) = VŽ sin(quz), 4 = (x+) (9) where k = 0,1,.... Show that these functions satisfy the boundary conditions 9 (0) = 0 and g(a) = 0, that they form an ortho-normal system relative to the standard scalar product on L([0,a]) and that they are eigenfunctions of with dc29k = -29 (10)