5, MATH 525 FALL 2015 This homework is due in class Nov 18th. Please write up your solutions on your own and give complete arguments. I. Let u(x, t) be the solution to the heat equation with diusion constant k and Dirichlet boundary conditions for (x, t) (0, L)(0, ). Let QT = {(x, t) : 0 x L, 0 t T }. (a) Because u is a solution to the heat equation, we have (ut kuxx )dxdt. QT Integrate the derivatives and use the fact that u(x, t) 0, as t to show that L ux (0, t)dt k f (x)dx = k 0 0 ux (L, t)dt. 0 (b) What physical interpretation does this equation have in the context of heat ow? II. Consider (1) ut kuxx = u in (0, L) (0, ), where u(0, t) = u(L, t) = 0 for all t > 0 and f (x) = u(x, 0) on (0, L). (a) Use separation of variables to nd the series solution u(x, t) of (1). [Note: you shouldn't have to go thru all the cases. Use the fact that we know the complete answer to the Dirichlet problem (i.e. when = 0).] (b) When > 0, the right side of the (1) represents a heat source which is proportional to the temperature. How large must be so that the solution grows with t instead of decaying to zero? III. Prove the Lagrange Identity: vLu uLv = {p(x)(u v v u)} for u, v C 2 (J) where J = [a, b] and Lu = (p(x)u ) + q(x)u. IV. Use the Lagrange Identity to show that b (vLu uLv)dx = 0 a 1 2 HOMEWORK 5, MATH 525 FALL 2015 if Ri u = Ri v = 0, for i = 1, 2 and where R1 and R2 are dened as follows: R1 y = 1 y(a) + 2 p(a)y (a) and R2 y = 1 y(b) + 2 p(b)y (b) 2 2 2 2 where p(x) > 0 on J, 1 + 2 > 0 and 1 + 2 > 0. V. Consider the following BVP: (y + y) = f (x) on (0, 1) y(0) = y(1) = 0 a) Verify that y1 (x) = sin(x) and y2 (x) = sin(1 x) are independent solutions of y + y = 0. [Hint: Show W (y1 , y2 ) = 0] b) Verify that y1 satises the left boundary condition y(0) = 0 and y2 satises the right boundary condition y(1) = 0. c) The Green's function is then given by G(x, s) = y1 (s)y2 (x)/W (y1 , y2 ) y1 (x)y2 (s)/W (y1 , y2 ) 0 s x, x s 1. Verify that 1 y(x) = G(x, s)f (s)ds 0 is a solution to the BVP. Check the ODE and the boundary conditions