Math 2280 Autumn 2016 Homework 9 Numbered problems due Wednesday 30th November class 1. Show that X 00 = X X 0 () = X 0 () X() = X() has no non-zero solution when < 0. 2. Solve the problem 4u = 0 u(x, H) = Lx x2 u(x, 0) = 0 u(0, y) = 0 u(L, y) = 0 on the rectangle 0 < x < L, 0 < y < H using the general solution from class. 3. (c.f. 9.7.11) Solve the problem 4u = 0 u(x, 0) = 0 u(x, H) = f (x) u(0, y) = 0 u (L, y) = 0 x on the rectangle 0 < x < L, 0 < y < H. (Hint: Make use of Problem 4 from Homework 8.) 4. Solve the problem 4u = 0 u(, a) = 0 u(, b) = f () on the annulus 0 < a2 < x2 + y 2 < b2 . (Hint: Use polar coordinates.) 5. A number is an eigenvalue of the Laplacian if there is a non-zero function called an eigenfunction such that 4 = . Verify that (x, y) = sin (x) sin (y) is an eigenfunction of the Laplacian on R2 . What is its eigenvalue? 6. The eigenfunctions of the Laplacian on the rectangle can be used to solve the heat problem u = k4u t u(x, 0, t) = 0 u(x, H, t) = 0 u(0, y, t) = 0 u(L, y, t) = 0 on the rectangle 0 < x < L, 0 < y < H. a) Verify that if is an eigenfunction of the Laplacian with eigenvalue then u(x, y, t) = ekt (x, y) is a solution of the heat equation u = k4u t in two dimensions. (1) b) For which values of and does the function (x, y) = sin (x) sin (y) satisfy all four of the boundary conditions above? c) Combine your answers to part b) with the method described in a) to form a general solution to the heat equation on the rectangle 0 < x < L, 0 < y < H. 7. Devise (but do not carry out) a strategy to solve the problem 4u = 0 u(x, 0) = L x u(0, y) = L y u(x, L x) = 0 on the triangle with vertices (0, 0), (L, 0), (0, L). A. Using your answer to Problem 5 solve (1) on the rectangle 0 < x < L, 0 < y < H with initial condition u(x, y, 0) = x(x L)y(y H). B. Does the function u(x, y) = 1 (x2 + (sin y)2 )300 satisfy the Laplace equation on R2 ? C. (An alternative to separation of variables) Consider the problem 2u u =k 2 t x u(0, t) = 0 u(L, t) = 0 on [0, L]. a) Fix a solution u(x, t) of the above problem and consider 2 V (n, t) = L ZL n u(x, t) sin x L \u0012 \u0013 dx 0 which is the nth Fourier sine coefficient of u(x, t). Write the nth Fourier sine coefficient u of (x, t) in terms of V (n, t). t 2u b) Write the nth Fourier sine coefficient of (x, t) in terms of V (n, t). (Hint: Use intex2 gration by parts.) c) What differential equation does V (n, t) satisfy? (Use the fact that u(x, t) satisfies the heat equation.) d) Solve the V (n, t) equation. e) What is u(x, t)