MATH 4340, Section 1, Homework Assignment 9 Heat Equation Worksheet Due: 9:55 am, Wednesday, April 6, 2016 Point value: 20 The purpose of this activity/homework assignment is to help you 1. understand the physics behind heat (diusion) equations; 2. compute solutions to the heat equation. 1. (8 pts.) If heat is lost from the right-hand end of the rod to the surrounding environment, then the right boundary condition becomes u x = h (u(L, t) um ) , x=L where h > 0 and um are constants. This condition states that the outward ux of heat from the rod is proportional to the dierence between the temperature at the end of the rod, u(L, t), and the constant temperature um of the surrounding environment. Write and solve an initial boundary value problem that describes the ow of heat through a rod of length 4 whose left endpoint is insulated and whose right endpoint is in contact with a medium held at a xed temperature of 0. The initial temperature distribution in the rod is u(x, 0) = x(4 x). 2. (6 pts.) Consider the heat equation with the non-zero (that is, non-homogeneous), xed temperature boundary conditions. 2u u = 3 2 , 0 < x < , t > 0 t x u(0, t) = 30, u(, t) = 50, t > 0 x u(x, 0) = 30 cos (x) + 80 sin , 0 < x < . 2 (a) Describe the boundary and initial conditions for this problem. What do you expect the long-term behavior of this system to be? 20 (b) Make a change of variable v(x, t) = u(x, t) 30 x, and rewrite the system above using v(x, t). (c) Use separation of variables to nd the solution to (b) (which is in terms of v(x, t)). (d) Recover u(x, t) by using the relationship in (b). 1 3. (6 pts.) Consider the initial boundary value problem u 2u = 2 , 0 < x < 2, t x ux (0, t) = 0, u(2, t) = 0, u(x, 0) = x, 0 < x 1 2 x, 1 < x < 2 t>0 (1) t>0 (2) 0