MATH400-101 Homework Assignment 4 (Due Date: October 28, 2016, by 5:30pm) 1. (20pts) Solve utt = c2 uxx + t for 0 < x < +, u(0, t) = et , u(x, 0) = 0, ut (x, 0) = 1. 2. (10pts) Find the general solutions to utt = c2 uxx , x > 0, t > 0 ux (0, t) = h(t), t > 0 u(x, 0) = 0, ut (x, 0) = 0, x > 0 3.(15pts) Find the solution to utt + utx 2uxx = 0, x > 0, t > 0 u(0, t) = sin t u(x, 0) = 0, ut (x, 0) = 0, x > 0 4. (15pts) Consider the following wave equation: utt = uxx , 0 < x < 1 u(x, 0) = x, ut (x, 0) = 1, 0 < x < 1 u(0, t) = u(1, t) = 0 Find u(1, 5 2 ). 5. (10pts) Find the ordinary dierential equation satised by f , if the function x 1 u(x, t) = f ( ) t t satises ut = uxx . 6.(20pts) Solve ut = kuxx + x, < x < u(x, 0) = e|x| , < x < + 7. (10pts) Consider the following diusion equation ut = kuxx + f (x, t), 0 < x < +, t > 0 u(x, 0) = (x), 0 < x < + ux (0, t) + au(0, t) = g(t), t > 0 Suppose a < 0. Use the energy method to show that the solution to the above problem is unique. 1