Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Homework 5

Math 442: Intro to Partial Differential Equations

(Exercises are taken from Partial Differential Equations An Introduction, Second Edition by Walter A. Strauss.)

1. Solve ut = uxxx with Dirichlet boundary condition on [−π, π] with u(x, 0) = sin(x) + cos(x)

2. Exercise §2.5 #4. Here is a direct relationship between the wave and diffusion equations. Let u(x,t) solve the wave equation on the whole line with bounded second derivatives. Let

(a) Show that v(x,t) solves the diffusion equation!

(b) Show that limt−→0v(x, t) = u(x, 0).

(Hint: (a) Write the formula as v(x, t) = R∞−∞ H(s, t)u(x, s)ds, where H(x,t) solves the diffusion equation with constant k/c2 for t > 0. Then differentiate v(x,t) using Section A.3. (b) Use the fact that H(s,t) is essentially the source function of the diffusion equation with the spatial variable s.)

3. Exercise §4.2 #1. Solve the diffusion problem ut = kuxx in 0 < x < l, with the mixed boundary conditions u(0, t) = ux(l, t) = 0 (Neumann at the left, Dirichlet at the right).