Spring 2014 MATH 8260 Homework ]1

Please write down all your work and solutions for the following problems. This homeworkis due on Tuesday, February 11, in class.

In the following problems, all given functions are assumed enough smoothness.

1. Find the solution of the initial value problem{ut + b u + cu = 0 (x, t) Rn (0,)u(x, 0) = g(x) x Rn

where b Rn is a constant vector and c is a constant.2. Find the general solution of the equation

tut xux + x2u = x2, x, t 6= 0.

3. Find the solution of the following 1-D wave equation on R+ = {x > 0} with Neumannboundary condition at x = 0

utt c2uxx = 0 (x, t) R+ (0,)u(x, 0) = g(x), ut(x, 0) = h(x) x R+ux(0, t) = 0 t (0,)

where g, h are given and satisfy f (0) = g(0) = 0 (compatibility condition).

4. Solve the following 1-D wave equation on R+utt uxx = 0 (x, t) R+ (0,)u(x, 0) = g(x), ut(x, 0) = 0 x R+ux(0, t) kut(0, t) = 0 t (0,)

where k > 0 is a constant, g(x) is given and satisfies g(0) = 0.

5. Solve the following nonhomogeneous 1-D wave equation{utt uxx = t sinx (x, t) R (0,)u(x, 0) = 0, ut(x, 0) = sin x x R.

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