Math Partial Differential Equations

Be sure that this examination has 1 page.

The University of Oklahoma

Midterm Examinations - April 3-4, 2020

Introduction to PDEs

Dr. M. Zhu

Open book examination Time: Due at 12:00pm, April 5, 2020. minutes

Marks

[20] 1. Let f(x) = sin x in (0,π). (1). Find its sine series, and the first two terms of its cosine series. (2). Solve the following diffusion problem with insulated boundary:

ut −uxx = 0, x ∈ (0,π), t > 0,

ux(0, t) = ux(π,t) = 0, t > 0,

u(x, 0) = sin x, x ∈ (0,π).

You can just write down first two terms if the solution is given via a Fourier series.

[10] 2. Solve utt = uxx, x > 0, t > 0

ux(0, t) = 0, t > 0,

u(x, 0) = cos x, ut(x, 0) = x 2, x > 0;

And find u(π, 1) =?.

[10] 3. Solve ut −uxx = e−t, x ∈ R, t > 0

u(x, 0) = e−x, x ∈ R.

[10] 4. Prove: for any constant k, u = 0 is the only solution to

uxx −k2u3 = 0, 0 < x < π

u(0) = u(π) = 0.

[50] Total Marks

The End