Scribd
Upload
39 views1 page

Ass 4

This document contains 7 problems related to partial differential equations for an assignment. It involves solving partial differential equations describing wave motion, including the wave equation. Several problems involve solving Cauchy problems with initial conditions specified. One problem evaluates the solution to a PDE at specific points and another proves properties of the solution if the initial functions are odd or periodic.

Uploaded by

Sketchily Sebente
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
39 views1 page

Ass 4

This document contains 7 problems related to partial differential equations for an assignment. It involves solving partial differential equations describing wave motion, including the wave equation. Several problems involve solving Cauchy problems with initial conditions specified. One problem evaluates the solution to a PDE at specific points and another proves properties of the solution if the initial functions are odd or periodic.

Uploaded by

Sketchily Sebente
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 1

The University of Zambia

Department of Mathematics and Statistics


MAT 5122 - Partial Differential Equations
Assignment four
2020

1. Solve the problem

utt − c2 uxx = f (x, t), 0 < x < ∞, u(x, 0) = x, ut (x, 0) = 0, u(0, t) = t2 .

2. Solve the Neumann problem for the wave equation on the half-line 0 < x < ∞.
3. Solve the problem

utt − uxx = 0, 0 < x < ∞, u(0, t) = t2 , t > 0

u(x, 0) = x2 , ut (x, 0) = 6x, x > 0,


and evaluate u(4, 1) and u(1, 4).
4. Solve the problem

utt − uxx = xt, −∞ < x < ∞, t > 0, u(x, 0) = 0, ut (x, 0) = ex

5. Solve the problem


utt = c2 uxx + cos x, −∞ < x < ∞, t > 0
u(x, 0) = sin x, ut (x, 0) = 1 + x.

6. Consider the cauchy problem

utt − 4uxx = −4ex , −∞ < x < ∞, t > 0



x, 0 < x < 1
 (
1 − x2 , |x| < 1

1, 1 < x < 2
u(x, 0) = f (x) = ut (x, 0) = g(x) =


 3 − x, 2 < x < 3 0, |x| > 1.

0, x > 3, x < 0.
(a) Is the d’Alembert solution of the problem classical? If not find all the points where
the solution is singular.
(b) Evaluate the solution at (1, 1).
7. Prove that if f, g, F are odd functions or periodic functions with period p then the
solution u(x, t) of the Cauchy problem

utt − c2 uxx = F (x, t), −∞ < x < ∞, t > 0, u(x, 0) = f (x), ut (x, 0) = g(x)

is also odd or periodic with period p, respectively.

You might also like