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MA 242 : Partial Differential EquationsAugust-December 2015

Department of Mathematics, Indian Institute of ScienceExercises 6

30 October, 2015

1. Consider the one-dimensional wave equation in R [0,)

utt uxx= 0, with initial data u(x, 0) =g(x), ut(x, 0) =h(x).

Let g, h have compact support. Show that the solution u(x, t) has com-pact support for each fixed t. Show that the functions F, G in the de-composition

u(x, t) =F(x+t) +G(x t)for u can be of compact support only when

h(y)dy= 0.

2. Consider the one-dimensional wave equation in R [0,)

utt uxx= 0, with initial data u(x, 0) =g(x), ut(x, 0) =h(x).

Let g, h have compact support. The kinetic energy is k(t) := 12

u2t (x, t)dx

the potential energy is p(t) := 12

u2x(x, t)dx. Prove thatk(t) +p(t) is

constant in t.

3. Solve the initial-boundary-value problem

utt =uxx for 0< x < , t > 0,

u= 0 forx= 0, ; t >0,

u= 1, ut= 0 for 0 < x < , t= 0.

4. Solveutt uxx=x

2 fort >0, x R

u(x, 0) =x, ut(x, 0) = 0.

5. Consider the initial value problem

utt u= 0, with initial data u(x, 0) = g(x), ut(x, 0) =h(x)

for the wave equation in n = 5. Let Mu(x,r,t) denote spherical mean ofu(x, t) on the sphere B (x; r). Set

N(x,r,t) =r2

rMu(x,r,t) + 3Mu(x,r,t).

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(a) Show that N(x,r,t) is a solution of

2

t2N=

2

r2N

and find N from its initial data in terms ofMg and Mh.

(b) Show that

u(x, t) = limr0

N(x,r,t)

3r = (

1

3t2

t+t)Mh(x, t)+

t(

1

3t2

t+t)Mg(x, t).

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