Homework #2. Due February 13 |

1. Riemann-Lebesgue Theorem (see Royden problem 16 p. 94)

Prove that, if f is an integrable function on (- , + ), then

lim

2-3. In the next two exercises you will to prove the Lax equivalence theorem, one on the main tools in Numerical analysis.

Let V be a Banach space, V

du(t)/dt = L u(t), 0 t T, (1)

u(0)= u

Definition: A function u: [0,T] V is a solution of the initial problem (1), (2) if for any t [0, T], u(t) V

lim

Definition: The initial value problem (1),(2) is well-posed if for any u

sup

Note: in general well-posedness means sup

where f

2. If u(t) is the solution of (1), (2), denote u(t) = S(t)u

a. Show that there exists a unique extension S(t) on the whole space V.

Definition: For u

b. Show that the generalized solutions u(t), that arise from this extension are continuous in t.

c. Using well-posedness show that S(t) is a semi-group, that is

S(t+s) = S(t)S(s)

Definition: A difference method is a one-parameter family of operators

C( t): V V,

and there exists t

Definition: The approximate solution is defined by

u

Definition: (Consistency) The difference method is consistent if there exists a dense subspace V

lim

Definition: (Convergence) The difference method is convergent if for any fixed t [0, T], and any u

lim

where {m} is a sequence of integers and { t} is a sequence of step sizes such that m t t.

Definition: (Stability) The difference method is not stable if the operators

{C( t)

are uniformly bounded.

3. Lax equivalence theorem

Suppose that if the initial value problem (1),(2) is well-posed. Prove that for a consistent difference method, stability is equivalent to convergence.

Hints: i. For the direction consider the error of the method and study this error for u

ii. The other direction can be proved by contradiction. Suppose the method is stable, but there is a sequence {m} and { t} such that m t < T but

lim

Using convergence you will be able to apply the principle of uniform boundedness here.

4. Assume that the heat equation:

u

u(0,t)= u(, t) = 0

u(x,0)=u

is a well-posed initial value problem.

Let N

A forward difference scheme is

(v(x

v(0,t

v(x

a. Choose

V= C

and

V

Show that V a Banach space. Show that V

b. For the forward scheme on the subspace V

C( t, r)= C(h

Show that the forward scheme is consistent if |h

c. Show that r 1/2 is a necessary and sufficient condition for both stability and convergence.

d. (Optional) Show that, indeed, the heat equation is well-posed. Use here the maximum principle.

5. The geometric series theorem.

a. Let V be a Banach space and L: V V with ||L|| < 1. Prove that I-L is a bijection on V, its inverse is a bounded linear operator and ||(1-L)

b. Consider the linear integral equation of the second kind:

u(x) -

Show that if

max

then there exists a unique solution of the linear integral equation of the second kind.

6. Different Banach spaces with the same norm. (see Royden problems 14-15 p. 126 and problem 23 p. 135)

a. Let R

(a

Define the norm

||a ||

b. Show that c, a subspace of l

c. Find a representation for the bounded linear functionals on c and on c