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 o - + f(x) cos(x/) dx =0.

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, Vo V a dense subspace of V. Let L: Vo V be a linear operator. The operator L is usually unbounded and can be thought of as a differential operator. Consider the initial value problem
du(t)/dt = L u(t), 0 t T, (1)
u(0)= uo. (2)
Definition: A function u: [0,T] V is a solution of the initial problem (1), (2) if for any t [0, T], u(t) Vo,
lim t o ||[u(t + t) - u(t)] / t- Lu(t)|| =0, and u(0)= uo.
Definition: The initial value problem (1),(2) is well-posed if for any uo Vo, there is a unique solution u=u(t) and the solution depends continuously on the initial value: there exists a constant Co > 0 such that if u1(t), u2(t) are the solutions for the initial values u1o, u2o Vo; then
sup0 t T || u1(t)- u2(t)||V Co || u1o - u2o ||V
Note: in general well-posedness means sup0 t T || u1(t)- u2(t)||V Co (|| u1o - u2o ||V + ||f1- f2||W )
where f1 and f2 are "forcing" terms for u1(t) and u2(t) respectively, and for them a different Banach space W is defined.

2. If u(t) is the solution of (1), (2), denote u(t) = S(t)uo.
a. Show that there exists a unique extension S(t) on the whole space V.
Definition: For uo V, uo Vo we call S(t)u o the generalized solution of the initial value problem (1), (2).
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 to > 0 such that for all 0 < t to C( t) are uniformly bounded: || C( t)|| c.
Definition: The approximate solution is defined by
u t (m t) = C( t)m uo, m=1,2,...
Definition: (Consistency) The difference method is consistent if there exists a dense subspace Vc of V such that for all uo Vc, for the corresponding solution u of the initial value problem (1),(2) we have
lim t o ||u(t + t) - C( t)u(t)|| / t =0, uniformly in [0, T].
Definition: (Convergence) The difference method is convergent if for any fixed t [0, T], and any uo V, we have
lim t o ||(C( t)m - S(t)) uo|| =0
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)m} | 0 < t to, m t 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 uo Vc first. Continuity is to be used in your estimates.
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 t o ||C( t)m || =
Using convergence you will be able to apply the principle of uniform boundedness here.

4. Assume that the heat equation:
ut = uxx + f(x,t) in [0, ]* [0, T],
u(0,t)= u(, t) = 0
is a well-posed initial value problem.
Let Nx, Nt be positive integers, hx = / Nx , ht = T/Nt, xj = j hx, tm = m ht , r= ht/hx2.
A forward difference scheme is
(v(xj,tm+1) - v(xj,tm)/ht= (v(xj+1,tm) - 2 v(xj,tm) +v(xj-1,tm)/hx2 + f(xj,tm),
v(0,tm)= v(,tm ) =0,
v(xj, 0)= uo(xj).
a. Choose
V= Co[0, ] = {v C[0, ], v(0)=v()=0}
Vo ={v| v(x) = j=1 n aj sin(j x)}
Show that V a Banach space. Show that Vo is a dense subspace of V.
b. For the forward scheme on the subspace Vo define a corresponding operator
C( t, r)= C(ht,hx), ht= t, hx = g(r) t.
Show that the forward scheme is consistent if |hx/ht| c as t 0.
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)-1 || 1/(1- ||L||).
b. Consider the linear integral equation of the second kind:
u(x) - a b k(x,y) u(y) dy = f(x), a x b.
Show that if
maxx a b |k(x,y)| dy < ||
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 be the space of infinite sequences
Define the norm
||a || = supi|ai|. Show that l, the space of of bounded sequences of real numbers with the || || -norm, is a Banach space.
b. Show that c, a subspace of l, the space of all convergent sequences of real numbers and co, a subspace of l, the space of all sequences which converge to zero are Banach spaces (with the || || -norm).
c. Find a representation for the bounded linear functionals on c and on co. Note: these representations are different!