# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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where Rn is given by Eq. (20).

> 2. Let f(x) = x for 0 < x < 1 and let ôò (x) = V^sin mn x.

(a) Find the coefficients bm in the expansion of f (x) in terms of ô1 (x), ô2^), ....

(b) Calculate the mean square error Rn for several values of n and plot the results.

(c) Find the smallest value of n for which Rn < 0.01.

> 3. Follow the instructions for Problem 2 using f (x) = x(1 — x) for 0 < x < 1.

4. In this problem we show that pointwise convergence of a sequence Sn(x) does not imply

mean convergence, and conversely.

(a) Let Sn (x) = n^/xe—nx /2, 0 < x < 1. Show that Sn (x) — 0 as n — æ for each x in 0 < x < 1. Show also that

Ã1 n

Rn = [0 — Sn(x)]2 dx = - (1 — e—n),

J0 2

and hence that Rn — æ as n -æ. Thus pointwise convergence does not imply mean convergence.

(b) Let Sn(x) = x1 for 0 < x < 1 and let f (x) = 0 for 0 < x < 1. Show that

Rn = f\ f(x) — Sn(x)]2 dx = Ë+J.

and hence Sn (x) converges to f (x) in the mean. Also show that Sn (x) does not converge to f (x) pointwise throughout 0 < x < 1. Thus mean convergence does not imply pointwise convergence.

5. Suppose that the functions ôõ,...,ôï satisfy the orthonormality relation (1), and that a given function f is to be approximated by Sn(x) = c^x(x) + ••• + cnôï(x), where the coefficients ct are not necessarily those of Eq. (9). Show that the mean square error Rn given by Eq. (6) may be written in the form

Ã1 n n

Rn = j r(x) f2(x) dx —J2 aj + J2 c — ai)2,

J0 i=1 i=1

where the at are the Fourier coefficients given by Eq. (9). Show that Rn is minimized if Cj = a. for each i.

6. In this problem we show by examples that the (Riemann) integrability of f and f2 are independent.

1 L nn x

L J L f(x) cos L

1 L nn x

Lj L f(x) sin L

dx,

bn = - I f (x) sin^- dx,

an =

676

Chapter 11. Boundary Value Problems and Sturm-Liouville Theory

Show that f (x) dx exists as an improper integral, but f2(x) dx does not.

0

1, x rational

(b) Let f (x) = .

I — 1, x irrational.

Show that f2(x) dx exists, but f (x) dx does not.

7. Suppose that it is desired to construct a set of polynomials f0(x), fj(x), f2(x), fk (x), ..., where fk (x) is of degree k, that are orthonormal on the interval 0 < x < 1. That is, the set of polynomials must satisfy

1

0

(f,, fk)) =\ f ,(x) fk(x) dx = j.

(a) Find f0(x) by choosing the polynomial of degree zero such that (f0, f0) = 1.

(b) Find fx(x) by determining the polynomial of degree one such that (f0, fj) = 0

and ( f1 , f1 )1 = 1. 0 1

(c) Find f2(x).

(d) The normalization condition (fk, fk) = 1 is somewhat awkward to apply. Let g0(x), gj(x), g) (x),... be the sequence of polynomials that are orthogonal on 0 < x < 1 and that are normalized by the condition gk(1) = 1. Find g0(x), g1(x), and g2(x) and compare them with f0(x), fj(x),and f2(x).

8. Suppose that it is desired to construct a set of polynomials P0(x), Pj(x), Pk (x),. where p (x) is of degree k, that are orthogonal on the interval — 1 < x < 1; see Problem 7. Suppose further that Pk(x) is normalized by the condition p(1) = 1. Find P0(x), Pj (x), P2(x), and P3(x). Note that these are the first four Legendre polynomials (see Problem 24 of Section 5.3).

9. This problem develops some further results associated with mean convergence. Let Rn(aj,..., an), Sn(x), and at be defined by Eqs. (6), (2), and (9), respectively.

(a) Show that

r 1 n

a2

1n

Rn = r(x) f2(x) dx —T a2.

J0 =

0

Hint: Substitute for Sn (x) in Eq. (6) and integrate, using the orthogonality relation (1).

n

n

(b) Showthat^] a2 < I r (x) f2(x) dx. This result is known as Bessel’s inequality. /0

n1

n ai2 < i=1 J0

æ

(c) Show that ai2 converges.

³=1

f j æ

(d) Showthat lim R = I r(x) f2(x) dx — a2.

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