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

**Download**(direct link)

**:**

**363**> 364 365 366 367 368 369 .. 609 >> Next

bounded at all points in the disk, that is periodic in 9 with period 2n, and that satisfies the

boundary condition v(c, 9) = f (9), where f is a given function on 0 < 9 < 2n.

Hint: The equation for R is a Bessel equation. See Problem 3 of Section 11.4.

8. Consider the flow of heat in an infinitely long cylinder of radius 1:0 < r < 1,0 < 9 < 2n, -æ < z < æ. Let the surface of the cylinder be held at temperature zero, and let the initial temperature distribution be a function of the radial variable r only. Then the temperature u is a function of r and t only, and satisfies the heat conduction equation

a2[urr + (1jr)ur] = ut, 0 < r < 1, t > 0,

and the following initial and boundary conditions:

u(r, 0) = f(r), 0 < r < 1,

u(1, t) = 0, t > 0^

Show that

æ

u(r, t) = J2 cnJ0(knr)e-a2k2nt,

n=1

where J0(kn) = 0. Find a formula for cn.

9. In the spherical coordinates p, 9, ô (p > 0, 0 < 9 < 2n, 0 < ô < n) defined by the equations

x = p cos 9 sinô, y = p sin9 sinô, z = p cos ô,

Laplace’s equation is

p2upp + 2pup + (csc2 ô)ïââ + uôô + (cot ô)öô = 0¦

(a) Show that if u(p, 9,ô) = P(p)©(9)®^),thenP, ©, and Ô satisfy ordinary differential equations of the form

p2P" + 2pP' - ij?P = 0,

©" + k2© = 0,

(sin2 ô)Ô" + (sinô cos ô)Ô' + (p,2 sin2 ô - ^)Ô = 0^

11.6 Series of Orthogonal Functions: Mean Convergence

669

The first of these equations is of the Euler type, while the third is related to Legendre’s equation.

(b) Show that if u(p, 9, ô) is independent of 9, then the first equation in part (a) is unchanged, the second is omitted, and the third becomes

(sin2 ô)Ô" + (sinô cos ô)Ô' + (p2 sin2 ô)Ô = 0.

(c) Show that if a new independent variable is defined by s = cos ô, then the equation for Ô in part (b) becomes

2 ¸2Ô dÔ 2

(1 — s2)—^ — 2s-+ ð2Ô = 0, —1 < s < 1.

ds2 ds

Note that this is Legendre’s equation.

10. Find the steady-state temperature u(p, ô) in a sphere of unit radius if the temperature is independent of 9 and satisfies the boundary condition

u(1, ô) = f (ô), 0 < ô < ï.

Hint: Refer to Problem 9 and to Problems 22 through 29 of Section 5.3. Use the fact that the only solutions of Legendre’s equation that are finite at both ±1 are the Legendre polynomials.

11.6 Series of Orthogonal Functions: Mean Convergence

In Section 11.2 we stated that under certain restrictions a given function f can be expanded in a series of eigenfunctions of a Sturm-Liouville boundary value problem, the series converging to [ f (x+) + f (x —)]/2 at each point in the open interval. Under somewhat more restrictive conditions the series converges to f (x) at each point in the closed interval. This type of convergence is referred to as pointwise convergence. In this section we describe a different kind of convergence that is especially useful for series of orthogonal functions, such as eigenfunctions.

Suppose that we are given the set of functions ô1,ô2,... ,ôï, which are continuous on the interval 0 < x < 1 and satisfy the orthonormality condition

^ r(õ)ô1 (x^j(x) dx = j!’ i = j (1)

where r is a nonnegative weight function. Suppose also that we wish to approximate a given function f, defined on 0 < x < 1, by a linear combination of ôõ,... ,ôï. That

is, if 1 n

n

Sn (x) = Yh aiô³ (x) ’ (2)

i=1

we wish to choose the coefficients ax,..., an so that the function Sn will best approximate f on 0 < x < 1. The first problem that we must face in doing this is to state precisely what we mean by “best approximate f on 0 < x < 1.” There are several reasonable meanings that can be attached to this phrase.

670

Chapter 11. Boundary Value Problems and Sturm-Liouville Theory

1. We can choose n points xv ..., xn in the interval 0 < x < 1, and require that Sn (x) have the same value as f (x) at each of these points. The coefficients aj7..., an are found by solving the set of linear algebraic equations

**363**> 364 365 366 367 368 369 .. 609 >> Next