401, Math. Sec. 500 Homework 13, due April 29
Spring, 2005
1. [Schaum's, p. 64, Ex. 3.22] Let r be any three-dimensional vector, and {i, j, k} the standard unit vectors along the coordinate axes. Prove the following finite-dimensional analogues of the theorems about orthonormal bases of functions: (a) (Parseval's equation) (b) (Bessel's inequality) (r i)2 + (r j)2 + (r k)2 = r r. (r i)2 + (r j)2 r r.
(c) (least squares approximation) p (r i)i + (r j)j is, of all vectors in the xy plane, the closest to r; furthermore, p is perpendicular to r - p. 2. [Schaum's, p. 64, Ex. 3.23] Suppose that one term in any orthonormal basis (i.e., a complete sequence of orthonormal functions) is omitted. (For example, leave sin(10x) out of the basis functions for the Fourier series on (-, ).) Call the resulting (amputated) sequence {n (x)}. (a) Can we expand an arbitrary function f (x) as a series (c) Is Bessel's inequality always satisfied? Justify your answers. 3. [Logan, p. 195, Ex. 3.1(b)] Find the eigenvalues and eigenfunctions for the problem y + y = 0, y(0) = 0, 4. [Schaum's, p. 64, Ex. 3.43] 0 < x < 1,
n cn n (x) ?
(b) Is Parseval's identity always satisfied? Is it ever satisfied?
y(1) - y (1) = 0.
Show that the solution of the boundary value problem (0 < x < L, 0 < t), u(0, x) = f (x),
2u u = 2 t x u (t, 0) = hu(t, 0), x where h and L are constants, is
u (t, L) = -hu(t, L), x
2
u(t, x) =
n=1
e-n t
n cos(n x) + h sin(n x) cn (n2 + h2 )L + 2h
where cn = and the are n solutions of
0
L
f (x)(n cos(n x) + h sin(n x)) dx tan(L) = 2h . 2 - h2
5. Let r and be polar coordinates, defined in the usual way: x = r cos , Show that
2
y = r sin .
1 2u 2u 2u 2 u 1 u + 2 2. + 2 = 2 + x2 y r r r r Hint: First show that (acting on any function) u sin = cos - , x r r cos = sin + . y r r Hint for hint: The calculations are easier if you use some kind of implicit differentiation. For example, a quick way to show that cos = y r
is to differentiate the equation
y x with respect to y (with x fixed) and perform some necessary algebra on the result. Remember that (for example) a y derivative with x fixed is not the same as a y derivative with r fixed! tan = In three dimensions the wave equation is 2u - c2 2 u = 0 t2 is the Laplacian. For waves with spherical symmetry, u = u(r, t) and
2
6. [Logan, p. 304, Ex. 3.13]
where
2
2 u 2 u . + r 2 r r By introducing the variable U = ru, show that the general solution for the spherically symmetric wave equation is 1 1 u = f (r - ct) + g(r + ct). r r u= 7. [Logan, p. 159, Ex. 1.3] Determine in which regions each equation is hyperbolic, elliptic, or parabolic. Subscripts indicate partial differentiation with respect to the indicated variables. [Read ahead in the notes if necessary (the section "Classification . . . "). Everybody should be prepared to answer questions like this on the final exam.] (a) (b) (c) (d) t utt + uxx = 0 utt - uxx = 0 utt + (1 + x2 )ux - ut = et uxx + uyy = f (x, y)