 12.1: Verify directly from the denition that f(x)(x)dx is a distribution ...
 12.2: Let f be any distribution. Verify that the functional f dened by (f...
 12.3: Verify that the derivative is a linear operator on the vector space...
 12.4: Denoting p(x)= x+, show that p = H and p = .
 12.5: Verify, directly from the denition of a distribution, that the disc...
 12.6: UseChapter5directlytoprove(19)forallC1 functions (x)thatvanish near.
 12.7: Let a sequence of L2 functions fn(x) converge to a function f(x) in...
 12.8: (a) Show that the product (x)(y)(z) makes sense as a threedimension...
 12.9: Show that no sense can be made of the square [(x)]2 as a distribution.
 12.10: Verify that Example 11 is a distribution.
 12.11: Verify that Example 12 is a distribution.
 12.12: Let a(x)=1/2a for a < x < a, and a(x)=0 for x > a. Show that a we...
 12.1: Give an interpretation of G(x,x0) as a stationary wave or as the st...
 12.2: An innite string, at rest for t < 0, receives an instantaneous tran...
 12.3: Asemiinnitestring(0 < x < ),atrestfort < 0andheldatu=0at theend,re...
 12.4: Let S(x, t) be the source function (Riemann function) for the onedi...
 12.5: Aforceactingonlyattheoriginleadstothewaveequationutt =c2u+ (x)f(t) ...
 12.6: Find the formula for the general solution of the inhomogeneous wave...
 12.7: Let R(x,t)= S(x x0,t t0) for t > t0 and let R(x,t)0 for t < t0. Let...
 12.8: (a) Prove that (a2 r2)= (ar)/2a for a > 0 and r > 0. (b) Deduce tha...
 12.9: Derive the formula (12) for the Riemann function of the wave equati...
 12.10: Consider an applied force f(t) that acts only on the z axis and is ...
 12.11: For any a =b, derive the identity [(a)(b)]= 1 ab[(a)+(b)].
 12.12: A rectangular plate {0 x a, 0 y b} initially has a hot spot at its ...
 12.13: Calculate the distribution (logr) in two dimensions.
 12.1: VerifyeachentryinthetableofFouriertransforms.(Use(15)asneeded.)
 12.2: Verify each entry in the table of properties of Fourier transforms.
 12.3: Show that 1 22cr 0sinkctsinkr dk =1 82cr [eik(ctr) eik(ct+r)]dk=1 4...
 12.4: Prove the following properties of the convolution. (a) f g = g f. (...
 12.5: a) Show that f = f for any distribution f, where is the delta funct...
 12.6: Let f(x) be a continuous function dened for < x < such that its Fou...
 12.7: (a) Letf(x)beacontinuousfunctionontheline(,)thatvanishes for large...
 12.8: Let a(x) bethefunctioninExercise12.1.12.ComputeitsFouriertransform ...
 12.9: Use Fourier transforms to solve the ODE uxx+a2u = , where = (x) is ...
 12.1: Use the Fourier transform directly to solve the heat equation with ...
 12.2: UsetheFouriertransforminthexvariabletondtheharmonicfunctionin theha...
 12.3: Use the Fourier transform to nd the bounded solution of the equatio...
 12.4: If p(x) is a polynomial and f(x) is any continuous function on the ...
 12.5: In the threedimensional halfspace{(x, y,z):z > 0}, solve the Lapl...
 12.6: Use the Fourier transform to solve uxx+uyy =0 in the innite strip {...
 12.1: Verify the entries (2)(9) in the table of Laplace transforms.
 12.2: Verify each entry in the table of properties of the Laplace transform.
 12.3: Find f(t) if its Laplace transform is F(s)=1/[s(s2 +1)].
 12.4: Show that the Laplace transform of tk is (k+1)/sk+1 for any k > 1, ...
 12.5: Use the Laplace transform to solve utt =c2uxx for 0 < x < l,u(0,t)=...
 12.6: Use the Laplace transform to solve utt =c2uxx+cost sinx for 0 < x <...
 12.7: UsetheLaplacetransformtosolveut =kuxx in(0,l),withux(0,t)=0, ux(l,t...
Solutions for Chapter 12: DISTRIBUTIONS AND TRANSFORMS
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 12: DISTRIBUTIONS AND TRANSFORMS
Get Full SolutionsSince 47 problems in chapter 12: DISTRIBUTIONS AND TRANSFORMS have been answered, more than 5847 students have viewed full stepbystep solutions from this chapter. Chapter 12: DISTRIBUTIONS AND TRANSFORMS includes 47 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Partial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Column space C (A) =
space of all combinations of the columns of A.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.