- 188.8.131.52.1: (a) Show that for G(x, t; x0, t0), G/t = G/t0. (b) Use part (a) to ...
- 184.108.40.206.2: Express (11.2.24) for a one-dimensional problem.
- 220.127.116.11.3: If G(x, t; x0, t0) = 0 for x on the boundary, explain why the corre...
- 18.104.22.168.4: For the one-dimensional wave equation, sketch G(x, t; x0, t0) as a ...
- 22.214.171.124.5: (a) For the one-dimensional wave equation, for what values of x0 (x...
- 126.96.36.199.6: (a) For the one-dimensional wave equation, for what values of x0 (x...
- 188.8.131.52.7: Reconsider Exercise 11.2.6 if Q(x, t) = g(x)eit. *(a) Solve for u(x...
- 184.108.40.206.8: (a) In three-dimensional infinite space, solve 2u t2 = c 22u + g(x)...
- 220.127.116.11.9: Consider the Greens function G(x, t; x0, t0) for the wave equation....
- 18.104.22.168.10: Consider 2u t2 = c2 2u x2 + Q(x, t), x> 0 u(x, 0) = f(x) u t (x, 0)...
- 22.214.171.124.11: Reconsider Exercise 11.2.10: (a) if Q(x, t) = 0, but f(x) = 0, g(x)...
- 126.96.36.199.12: Consider the Greens function G(x, t; x1, t1) for the two-dimensiona...
- 188.8.131.52.13: Consider the three-dimensional wave equation. Determine the respons...
- 184.108.40.206.14: Consider the three-dimensional wave equation. Determine the respons...
- 220.127.116.11.15: Derive the one-dimensional Greens function for the wave equation by...
Solutions for Chapter 11.2: Greens Functions for Wave and Heat Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Remove row i and column j; multiply the determinant by (-I)i + j •
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Invert A by row operations on [A I] to reach [I A-I].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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!
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).