- 126.96.36.199.1: Show that for the Greens functions defined by (11.3.2) with (11.3.3...
- 188.8.131.52.2: Consider u t = k 2u x2 + Q(x, t), x> 0 u(0, t) = A(t) u(x, 0) = f(x...
- 184.108.40.206.3: Determine the Greens function for u t = k 2u x2 + Q(x, t), x> 0 u x...
- 220.127.116.11.4: Consider (11.3.34), the Greens function for (11.3.31). Show that th...
- 18.104.22.168.5: Consider u t = k 2u x2 + Q(x, t) u(x, 0) = f(x) u x(0, t) = A(t) u ...
- 22.214.171.124.6: Determine the Greens function for the heat equation subject to zero...
- 126.96.36.199.7: Derive the two-dimensional infinite space Greens function by taking...
- 188.8.131.52.8: Derive the three-dimensional infinite space Greens function by taki...
Solutions for Chapter 11.3: Greens Functions for Wave and Heat Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A sequence of steps intended to approach the desired solution.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.