- 220.127.116.11.1: Consider 2u t2 = c2 2u x2 ,
- 18.104.22.168.2: (a) Using the results of this section, invert (13.5.8) based on the...
- 22.214.171.124.3: Solve for u(x, t) using Laplace transforms: 2u t2 = c 2 2u x2 u(0, ...
- 126.96.36.199.4: Reconsider Exercise 13.5.3 if instead (a) u x (0, t) = 0 and u(L, t...
- 188.8.131.52.5: Solve for u(x, t) using Laplace transforms: u t = k 2u x2 ,
- 184.108.40.206.6: Solve for u(x, t) using Laplace transforms: u t = k 2u x2 subject t...
Solutions for Chapter 13.5: Laplace Transform Solution of Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Solutions for Chapter 13.5: Laplace Transform Solution of Partial Differential EquationsGet Full Solutions
peA) = det(A - AI) has peA) = zero matrix.
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Solvable system Ax = b.
The right side b is in the column space of A.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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·
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.