 13.5.13.5.1: Consider 2u t2 = c2 2u x2 ,
 13.5.13.5.2: (a) Using the results of this section, invert (13.5.8) based on the...
 13.5.13.5.3: Solve for u(x, t) using Laplace transforms: 2u t2 = c 2 2u x2 u(0, ...
 13.5.13.5.4: Reconsider Exercise 13.5.3 if instead (a) u x (0, t) = 0 and u(L, t...
 13.5.13.5.5: Solve for u(x, t) using Laplace transforms: u t = k 2u x2 ,
 13.5.13.5.6: 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
ISBN: 9780321797056
Solutions for Chapter 13.5: Laplace Transform Solution of Partial Differential Equations
Get Full SolutionsSince 6 problems in chapter 13.5: Laplace Transform Solution of Partial Differential Equations have been answered, more than 7821 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. Chapter 13.5: Laplace Transform Solution of Partial Differential Equations includes 6 full stepbystep solutions. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5.

CayleyHamilton Theorem.
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. Blockdiagonal: 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.

Fundamental Theorem.
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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
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 = Q1 = 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 AI 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.