- 10.2.1: ut D kuxx (00); ux.0; t / D hu.L; t / C ux.L; t / D 0, u.x; 0/ D f ...
- 10.2.2: Find formal series solutions of the boundary value problems in 1 th...
- 10.2.3: Find formal series solutions of the boundary value problems in 1 th...
- 10.2.4: Find formal series solutions of the boundary value problems in 1 th...
- 10.2.5: Find formal series solutions of the boundary value problems in 1 th...
- 10.2.6: Find formal series solutions of the boundary value problems in 1 th...
- 10.2.7: Let u.x; y/ denote the bounded steady-state temperature in an infin...
- 10.2.8: If the bar in Example 2 has no mass attached to the end x D L, then...
- 10.2.9: (a) Show that D 0 is not an eigenvalue of the problem in (19). (b) ...
- 10.2.10: Calculate the speed (in miles per hour) of longitudinal sound waves...
- 10.2.11: Calculate the speed (in miles per hour) of longitudinal sound waves...
- 10.2.12: Suppose that the free end of the bar of Example 2 is attached to a ...
- 10.2.13: If a bar has natural length L, cross-sectional area A, and Youngs m...
- 10.2.14: Show that the eigenfunctions fXn.x/g 1 1 of the problem in (19) are...
- 10.2.15: Show that the eigenfunctions fsin nx=Lg 1 1 of the problem in (19) ...
- 10.2.16: According to of Section 9.7, the temperature u.r; t / in a uniform ...
- 10.2.17: A problem concerning the diffusion of gas through a membrane leads ...
- 10.2.18: Suppose that the simply supported uniform bar of Example 3 has, ins...
- 10.2.19: To approximate the effect of an initial momentum impulse P applied ...
- 10.2.20: To approximate the effect of an initial momentum impulse P applied ...
Solutions for Chapter 10.2: Applications of Eigenfunction Series
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of 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.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Outer product uv T
= column times row = rank one matrix.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.