 5.3.5.3.1: Do Exercise 4.4.2(b). Show that the partial differential equation m...
 5.3.5.3.2: Consider 2u t2 = T0 2u x2 + u + u t . (a) Give a brief physical int...
 5.3.5.3.3: Consider the nonSturmLiouville differential equation d2 dx2 + (x) ...
 5.3.5.3.4: Consider heat flow with convection (see Exercise 1.5.2): u t = k 2u...
 5.3.5.3.5: For the SturmLiouville eigenvalue problem, d2 dx2 + = 0 with d dx(0...
 5.3.5.3.6: Redo Exercise 5.3.5 for the SturmLiouville eigenvalue problem d2 dx...
 5.3.5.3.7: Which of statements 15 of the theorems of this section are valid fo...
 5.3.5.3.8: Show that 0 for the eigenvalue problem d2 dx2 + ( x2) = 0 with d dx...
 5.3.5.3.9: Consider the eigenvalue problem x2 d2 dx2 + xd dx + = 0 with (1) = ...
 5.3.5.3.10: Reconsider Exercise 5.3.9 with the boundary conditions d dx(1) = 0 ...
Solutions for Chapter 5.3: SturmLiouville Eigenvalue Problems
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 5.3: SturmLiouville Eigenvalue Problems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.3: SturmLiouville Eigenvalue Problems includes 10 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. Since 10 problems in chapter 5.3: SturmLiouville Eigenvalue Problems have been answered, more than 8145 students have viewed full stepbystep solutions from this chapter. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
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 nl  An).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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.