 4.4.4.4.1: Consider vibrating strings of uniform density 0 and tension T0. *(a...
 4.4.4.4.2: In Section 4.2 it was shown that the displacement u of a nonuniform...
 4.4.4.4.3: Consider a slightly damped vibrating string that satisfies 0 2u t2 ...
 4.4.4.4.4: Redo Exercise 4.4.3(b) by the eigenfunction expansion method
 4.4.4.4.5: Redo Exercise 4.4.3(b) if 420T0/L2 < 2 < 1620T0/L2
 4.4.4.4.6: For (4.4.1)(4.4.3), from (4.4.11) show that u(x, t) = R(x ct) + S(x...
 4.4.4.4.7: If a vibrating string satisfying (4.4.1)(4.4.3) is initially at res...
 4.4.4.4.8: If a vibrating string satisfying (4.4.1)(4.4.3) is initially unpert...
 4.4.4.4.9: From (4.4.1), derive conservation of energy for a vibrating string,...
 4.4.4.4.10: What happens to the total energy E of a vibrating string (see Exerc...
 4.4.4.4.11: Show that the potential and kinetic energies (defined in Exercise 4...
 4.4.4.4.12: Using (4.4.15), prove that the solution of (4.4.1)(4.4.3) is unique.
 4.4.4.4.13: (a) Using (4.4.15), calculate the energy of one normal mode. (b) Sh...
Solutions for Chapter 4.4: Fourier Series
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 4.4: Fourier Series
Get Full SolutionsApplied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. Since 13 problems in chapter 4.4: Fourier Series have been answered, more than 8051 students have viewed full stepbystep solutions from this chapter. Chapter 4.4: Fourier Series includes 13 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. This expansive textbook survival guide covers the following chapters and their solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.