 4.4.1: (a) Use the Fourier expansion to explain why the note produced by a...
 4.4.2: Consider a metal rod (0 < x < l), insulated along its sides but not...
 4.4.3: A quantummechanical particle on the line with an innite potential ...
 4.4.4: Consider waves in a resistant medium that satisfy the problem utt =...
 4.4.5: Do the same for 2c/l < r < 4c/l.
 4.4.6: Separate the variables for the equation tut =uxx +2u with the bound...
 4.4.2.1: Solve the diffusion problem ut =kuxx in0 < x < l, with the mixed bo...
 4.4.2.2: Consider the equation utt =c2uxx for0 < x < l, with the boundary co...
 4.4.2.3: Solve the Schrodinger equation ut =ikuxx for real k in the interval...
 4.4.2.4: Considerdiffusioninsideanenclosedcirculartube.Letitslength(circumfe...
 4.4.3.1: Find the eigenvalues graphically for the boundary conditions X(0)=0...
 4.4.3.2: Consider the eigenvalue problem with Robin BCs at both ends: X = X ...
 4.4.3.3: Derive the eigenvalue equation (16) for the negative eigenvalues = ...
 4.4.3.4: Consider the Robin eigenvalue problem. If a0 < 0, al < 0 and a0 al ...
 4.4.3.5: InExercise4(substantialabsorptionatbothends)showgraphicallythat the...
 4.4.3.6: If a0 =al =a in the Robin problem, show that: (a) There are no nega...
 4.4.3.7: Ifa0 =al =a,showthatasa +,theeigenvaluestendtotheeigenvalues of the...
 4.4.3.8: Consider again Robin BCs at both ends for arbitrary a0 and al. (a) ...
 4.4.3.9: Ontheinterval0 x 1oflengthone,considertheeigenvalueproblem X = X X(...
 4.4.3.10: Solve the wave equation with Robin boundary conditions under the as...
 4.4.3.11: (a) Provethatthe(total)energyisconservedforthewaveequationwith Diri...
 4.4.3.12: Consider the unusual eigenvalue problem vxx = v for 0 < x < l vx(0)...
 4.4.3.13: Considerastringthatisxedattheendx=0andisfreeattheendx=l except that...
 4.4.3.14: Solve the eigenvalue problem x2u+3xu+u =0for1 < x < e, with u(1)=u(...
 4.4.3.15: Find the equation for the eigenvalues of the problem ((x)X)+(x)X =0...
 4.4.3.16: Find the positive eigenvalues and the corresponding eigenfunctions ...
 4.4.4.130: Solvethefourthordereigenvalueproblem X = X in0 < x < l,with the fo...
 4.4.4.131: A tuning fork may be regarded as a pair of vibrating exible bars wi...
 4.4.4.132: Show that in Case 1 (radiation at both ends)lim nn n22 l2 = 2 l(a0 ...
Solutions for Chapter 4: BOUNDARY PROBLEMS
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 4: BOUNDARY PROBLEMS
Get Full SolutionsPartial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567. Since 29 problems in chapter 4: BOUNDARY PROBLEMS have been answered, more than 5740 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. Chapter 4: BOUNDARY PROBLEMS includes 29 full stepbystep solutions.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.