 9.5.1: Solve the boundary value problems in 1 through 12.
 9.5.2: Solve the boundary value problems in 1 through 12.
 9.5.3: Solve the boundary value problems in 1 through 12.
 9.5.4: Solve the boundary value problems in 1 through 12.
 9.5.5: Solve the boundary value problems in 1 through 12.
 9.5.6: Solve the boundary value problems in 1 through 12.
 9.5.7: Solve the boundary value problems in 1 through 12.
 9.5.8: Solve the boundary value problems in 1 through 12.
 9.5.9: Solve the boundary value problems in 1 through 12.
 9.5.10: Solve the boundary value problems in 1 through 12.
 9.5.11: Solve the boundary value problems in 1 through 12.
 9.5.12: Solve the boundary value problems in 1 through 12.
 9.5.13: Suppose that a rod 40 cm long with insulated lateral surface is hea...
 9.5.14: A copper rod 50 cm long with insulated lateral surface has initial ...
 9.5.15: The two faces of the slab 0 5 x 5 L are kept at temperature zero, a...
 9.5.16: Two iron slabs are each 25 cm thick. Initially one is at temperatur...
 9.5.17: Two iron slabs are each 25 cm thick. Initially one is at temperatur...
 9.5.18: Suppose that a laterally insulated rod with length L D 50 and therm...
 9.5.19: Suppose that heat is generated within a laterally insulated rod at ...
 9.5.20: Suppose that current flowing through a laterally insulated rod gene...
 9.5.21: The answer to part (a) of is uss.x/ D Cx.L x/=2k. If f .x/ 0 in 20,...
 9.5.22: Consider the temperature u.x; t / in a bare slender wire with u.0; ...
 9.5.23: Consider the temperature u.x; t / in a bare slender wire with u.0; ...
 9.5.24: Suppose that a laterally insulated rod with length L, thermal diffu...
Solutions for Chapter 9.5: Heat Conduction and Separation of Variables
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 9.5: Heat Conduction and Separation of Variables
Get Full SolutionsDifferential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.5: Heat Conduction and Separation of Variables includes 24 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Since 24 problems in chapter 9.5: Heat Conduction and Separation of Variables have been answered, more than 15156 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

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.

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.