 17.17.1: Formulate an initialboundary value problem modeling heat conductio...
 17.17.4: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.27: In each of 1 through 4, solve the problem u t = k 2 u x 2 for < x <...
 17.17.38: Solve u t = k 2 u x 2 for 0 < x < L,t > 0, u(x, 0) = 0, u(0,t) = 0,...
 17.17.42: Suppose R = 1, k = 1 and f (r) = r. Assume that U(1,t) = 0 for t > ...
 17.17.45: Write the solution for the general problem u t = k 2 u x 2 + 2 u y2...
 17.17.2: Formulate an initialboundary value problem modeling heat conductio...
 17.17.5: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.28: In each of 1 through 4, solve the problem u t = k 2 u x 2 for < x <...
 17.17.39: Solve u t = k 2 u x 2 for x > 0,t > 0 u(x, 0) = 0, u(0,t) = t 2 , l...
 17.17.43: Repeat the calculations of with k =16, R =3 and f (r) = er
 17.17.46: Solve this problem when k = 4, L = 2, K = 3, and f (x, y) = x 2 (L ...
 17.17.3: Formulate an initialboundary value problem for the temperature dis...
 17.17.6: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.29: In each of 1 through 4, solve the problem u t = k 2 u x 2 for < x <...
 17.17.40: u t = k 2 u x 2 for x > 0,t > 0, u(x, 0)= ex , u(0,t) = 0, lim x u(...
 17.17.44: Repeat the calculations of with k = 1/2, R = 3 and f (r) = 9 r 2 .
 17.17.47: Solve this problem when k = 1, L = , K = , and f (x, y) = sin(x) co...
 17.17.7: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.30: In each of 1 through 4, solve the problem u t = k 2 u x 2 for < x <...
 17.17.41: Apply the Laplace transform with respect to t to the problem u t = ...
 17.17.8: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.31: In each of 5 through 8, solve the problem u t = k 2 u x 2 for x > 0...
 17.17.9: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.32: In each of 5 through 8, solve the problem u t = k 2 u x 2 for x > 0...
 17.17.10: In each of 1 through 7, write a solution of the initialboundary va...
 17.17.33: In each of 5 through 8, solve the problem u t = k 2 u x 2 for x > 0...
 17.17.11: A thin, homogeneous bar of length L has insulated ends and initial ...
 17.17.34: In each of 5 through 8, solve the problem u t = k 2 u x 2 for x > 0...
 17.17.12: A thin homogeneous bar of length L has initial temperature f (x) = ...
 17.17.35: In each of 9 and 10, use a Fourier transform on the halfline to so...
 17.17.13: A thin, homogeneous bar having thermal diffusivity of 9 and a lengt...
 17.17.36: In each of 9 and 10, use a Fourier transform on the halfline to so...
 17.17.14: Show that the partial differential equation u t = k 2 u x 2 + A u x...
 17.17.37: . Derive equation (17.13). Hint: This can be done using complex fun...
 17.17.15: Use the idea of to solve u t = 2 u x 2 + 4 u x + 2u for 0 < x < ,t ...
 17.17.16: Solve u t = 2 u x 2 + 6 u x for 0 < x < 4,t > 0, u(0,t) = u(4,t) = ...
 17.17.17: Solve u t = 2 u x 2 6 u x for 0 < x < ,t > 0, u(0,t) = u(,t) = 0 fo...
 17.17.18: Solve u t = 16 2u x 2 for 0 < x < 1,t > 0, u(0,t) = 2, u(1,t) = 5 f...
 17.17.19: Solve u t = k 2 u x 2 for 0 < x < L,t > 0, u(0,t) = T, u(L,t) = 0 f...
 17.17.20: Solve u t = 4 2 u x 2 Au for 0 < x < 9,t > 0, u(0,t) = u(9,t) = 0 f...
 17.17.21: Solve u t = 9 2 u x 2 for 0 < x < L,t > 0, u(0,t) = T, u(L,t) = 0 f...
 17.17.22: In each of 19 through 23, solve the problem u t = k 2 u x 2 + F(x,t...
 17.17.23: In each of 19 through 23, solve the problem u t = k 2 u x 2 + F(x,t...
 17.17.24: In each of 19 through 23, solve the problem u t = k 2 u x 2 + F(x,t...
 17.17.25: In each of 19 through 23, solve the problem u t = k 2 u x 2 + F(x,t...
 17.17.26: In each of 19 through 23, solve the problem u t = k 2 u x 2 + F(x,t...
Solutions for Chapter 17: The Heat Equation
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 17: The Heat Equation
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Since 47 problems in chapter 17: The Heat Equation have been answered, more than 7773 students have viewed full stepbystep solutions from this chapter. Chapter 17: The Heat Equation includes 47 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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).

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).