- 18.104.22.168.66: Use the Fourier integral transforms of this section to solve the gi...
- 22.214.171.124.67: Use the Fourier integral transforms of this section to solve the gi...
- 126.96.36.199.68: Find the temperature u(x, t) in a semi-infinite rod if u(0, t) u0, ...
- 188.8.131.52.69: Use the result , to show that the solution of can be written as
- 184.108.40.206.70: Find the temperature u(x, t) in a semi-infinite rod if u(0, t) 0, t...
- 220.127.116.11.71: Solve if the condition at the left boundary is , where A is a const...
- 18.104.22.168.72: Solve if the end x 0 is insulated.
- 22.214.171.124.73: ind the temperature u(x, t) in a semi-infinite rod if u(0, t) 1, t ...
- 126.96.36.199.74: (a) (b) If g(x) 0, show that the solution of part (a) can be writte...
- 188.8.131.52.75: Find the displacement u(x, t) of a semi-infinite string if
- 184.108.40.206.76: Solve the problem in Example 2 if the boundary conditions at x 0 an...
- 220.127.116.11.77: Solve the problem in Example 2 if the boundary condition at y 0 is ...
- 18.104.22.168.78: Find the steady-state temperature u(x, y) in a plate defined by x 0...
- 22.214.171.124.79: Solve if the boundary condition at x 0 is u(0, y) 0, y 0.
- 126.96.36.199.80: Use the Fourier integral transforms of this section to solve the gi...
- 188.8.131.52.81: Use the Fourier integral transforms of this section to solve the gi...
- 184.108.40.206.82: In 17 and 18 find the steady-state temperature in the plate given i...
- 220.127.116.11.83: In 17 and 18 find the steady-state temperature in the plate given i...
- 18.104.22.168.84: Use the result to solve the boundary-value problem
- 22.214.171.124.85: If , then the convolution theorem for the Fourier transform is give...
- 126.96.36.199.86: Use the transform given in to find the steady-state temperature in ...
- 188.8.131.52.87: The solution of can be integrated. Use entries 42 and 43 of the tab...
- 184.108.40.206.88: Use the solution given in to rewrite the solution of Example 1 in a...
- 220.127.116.11.89: The steady-state temperatures in a semi-infinit cylinder are descri...
- 18.104.22.168.90: Find the steady-state temperatures in the semiinfinite cylinder in ...
- 22.214.171.124.91: (a) Suppose where Find f(x). (b) Use part (a) to show that
- 126.96.36.199.92: Assume that u0 100 and k 1 in the solution in 23. Use a CAS to grap...
Solutions for Chapter 14.4: Integral Transforms
Full solutions for Differential Equations with Boundary-Value Problems, | 8th Edition
Tv = Av + Vo = linear transformation plus shift.
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
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.
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 •
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).