- 13.1: In 1 and 2, use separation of variables to find product solutions u...
- 13.2: In 1 and 2, use separation of variables to find product solutions u...
- 13.3: Find a steady-state solution c(x) of the boundary-value problem k 0...
- 13.4: Give a physical interpretation for the boundary conditions in 3.
- 13.5: At t 0 a string of unit length is stretched on the positive x-axis....
- 13.6: The partial differential equation 02 u 0x2 1 x2 5 02 u 0t 2 is a fo...
- 13.7: Find the steady-state temperature u(x, y) in the square plate shown...
- 13.8: Find the steady-state temperature u(x, y) in the semi- infinite pla...
- 13.9: Solve if the boundaries y 0 and y p are held at temperature zero fo...
- 13.10: Find the temperature u(x, t) in the infinite plate of width 2L show...
- 13.11: (a) Solve the boundary-value problem 02 u 0x2 5 0u 0t , 0 , x , p, ...
- 13.12: Solve the boundary-value problem 02 u 0x2 1 sin x 5 0u 0t , 0 , x ,...
- 13.13: Find a series solution of the problem 02 u 0x 2 2 0u 0x 02 u 0t 2 2...
- 13.14: The concentration c(x, t) of a substance that both diffuses in a me...
- 13.15: Solve the boundary-value problem 02 u 0x2 0u 0t , 0 , x , 1, t . 0 ...
- 13.16: Solve Laplaces equation for a rectangular plate subject to the boun...
- 13.17: Use the substitution u(x, y) v(x, y) c(x) and the result of to solv...
- 13.18: Solve the boundary-value problem 02 u 0x2 ex 0u 0t , 0 , x , p, t ....
- 13.19: A rectangular plate is described by the region in the xy-plane defi...
- 13.20: If the four edges of the rectangular plate in are simply supported,...
Solutions for Chapter 13: Boundary-Value Problems in Rectangular Coordinates
Full solutions for Advanced Engineering Mathematics | 6th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Remove row i and column j; multiply the determinant by (-I)i + j •
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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).
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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 r (A)
= number of pivots = dimension of column space = dimension of row space.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.