- 5.3.1: Find a solution u(x, t) of the following problems.
- 5.3.2: Find a solution u(x, t) of the following problems.
- 5.3.3: Use the method of separation of variables to solve the boundary-val...
- 5.3.4: Use the method of separation of variables to solve each of the foll...
- 5.3.5: Use the method of separation of variables to solve each of the foll...
- 5.3.6: Use the method of separation of variables to solve each of the foll...
- 5.3.7: Use the method of separation of variables to solve each of the foll...
- 5.3.8: Determine whether the method of separation of variables can be used...
- 5.3.9: The heat equation in two space dimensions is (a) Assuming that u(x,...
- 5.3.10: The heat equation in two space dimensions may be expressed in terms...
Solutions for Chapter 5.3: Introduction to partial differential equations
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition
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).
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 n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
lA-II = 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.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.