 Chapter 1: WHERE PDEs COME FROM
 Chapter 10: BOUNDARIES IN THE PLANE AND IN SPACE
 Chapter 11: GENERAL EIGENVALUE PROBLEMS
 Chapter 12: DISTRIBUTIONS AND TRANSFORMS
 Chapter 13: PDE PROBLEMS FROM PHYSICS
 Chapter 14: NONLINEAR PDES
 Chapter 2: WAVES AND DIFFUSIONS
 Chapter 3: REFLECTIONS AND SOURCES
 Chapter 4: BOUNDARY PROBLEMS
 Chapter 5: FOURIER SERIES
 Chapter 6: HARMONIC FUNCTIONS
 Chapter 7: GREENS IDENTITIES AND GREENS FUNCTIONS
 Chapter 8: COMPUTATION OF SOLUTIONS
 Chapter 9: WAVES IN SPACE
Partial Differential Equations: An Introduction 2nd Edition  Solutions by Chapter
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Partial Differential Equations: An Introduction  2nd Edition  Solutions by Chapter
Get Full SolutionsPartial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. Since problems from 14 chapters in Partial Differential Equations: An Introduction have been answered, more than 8377 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 14. The full stepbystep solution to problem in Partial Differential Equations: An Introduction were answered by , our top Math solution expert on 01/05/18, 06:22PM.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

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 .

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.