Partial Differential Equations Haberman Solution 7.7.3

Solutions for Chapter 7.7

Textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems

Edition: 5

Author: Richard Haberman

ISBN: 9780321797056

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. Since 16 problems in chapter 7.7: Higher-Dimensional Partial Differential Equations have been answered, more than 26901 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5. Chapter 7.7: Higher-Dimensional Partial Differential Equations includes 16 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

• Back substitution.

  Upper triangular systems are solved in reverse order Xn to Xl.

• Characteristic equation det(A - AI) = O.

  The n roots are the eigenvalues of A.

• Column space C (A) =

  space of all combinations of the columns of A.

• Diagonalization

  A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Echelon matrix U.

  The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Exponential eAt = I + At + (At)2 12! + ...

  has derivative AeAt; eAt u(O) solves u' = Au.

• 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.

• Hermitian matrix A H = AT = A.

  Complex analog a j i = aU of a symmetric matrix.

• Incidence matrix of a directed graph.

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

• Iterative method.

  A sequence of steps intended to approach the desired solution.

• Left inverse A+.

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

• Multiplier eij.

  The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Orthogonal matrix Q.

  Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Pascal matrix

  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).

• Reflection matrix (Householder) Q = I -2uuT.

  Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Row space C (AT) = all combinations of rows of A.

  Column vectors by convention.

• Semidefinite matrix A.

  (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Spanning set.

  Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Symmetric factorizations A = LDLT and A = QAQT.

  Signs in A = signs in D.

• Trace of A

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

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Partial Differential Equations Haberman Solution 7.7.3

Source: https://studysoup.com/tsg/math/284/applied-partial-differential-equations-with-fourier-series-and-boundary-value-problems/chapter/12496/7-7

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