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# Solutions for Chapter 6.4: Finite Difference Numerical Methods for Partial Differential Equations

## Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition

ISBN: 9780321797056

Solutions for Chapter 6.4: Finite Difference Numerical Methods for Partial Differential Equations

Solutions for Chapter 6.4
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##### ISBN: 9780321797056

Since 4 problems in chapter 6.4: Finite Difference Numerical Methods for Partial Differential Equations have been answered, more than 8112 students have viewed full step-by-step solutions from this chapter. Chapter 6.4: Finite Difference Numerical Methods for Partial Differential Equations includes 4 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056.

Key Math Terms and definitions covered in this textbook
• Back substitution.

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

• Covariance matrix:E.

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.

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• 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.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• 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).

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Schwarz inequality

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

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• 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.

• Volume of box.

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

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