 13.2.1: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.2: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.3: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.4: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.5: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.6: In 1 6, a rod of length L coincides with the interval [0, L] on the...
 13.2.7: In 710, a string of length L coincides with the interval [0, L] on ...
 13.2.8: In 710, a string of length L coincides with the interval [0, L] on ...
 13.2.9: In 710, a string of length L coincides with the interval [0, L] on ...
 13.2.10: In 710, a string of length L coincides with the interval [0, L] on ...
 13.2.11: In 11 and 12, set up the boundaryvalue problem for the steadystat...
 13.2.12: In 11 and 12, set up the boundaryvalue problem for the steadystat...
Solutions for Chapter 13.2: Classical PDEs and BoundaryValue Problems
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 13.2: Classical PDEs and BoundaryValue Problems
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since 12 problems in chapter 13.2: Classical PDEs and BoundaryValue Problems have been answered, more than 39884 students have viewed full stepbystep solutions from this chapter. Chapter 13.2: Classical PDEs and BoundaryValue Problems includes 12 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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).

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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

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.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.