 13.2.13.2.1: From the definition of the Laplace transform (i.e., using explicit ...
 13.2.13.2.2: The gamma function (x) was defined in Exercise 10.3.14. Derive that...
 13.2.13.2.3: Derive the following fundamental properties of Laplace transforms: ...
 13.2.13.2.4: Using Table 13.2.1, determine the Laplace transform of t 0 f(t) dt ...
 13.2.13.2.5: Using Table 13.2.1, determine the Laplace transform of f(t) =: (a) ...
 13.2.13.2.6: Using Table 13.2.1, determine the inverse Laplace transform of F(s)...
 13.2.13.2.7: Using Table 13.2.1, determine the inverse Laplace transform of F(s)...
 13.2.13.2.8: Derive the convolution theorem for Laplace transforms without using...
 13.2.13.2.9: In this exercise we will determine I = L{t 3/2e a2/4t } and J = L{t...
Solutions for Chapter 13.2: Laplace Transform Solution of Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 13.2: Laplace Transform Solution of Partial Differential Equations
Get Full SolutionsSince 9 problems in chapter 13.2: Laplace Transform Solution of Partial Differential Equations have been answered, more than 8145 students have viewed full stepbystep solutions from this chapter. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 13.2: Laplace Transform Solution of Partial Differential Equations includes 9 full stepbystep solutions.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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 .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.