- 10.4.10.4.1: Using Greens formula, show that F d2f dx2 = 2F() + eix 2 df dx if .
- 10.4.10.4.2: For the heat equation, u(x, t) is given by (10.4.1). Show that u 0 ...
- 10.4.10.4.3: (a) Solve the diffusion equation with convection: u t = k 2u x2 + c...
- 10.4.10.4.4: (a) Solve u t = k 2u x2 u,
- 10.4.10.4.5: (a) Solve u t = k 2u x2 u, (b) Determine U. *(c) Solve for u(x, t) ...
- 10.4.10.4.6: The Airy function Ai(x) is the unique solution of d2y dx2 xy = 0 th...
- 10.4.10.4.7: (a) Solve the linearized KortewegdeVries equation u t = k 3u x3 , 0.
- 10.4.10.4.8: Solve 2u x2 + 2u y2 = 0, 0
- 10.4.10.4.9: Solve 2u x2 + 2u y2 = 0, y > 0
- 10.4.10.4.10: Solve 2u t2 = c2 2u x2 , 0 u t (x, 0) = 0. (Hint: If necessary, see...
- 10.4.10.4.11: Derive an expression for the Fourier transform of the product f(x)g(x)
- 10.4.10.4.12: Solve the heat equation, 0 subject to the condition u(x, 0) = f(x).
Solutions for Chapter 10.4: Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Solutions for Chapter 10.4: Infinite Domain Problems: Fourier Transform Solutions of Partial Differential EquationsGet Full Solutions
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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.
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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.