- •Introduction to MatLab
- •Exercises
- •First Order Equations
- •Linear First Order Equations
- •Separation of Variables Applied to Mechanics
- •Exercises
- •Second Order Linear Equations
- •Theory of Second Order Equations
- •Reduction of Order
- •Exercises
- •Introduction
- •Exercises
- •The Matrix Exponential
- •Relation to Earlier Methods of Solving Constant Coefficient DEs
- •Inhomogenous Matrix Equations
- •Exercises
- •Weighted String
- •Reduction to an Eigenvalue Problem
- •The Eigenvectors
- •Determination of constants
- •Continuum Limit: The Wave Equation
- •Inhomogeneous Problem
- •Vibrating Membrane
- •Exercises
- •Quantum Harmonic Oscillator
- •Harmonic Oscillator
- •Some properties of the harmonic oscillator
- •The Heisenberg Uncertainty Principle
- •Exercises
- •Laplace Transform
- •Exercises
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CHAPTER 8. |
LAPLACE TRANSFORM |
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(sIn − A) = |
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−1P −1. |
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independent of s, |
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Since P and |
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diagonal matrix |
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matrix (sIn − A) |
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(sIn − A)−1 = |
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j=1 |
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where
Pj = P Ej P −1
and Ej is the diagonal matrix with all zeros on the main diagonal except for 1 at the (j, j)th entry. This follows from the fact that
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Note that Pj have the property that |
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P 2 |
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Such matrices are called projection operators.
In general, it follows from Cramer’s method of computing the inverse of a matrix, that the general structure of (sIn −A)−1 will be 1/p(s) times a matrix whose entries are polynomials of at most degree n − 1 in s. When an eigenvalue, say λ1, is degenerate and of (algebraic) multiplicity m1, then the characteristic polynomial will have a factor (s − λ1)m1 . We have seen that if the matrix is diagonalizable, upon a partial fraction decomposition only a single power of (s − λ1) will appear in the denominator of the partial fraction decompostion. Finally, we conclude by mentioning that when the matrix A is not diagonalizable, then this is reflected in the partial fraction decomposition of (sIn − A)−1 in that some powers of (s − λj ) occur to a higher degree than 1.
8.3Exercises
#1.
Use the Laplace transform to find the solution of the initial value problem
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#2.
Let A be a n × n matrix whose entries are real numbers and x Rn. Prove that
L(Ax) = AL(x)
where L denotes the Laplace transform.
8.3. EXERCISES |
121 |
#3.
Let Ej denote the diagonal n × n matrix with all zeros on the main diagonal except for 1 at the (j, j) entry.
•Prove that Ej2 = Ej .
•Show that if P is any invertible n × n matrix, then Pj2 = Pj where Pj := P Ej P −1.
#4.
It is a fact that you will learn in an advanced linear algebra course, that if a 2 × 2 matrix A is not diagonalizable, then there exists a nonsingular matrix P such that
A = P B P −1
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for some constant λ.
•Show that λ must be an eigenvalue of A with algebraic multiplicity 2.
•Find an eigenvector of A (in terms of the matrix P ), and show that A has no other eigenvectors (except, of course, scalar multiples of the vector you have already found).
•Show that
(sI2 − A)−1 |
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P E1P −1 |
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P E2P −1 + |
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P N P −1 |
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s − λ |
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where |
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• Relate what is said here to the remarks in the footnote in Exercise 5.5.2.
122 |
CHAPTER 8. LAPLACE TRANSFORM |
Bibliography
[1]V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., SpringerVerlag, N.Y. 1989.
[2]V. I. Arnold, Ordinary Di erential Equations, Second printing of the 1992 edition, Springer-Verlag, Berlin, 2006.
[3]G. Birkho and G-C Rota, Ordinary Di erential Equations 4th ed., John Wiley & Sons, 1989.
[4]W. E. Boyce and R. C. DiPrima, Elementary Di erential Equations and Boundary Value Problems, 7th–9th eds., John Wiley & Sons.
[5]G. Chen, P. J. Morris and J. Zhou, Visualization of special eigenmode shapes of a vibrating elliptical membrane, SIAM Review 36 (1994), 453–469.
[6]B. Temple and C. A. Tracy, From Newton to Einstein, American Math. Monthly 99 (1992), 507–512.
123
Index
Acceleration due to gravity, 1 Acoustic phonons, 94 Angular momentum, 26
Annihilation and creation operators, 113
Bessel functions, 86
Binet’s equation, 28
Catalan numbers, 50
Classical harmonic oscillator, 18, 34 Forced oscillations, 40 Friction, 25
Commutator, 99
Complex exponential function, 37 Addition formula, 38
Euler’s formula, 38
Di erence equations, 47 Constant coe cients, 47
Catalan numbers, 50 Fibonnaci numbers, 48
Di erential equation, definition, 9
Energy conservation, 14, 18
Fibonnaci numbers, 48
Half-life, 23
Harmonic oscillator, 18, 34, 61, 82 Forced oscillations, 40 Quantum, 98
Harmonic oscillator wave functions, 106 Heisenberg Uncertainty Principle, 111 Heisenberg Uncertainty Relation, 99 Helmholtz equation, 83, 93
Hermite polynomials, 101–103, 106, 113 Generating functions, 113
Hooke’s Law, 15
Forced oscillations, 40 Resonance, 42
Frequency, 18
Friction, 25
Nonlinear correction, 25
Integrating factor, 10
Kinetic energy, 14, 95, 99 Kolmogorov forward equations, 63
Laplace transform, 115
Application to radioactive decay, 63 Laplacian, 84
Lax pairs, 91
Linear first order ODE, 9 integrating factor, 10 Loan payments, 11
MatLab, 4, 21, 39
Symbolic Math Toolbox, 4 Matrix exponential, 53
Applied to matrix ODE, 56 Nilpotent matrix, 61
Matrix ODE, 56
Alternating masses-spring problem, 94 Inhomogeneous, 60
Radioactive decay, 63
Newton’s equations, 1, 14, 68, 89, 97 Central field, 26
Inverse square law, 28 Newton’s principle of determinacy, 1 Normal modes, 72
Optical phonons, 94
Order of a DE, 9
Pendulum, 1
Coupled, 88
Period, 18
Planck’s constant, 98
Population dynamics, 24
Potential energy, 14, 95, 99
124
INDEX |
125 |
Quantum harmonic oscillator, 98
Annihilation and creation operators, 113
Hermite polynomials, 102
Quantization of energy, 103
Quantum mechanics, 97
State space, 97
Reduction of order, 35
Rigid body motion, 28
Schr¨odinger equation, 97
Second order linear ODE, 31
Bessel functions, 86
Constant coe cients, 35
Vector space of solutions, 31
Wronskian, 33
Toda chain, 91
Vibrating membrane, 83
Circular domain, 85
Elliptical domain, 87
Rectangular domain, 85, 93
Vibrating string, 79
Energy, 95
Wave equation, 79
Helmholtz equation, 83
Relation to harmonic oscillators, 82
Separation of variables, 80
Vibrating membrane, 83
Vibrating string, 79
Weighted string, 67
Circle, 88
Friction, 93
Normal modes, 72
Wronskian, 33, 45