D YNAMIC E QUATIONS OF T IME S CALES L ECTURE 1, W EEK 10
Imanuel Costigan, UNSW
May 9, 2005
So far our interest has been on dynamic equations of rst order. Namely,y
f (t, y, y ) for f linear or non-linear.
We now consider (linear) second-order dynamic eq
D YNAMIC E QUATIONS OF T IME S CALES MATH5215
W EEK 4 L ECTURE N OTES (M ONDAY )
Dr. Chris Tisdell, UNSW
March 22, 2005
1
D ELTA D ERIVATIVE
From a previous lecture, we know that
x
=
T=R
T=Z
x
x
Note in the denitions of (t) and (t) that both (t) and (t) a
MATH5215 Notes: Week 5 Monday 04/04/05 by Smith Huang
Last Time: Regulated and Crd function.
Today: More on integration on T.
Denition: A continuous function f : T R is called pre-delta-diable on
a set D (with region of dbility D) if:
a) D T
b) T \D is co
MATH5215 CLASS NOTES FOR 05/4/2005 BY CHRISTIAN MOLDENHAUER
Last time we dealt with the Delta Integral. Today we will look at some examples and the Chain Rule on
Some special cases will be presented now.
T.
Theorem: Let a, b T and f Crd
b
b
a
(i) If
a
T =
11/4/05
Dynamic Equations on Time Scales
Last Time: Examples of delta integration
Today: Chain Rule on T
Dynamic Equations on T
Theorem (Chain Rule)
Let f: R R be continuously differentiable. (i.e. f is continuous) and suppose g: T
R is d-differentiable.
Time Scales1
Denition 1. A time scale T is an arbitrary closed, non-empty subset of R.
Example 1. Some examples of time scales are (see Diagram 1 for illustrations of (1) to (7):
1. The real numbers R
2. The integers Z
3. The natural numbers N = cfw_1, 2,
MATH5215, Some Questions Involving
March 23, 2005
Below are some very gentle exercises to get you working with the summation operator
.
1. Let C(t) = 0. Show that
cos at =
sin a(t 1/2)
+ C(t),
2 sin(a/2)
t+r
r
2. Use summation by parts to compute:
=
a =
MATH5215, Solutions to Some Questions Involving
March 23, 2005
1.
y(t)
z(t)
y(t + 1) y(t)
z(t + 1) z(t)
y(t + 1)z(t) y(t)z(t + 1)
=
z(t)z(t + 1)
z(t)Ez(t) y(t)Ez(t)
=
z(t)Ez(t)
=
2. We use the identity
sin u sin v = 2 sin
1
1
(u v) cos (u + v) .
2
2
sin
MATH5215, Solutions to Some Questions Involving
April 13, 2005
Throughout we assume C(t) = 0.
1. (a) We use the identity
sin u sin v = 2 sin
1
1
(u v) cos (u + v) .
2
2
sin a(t 1/2) = sin a(t + 1/2) sin a(t 1/2)
a
a
= 2 sin (t + 1/2) (t 1/2) cos (t + 1/
NAME OF CANDIDATE: . . . . . . . . . . . . . . . . . . .
STUDENT NUMBER: . . . . . . . . . . . . . . . . . . . . . .
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
June 2005
PRACTICE EXAM
MATH5215
Dynamic Equations on Time Scales
TIME ALLOWED 2 h
MATH5215, Some Questions Involving ep(t, t0)
May 22, 2005
1. Consider the following dynamic IVP on arbitrary T
y = p(t)y, p R,
y(t0 ) = 1, t0 T.
(0.1)
(0.2)
Show that the only solution to (0.1), (0.2) is ep (t, t0 ). (Hint: Let y be a solution to
the IVP
MATH5215 CLASS NOTES FOR 22/3/2005 BY A. COWLING
Last lecture we looked at the delta derivative. Today we will look at the delta
derivative and delta integral.
Example: Consider T = R, so all points in T are right dense. From iii),we have
that:
f (t) f (s
Time Scales1
We have previously considered how to get a priori bounds on the solutions to various dynamic
equations, without knowing anything on the existence of solutions. We now begin applying these
bounds to existence questions.
Consider the following
MATH5215 LECTURE 10/5/2005
Last time we solved the homogeneous problem y + 2y 3y = 0. What about
the inhomogeneous case?
Consider y + 2y 3y = f (t). If yh is a solution to the homogeneous problem
and if yp is a particular solution to the inhomogeneous pro
MATH5215 CLASS NOTES FOR 17/5/2005 BY CHRISTIAN MOLDENHAUER
Consider the dynamic IVP with arbitrary
T.
x = f (t, x)
x(t0 ) = x0
(1)
(2)
t [t0 , N ]T
x0 Rn , t0 R, N T
where f : [t, N ]T Rn Rn is continous and solutions to (1) and (2) are of type
x : [t0 ,
MATH5215 Notes: Week 11 Monday 16/05/05
Last Time: Consider the 2nd-order dynamic equation
(1)
y + by + cy = 0,
b, c = constants
when b2 4c > 0 (and c b ) we looked at example of (1)
Today: We partly answer what if b2 4c < 0?
Consider the simpler case
(2)