1. Chapter 5. Ordinary Differential Equation
수학과 김찬용 , 컴퓨터학과 김현우, 장한용

2. 5.1 The Elementary Theory of Initial-Value Problems
Definition 5.1
f(x,y) : Lipschitz condition on set D⊂R2 ,
∃L > 0 with
, (t,y1), (t,y2) ∈ D , L : Lipschitz constant
Definition 5.2
D⊂R2 : convex , (t1,y1), (t2,y2) ∈ D , λ ∈[0,1]
( (1- λ)t1 + λt2 , (1- λ) y1 + λy2 )∈ D
i.e. D = { (t , y) | a ≤ t ≤ b, | y | ＜ ∞ } : convex

3. 5.1 The Elementary Theory of Initial-Value Problems
Definition 5.3
f(x,y) is defined on a convex set D⊂R2
∃L > 0 with
=> f : Lipschitz condition on D with Lipschitz constant L.
Definition 5.4
, f(x,y) : continuous on D
If f satisfies a Lipschitz condition on D,
then y′(t) = f(t,y) , a ≤ t ≤ b, y(a) =a
has a unique solution y(t) for a ≤ t ≤ b.

4. 5.1 The Elementary Theory of Initial-Value Problems
Definition 5.5
: well-posed problem
if ∃y(t) : unique solution,
and ∃ε0 > 0 , ∃k > 0 s.t ∀ε, with ε0 > ε > 0, whenever
δ(t) : continuous with |δ(t)| < ε for all t in [a , b]
& when |δ0| < ε, dz/dt = z′(t) = f(t,z) + δ(t), a ≤ t ≤ b, z(a) = a + δ0 has unique solution z(t) s.t |z(t) - y(t)| < kε for all t in [a , b]
Definition 5.6
b = { (t,y) | a ≤ t ≤ b, |y| < ∞ }
f : continuous & Lipschitz condition
=> dy/dt = f(t,y) , a ≤ t ≤ b, y(a) = a : well-posed

5. 5.2 Euler’s Method dy/dy = y′(t) = f(t,y) , a ≤ t ≤ b , y(a) = a
ti∈[a,b] : mesh points.
ti = a + ih , for each i = 0,1,2,… , N ( h= (b-a)/N = ti+1 – ti : step size)
using Taylor’s Theorem,
y(t) ∈ C2[a,b] : unique solution,
∈[a,b]
since h= ti+1 – ti
Euler’s method :

6. 5.2 Euler’s Method Lemma 5.7 Lemma 5.8 Theorem 5.9
∀x ≥ -1 & ∀x > 0, 0≤ (1+x)m ≤ emx
Lemma 5.8
s,t ∈ R , :
then
Theorem 5.9
f : continuous & Lipschitz condition with L on D
& ∃M with |y˝(t)| ≤ M,for all t∈[a,b].
Let y(t) : unique solution, Euler’s method
=>

7. 5.2 Euler’s Method Theorem 5.10
let y(t) : unique solution & u0, u1, … , un : approximation,
& |y˝(t)| ≤ M
then
: minimal value of E(h)

8. 5.2 High-Order Taylor Methods
Definition 5.11
has local truncation error
for each i = 0, 1, … , N -1
Taylor method of order n
ω0 = a , ωi = ωi + hT(n)(ti, ωi), for each i = 0, 1, … , N -1
where
T(n)(ti, ωi) = f(ti, ωi) + h/2*f ′(ti, ωi) + … + hn-1/n!*f(n-1)(ti, ωi)
Note : Euler’s method is Taylor’s method of order one.

9. 5.2 High-Order Taylor Methods
Definition 5.12
using Taylor’s method’s of order n, h: step size.
if y ∈ Cn+1[a,b], then the local truncation error is O(hn).

10. 5.4 Runge-Kutta Methods Definition 5.13
f(t,y) & all its partial derivatives of order less than or equal to n+1 : continuous on
let , ∀ , ∃ ∈(t,t0), ∃ ∈(y,y0) with
where
: n th Taylor polynomial in two variables.

11. 5.4 Runge-Kutta Methods
≒ a1T(2)(t,y) + a1b1(t+a1,y+b1)
where

12. 5.4 Runge-Kutta Methods
=>
where

13. 5.4 Runge-Kutta Methods Specific Runge-Kutta method. Midpoint Method
Modified Euler Method
Heun’s Method

14. 5.4 Runge-Kutta Methods Runge-Kutta Order Four :
for each i = 0, 1, … ,N-1

15. 5.5 Error Control and the Runge-Kutta-Fehlberg Method
(n+1)st – order Taylor method of the form
Producing approximations
assume

16. 5.5 Error Control and the Runge-Kutta-Fehlberg Method≒

17. 5.5 Error Control and the Runge-Kutta-Fehlberg Method
Using runge-kutta method with local truncation error of order five,
estimate the local error in a runge-kutta method of order four
where the coefficient equation are

18. 5.5 Error Control and the Runge-Kutta-Fehlberg Method
The value of q determined at the i th step is used for two purpose
When q<1, to reject the initial choice of h at the i th step and repeat the calculations using qh, and
When q≥1, to accept the computed value at the i th step using the step size h and to change the step size to qh for (i + 1)st step.
n=4 runge-kutta-fehlberg method

19. 5.6 Multistep Methods Definition 5.14 m-step multistep method
are consistants.
when bm = 0 : explicit or open
bm ≠0 : implicit or closed

20. 5.6 Multistep Methods To begin the derivation of multistep method,
Since we can not integrate f(t,y(t)) without knowing y(t)
P(t) : interpolating polynomial, (t0, ω0) ….. (ti, ωi)
assume

21. 5.6 Multistep Methods Adams-Bashforth explicit m-step technique
Pm-1(t) : backward-difference polynomial,
…..
t = ti + sh ,dt = hds , error term

22. 5.6 Multistep Methods
Example
three-step Adams-Bashforth technique

23. 5.6 Multistep Methods
yi ≒ ωi

24. 5.6 Multistep Methods Definition 5.15
is the (i+1)st step in a multistep method,
local truncation error at this step is

25. 5.6 Multistep Methods Adams-Bashforth Adams-Moulton Two-step
τi+1(h) = 5/12y(3)(μi)h2
μi ∈ (ti-1, ti+1)
τi+1(h) = -1/24y(4)(μi)h3
Three-step
τi+1(h) = 3/8y(4)(μi)h3
μi ∈ (ti-2, ti+1)
τi+1(h) =
-19/720y(5)(μi)h4
Four-step
τi+1(h) = 251/720y(5)(μi)h4
μi ∈ (ti-3, ti+1)
τi+1(h) = -3/160y(6)(μi)h5

26. 5.7 Variable step-size Multistep Method
Adams-Bashforth four-step method
ω0, ω1, ... , ωi , μi ∈ (ti-3, ti+1)
Adams-Bashforth three-step method

27. 5.7 Variable step-size Multistep Method
new step size qh, generating new approximations
As a consequence, we commonly ignore the step-size change when the local truncation error is between ε/10 and ε that is when

28. 5.8 Extrapolation Methods
assume fixed step size h, y(ti)=y(a+h)
let h0 = h/2, use Euler's method with ω0=a , y(a + h0) = y(a+h/2)
apply Midpoint method
let h = h/4 use Euler's method ω0=a , y(a + h1) = y(a+h/4) with ω1, y(a + 2h1) = y(a+h/2) with ω2 , y(a + 3h1) = y(a+3h/4) with ω3

29. 5.8 Extrapolation Methods
approximation

30. 5.9 High-Order Equations and of Differential Equations
m th - order system

31. 5.9 High-Order Equations and of Differential Equations
Definition 5.16
, on
satisfy a Lipschitz condition on D, ∃L > 0 with
Definition 5.17
fi(t,u1, ... , um) : continuous on D & satisfy a Lipschitz condition.
The system of first-order differential equations, subject to the initial conditions has a unique solution u1(t), ... ,um(t) for a ≤ t ≤ b.

32. 5.9 High-Order Equations and of Differential Equations
for each i = 1, 2, ... , m :
for each i = 1, 2, ... , m : and then

33. 5.10 Stability Definition 5.18 Definition 5.19
A one-step difference-equation method with local truncation error τi(h) at the i th step is said to be consistent with the differential equation it approximates if
Definition 5.19
A one-step difference-equation method is said to be convergent with respect to the differential equation it approximates if
where yi = y(ti) : exact value of solution of differential equation
ωi : approximation obtained from difference method at the with step.

34. 5.10 Stability
Theorem 5.20
is approximated by a one-step difference method in the form
∃h0 > 0, φ(t,w,h) : continuous & satisfies a Lipschitz condition on
Then
ⅰ) The method is stable;
ⅱ) The difference method is convergent if and only if it is consistent,
which is equivalent to
ⅲ) ∃function and

35. 5.10 Stability Theorem 5.21 with local truncation error τi+1(h)
f(t,y) and fy(t,y) : conditinuous on
, fy is bounded then, the local truncation error of the predictor-corrector method is

36. 5.10 Stability
Definition 5.22
let λ1, λ2, ... , λm : root of the characteristic equation
associated with the multistep difference method
&
if |λi| ≤ 1 , i = 1, 2, ... , m, & all roots with absolute value 1 are simple roots, then the difference method is said to satisfy the root condition.

37. 5.10 Stability Definition 5.23 Theorem 5.24
ⅰ) Methods the satisfy the root condition and hand λ = 1 as the only root of the characteristic equation of magnitude one are called strongly stable.
ⅱ) Methods the satisfy the root condition and have more than one distinct root with magnitude one are called weakly stable.
ⅲ) Methods that do not satisfy the root condition are called instable.
Theorem 5.24
A multistep method of the form
where
: stable ⇔ root condition
moreover, if the difference method is consistant with the difference equation, then the method is stable if it is convergent.

38. 5.11 Stiff Differential Equations
: solution, which the transient solution
Euler's method applied to the test equation
let h = (b-a) / N , tj = jh , for j = 0, 1, 2, ... , N , so
absolute error
λ < 0 : (ehλ)j decays to zero as j increases
|1+hλ| < 1 : proerty approximation => -2 < hλ < 0
This effectively restricts the step size h for Euler's' method to
satisfy h < 2/| λ |

39. 5.11 Stiff Differential Equations
δ0 : round-off error
δ1 : (1+hλ)jδ0 : j th step the round-off error
since λ < 0 , the condition for the control of the growth of round-off error is the same as the condition for controlling the absolute error, |1+hλ| < 1, which implies that h < 2/| λ |
Definition 5.24
The region R of absolute stability for a one-step method is
R = { hλ∈C | |Q(hy)| < 1}, and for a multistep method,
it is R = { hλ∈C | |bk| < 1, for all zeros bk of Q(z,hy)}.