chapter 5. ordinary differential equation

NUMERICAL ANALYSIS

http://korea.ac.kr CSE503 응용 수학Ⅱ

Chapter 5. Ordinary Differential Equation

수학과 김찬용 , 컴퓨터학과 김현우 , 장한용

2 http://korea.ac.kr CSE503 응용 수학Ⅱ

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



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) =has a unique solution y(t) for a ≤ t ≤ b.



: 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) = 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) = : well-posed


5.2 Euler’s Method dy/dy = y′(t) = f(t,y) , a ≤ t ≤ b , y(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 :


5.2 Euler’s Method Lemma 5.7

∀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

=>


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)


5.2 High-Order Taylor Methods Definition 5.11

has local truncation error

for each i = 0, 1, … , N -1Taylor method of order n

ω0 = , ωi = ωi + hT(n)(ti, ωi), for each i = 0, 1, … , N -1

whereT(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.


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).


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.


≒ a1T(2)(t,y) + a11(t+1,y+1)

where

5.4 Runge-Kutta Methods



=>

where


Specific Runge-Kutta method. Midpoint Method

Modified Euler Method

Heun’s Method



Runge-Kutta Order Four :

for each i = 0, 1, … ,N-1



5.5 Error Control and the Runge-Kutta-Fehlberg Method

(n+1)st – order Taylor method of the form

Producing approximations

assume



≒ ≒



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



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


5.6 Multistep Methods Definition 5.14

m-step multistep method

are consistants.

when bm = 0 : explicit or open

bm ≠0 : implicit or closed


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


5.6 Multistep Methods Adams-Bashforth explicit m-step technique

Pm-1(t) : backward-difference polynomial,

…..

t = ti + sh ,dt = hds , error term



Examplethree-step Adams-Bashforth technique



yi ≒ ωi


5.6 Multistep Methods Definition 5.15

is the (i+1)st step in a multistep method,

local truncation error at this step is



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

μi (∈ ti-1, ti+1)

Three-step

τi+1(h) = 3/8y(4)(μi)h3

μi (∈ ti-2, ti+1)

τi+1(h) =

-19/720y(5)(μi)h4

μi (∈ ti-2, ti+1)

Four-step

τi+1(h) = 251/720y(5)(μi)h4

μi (∈ ti-3, ti+1)

τi+1(h) = -3/160y(6)(μi)h5

μi (∈ ti-3, ti+1)


5.7 Variable step-size Multistep Method

Adams-Bashforth four-step methodω0, ω1, ... , ωi , μi (∈ ti-3, ti+1)

Adams-Bashforth three-step method


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


5.8 Extrapolation Methods

assume fixed step size h, y(ti)=y(a+h)

let h0 = h/2, use Euler's method with ω0=y(a + h0) = y(a+h/2)

apply Midpoint method

let h = h/4 use Euler's method ω0=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


5.8 Extrapolation Methods

approximation


5.9 High-Order Equations and of Differential Equations

m th - order system


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.


5.9 High-Order Equations and of Differential Equations

for each i = 1, 2, ... , m :

for each i = 1, 2, ... , m :

for each i = 1, 2, ... , m : and then


5.10 Stability Definition 5.18

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.


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 toⅲ) ∃function and


5.10 Stability Theorem 5.21

with local truncation error τi+1(h)

with local truncation errorf(t,y) and fy(t,y) : conditinuous on

, fy is bounded then, the local truncation error of the predictor-corrector method is



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.



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 conditionmoreover, if the difference method is consistant with the difference equation, then the method is stable if it is convergent.


5.11 Stiff Differential Equations

: solution, which the transient solution .Euler's method applied to the test equationlet 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 < 0This effectively restricts the step size h for Euler's' method to satisfy h < 2/| λ |


5.11 Stiff Differential Equations

ω0 = + δ0

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 | |k| < 1, for all zeros k of Q(z,hy)}.

chapter 5. ordinary differential equation

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