1. 12 Ordinary Differential Equations: Initial-Value Problems
자연어처리 연구실
김혜겸
윤도상
최성원

2. Introduction
Techniques for solving first-order ordinary differential equations are the subject of this chapter
the differential equation is written in the form with the value of function given at ,i.e.,
Basic idea is to divide the internal of interest into discrete steps and find approximations to the function y at those values of
Taylor Method
Runge-Kutta Method
Multistep Method

3. 12.1 TAYLOR METHODS
Consider two methods that are based on the Taylor series representation of unknown function
Euler’s method
Uses only the first term in the Taylor series
Higher order Taylor method

4. 12.1 TAYLOR METHODS 12.1.1 Euler’s Method
The simplest method of approximating the solution of the differential equation
Start by treating the function as a constant, , and replace the derivative by the forward difference quotient. This gives
or ( where )
Geometrically, this corresponds to using the line tangent to the true solution curve to find the value of

5. 12.1 TAYLOR METHODS 12.1.1 Euler’s Method
Euler Method gives
, for
For example,

6. Example 12.1 Solving a Simple ODE with Euler’s Method(1/2)
, (i.e., a=0, b=1) ,
First, find the approximate solution for h =0.5 (n =2)
Approximation at is
Second, find the approximate solution at :

7. Example 12.1 Solving a Simple ODE with Euler’s Method(2/2)
To find a better approximate solution(use n=10 intervals)

8. MATLAB Function For Euler’s Method

9. Example 12.2 Another Example of the Use of Euler’s Method
Consider the differential equation
Interval with initial value
With intervals, the step size
Exact solution is
y1=y0+hf(x0,y0)
=0+0.2(1)=0.2
y2=y1+hf(x1,y1)
= (0.2(-0.4+1/2))=0.384 .

10. Discussion
How well a numerical technique for solving an initial-value ordinary differential equation works
Total truncation error for Euler’s method is
Related to the truncation error of the method is
Result of basic Euler’s method can be improved by using it with Richardson extrapolation
Step size
Extrapolation level
Computed directly from Euler’s method

11. 12.1 TAYLOR METHODS 12.1.2 Higher Order Taylor Method
Use more terms in the Taylor series for y, in order to obtain a higher order truncation error
However do not have a formula for
If is not too complicated
Differentiate with respect to to find a representation for

12. Example 12.3 Solving a Simple ODE with Taylor’s Method
Consider the differential equation
with initial condition
To apply the second order Taylor method
This gives the approximation formula or
For , we find

13. MATLAB Function For Taylor’s Method

14. 12.2 RUNGE-KUTTA METHOD 배경
Talyor 해법은 Euler해법에 비해서 정확하지만, f(x,y)의 고계도함수를 구해야 하는 어려움
Runge-kutta해법은 talyor 해법의 장점이 되는 계산 정밀도는 유지하되, 단점을 보완한 방법

15. 12.2 RUNGE-KUTTA METHOD 12.2.1 Midpoint Method공식
Runge-kutta 해법은 더 쉬운 계산법을 사용하여 Tailer방법을 근사시키는 전략을 사용함

16. 12.2 RUNGE-KUTTA METHOD 12.2.1 Midpoint Method
Example 12.4 Solving a Simple ODE with the Midpoint Method
에서 미분 방정식은 , y(0) = 2
1) h = 0.5 (n=2)  x1 = 0.5
<P406 참조>
Exact :
Euler’s :
2) x2 = h = 1.0
<P406 참조>
Exact :
Euler’s :

17. 12.2 RUNGE-KUTTA METHOD 12.2.1 Midpoint Method
MATLAB Function for the Midpoint Method
Function [x, y] = RK2 (f, tspan, y0, n)
% function [x, y] = RK2 (f, y0, a, b, n)
% solve y’ = f(x, y)
% with initial condition y(a) = y0
% using n steps of the midpoint (RK2) method;
a = trans(1); b = trans(2); h = (b - a) / n
x = (a+h: h : b);
k1 = h * feval (f, a, y0);
k2 = h * feval (f, a + h / 2, y0 + k1 / 2);
y(1) = y0 + k2;
for i = 1 : n-1
k1 = h * feval (f, x(i), y(i) );
k2 = h * feval (f, x(i) + h / 2, y(i) + k1 / 2);
y(i+1) = y(i) + k2;
end
x = [ a x ];
y = [ y0 y];

18. 12.2 RUNGE-KUTTA METHOD 12.2.2 Other Second-Order Runge-Kutta Mehods
Modified Euler’s method
Heun’s method
Midpoint Method c2
a21 w1 w2 1
1/2
2/3
1/4
3/4
1/2 1

19. 12.2 RUNGE-KUTTA METHOD 12.2.3 Third-Order Runge-Kutta Mehodsc3
a31 w1 w2 c2
a21
a32 w3
- Nystrom
- Nearly Optimal
2/3
2/8
3/8
3/4
2/9
3/9
1/2
4/9
- Classical
- Heun 1 -1
1/6
4/6
1/2 2
2/3
1/4
1/3
3/4

20. 12.2 RUNGE-KUTTA METHOD 12.2.4 Classic Runge-Kutta Method
2차나 3차보다 더 정확한 해를 구해 줌
다른 Runge-Kutta방법들 중에서 가장 널리 사용됨

21. 12.2 RUNGE-KUTTA METHOD 12.2.4 Classic Runge-Kutta Method
MATLAB Function for Classic Runge-Kutta Method
Function [x, y] = RK4 (f, tspan, y0, n)
% solve y’ = f(x, y) with initial condition y(a) = 0
% using n steps of the classic 4th order Runge-Kutta method;
a = trans(1); b = trans(2); h = (b - a) / n
x = (a+h: h : b);
k1 = h * feval (f, a, y0);
k2 = h * feval (f, a + h / 2, y0 + k1 / 2);
k3 = h * feval (f, a + h / 2, y0 + k2 / 2);
k4 = h * feval (f, a + h, y0 + k3);
y(1) = y0 + k1/6 + k2/3 + k3/3 + k4/6;
for i = 1 : n-1
k1 = h * feval (f, x(i), y(i) );
k2 = h * feval (f, x(i) + h/2, y(i) + k1 / 2);
k3 = h * feval (f, x(i) + h/2, y(i) + k2 / 2);
k4 = h * feval (f, x(i) + h, y(i) + k3);
y(i+1) = y(i) + k1/6 + k2/3 + k3/3 + k4/6;
end
x = [ a x ];
y = [ y0 y];

22. 12.2 RUNGE-KUTTA METHOD 12.2.4 Classic Runge-Kutta Method
Example 12.6 Solving a Simple ODE with the Classic Runge-Kutta Method x
Exact solution
Approximate Solution
Absolute error
0.2
2.4642
8.2745e-06
0.4
3.0755
2.0213e-05
0.6
3.8664
3.8663
3.7032e-05
0.8
4.8766
6.0308e-05
1.0
6.1548
9.2076e-05
<P406 참조>
Exact :
Euler’s :
Midpoint : 5.922

23. 12.2 RUNGE-KUTTA METHOD 12.2.5 Other Runge-Kutta Methods
General Runge-Kutta Mothod (1/2) c3
a31 w1 w2 c2
a21
a32 w3 c4
a41
a42
a43 w4

24. 12.2 RUNGE-KUTTA METHOD 12.2.5 Other Runge-Kutta Methods
General Runge-Kutta Mothod (2/2)
- Classic fourth-order Runge-Kutta
- Kutta’s method
1/2
1/6
2/6 1
1/2
1/6
2/6 1
- Lawson’s fifth-order Runge-Kutta Method
- Butcher’s sixth-order Runge-Kutta method
2/3
11/120
1/3
27/40
1/12
-1/12
1/2
-1/16
9/8
-3/16
-3/8
-3/4
-4/15 1
9/44
-9/11
63/44
18/11
-16/11
1/4
3/16
7/90
1/2
1/16
32/90
12/90
3/4
-3/16
6/16 1
1/7
4/7
6/7
9/16
-12/7
8/7

25. 12.3 Multistep methods Multistep methods
앞에서의 방법론들은 단일점 x = xi 에서의 정보를 이용하여, x = xi+1의 해 yi+1의 값을 계산했다. – one step methods
단계별로 계산량이 적기 때문에 다른 방법에 비해 효율적
General form of two-step methods
a1,a2,b0,b1 등은 methods에 따라 달라진다.
앞으로는 다음과 같은 notation으로 표현하겠다.

26. 12.3 Multistep methods Multi-step methods (cont’)
Multi-step methods는 fi+1의 계수 b의 값에 따라
b = 0인 경우 명시적(explicit) 또는 개식(open) methods라 부르고,
b ≠0 인 경우에는 암묵적(implicit) 또는 폐식(closed) method라 부른다.
Multi-step methods는 시작 값을 필요로 하므로, 보통 Runge-Kutta와 같은 기존의 method를 이용해서 초기값 y1,y2,…을 구한다.

27. 12.3 Multistep methods 12.3.1 Adams-Bashforth Methods
Most popular explicit multistep methods
Two-step method formula
Local truncation error O(h3)
Three-step method formula
y0 는 주어져 있고, y1&y2 는 one-step method에서 얻는다.
Local truncation error O(h4)
Step이 올라가면 그만큼 truncation error가 줄어든다.

28. 12.3 Multistep methods 12.3.1 Adams-Bashforth Methods
Mathlab code

29. 12.3 Multistep methods 12.3.1 Adams-Bashforth Methods
Example 12.8
Y’ = x+y, y(0) = 2 x
0.2
0.4
0.6
0.8 1 y 2
2.46
3.0652
3.8509
4.8546
6.1241
(real)y
2.4642
3.0755
3.8664
4.8766
6.1548

30. 12.3 Multistep methods 12.3.2 Adams-Moultion Methods
Most popular implicit method
Two-step method formula
y0 is given, y1 is found from a one step methods
Local truncation error : O(h4)
Adams-Bashforth Methods의 local truncation error : O(h3)
Adams-Bashforth Methods에 비해서 error가 작다

31. 12.3 Multistep methods 12.3.2 Adams-Moultion Methods
Matlab code

32. 12.3 Multistep methods 12.3.2 Predictor-Corrector Methods
Procedure
각 구간에서 predictor step으로 implicit method를 사용한다
Corrector step으로 explicit method를 사용한다. (predictor가 새 점에서 문제를 예측하고 corrector가 그 정확도를 개선하는 것을 의미한다.)
Adams Third-Order Predictor-Corrector Method
y0는 초기값으로 주어지고, y1, y2는 one-step method로 구한다.
Predictor
Corrector

33. 12.4 Stability Definition Root condition :
M-step method
에 대한 characteristic polynomial
의 근을 λ1, λ2,… λm 로 나타낼때, | λi| ≤ 1, i = 1,2,…,m 이고, 절대값이 1인 모든 근이 simple root이면, root condition을 만족한다 라고 한다.
Strongly stable : root condition을 만족하고 크기가 1인 특성방정식의 유일한 근으로 λ = 1 인 방법
Weakly stable : root condition 을 만족하고 크기가 1인 근을 두개 이상 갖는 방법
Unstable : root condition 을 만족하지 않는 방법

34. 12.4 Stability Example 12.11 – Weakly Stable method
Simple two step method yi+1 = yi-1 +2hfi
Differential equation y’ = -4y, y(0) = 1 : exact solution is y = e-4x
P(λ) = λ2 – (a1λ1 + a2λ0 ) = λ2 – 1 = 0 : λ = 1, -1
ㅡ> weakly stable

35. 12.4 Stability
Example 12.11(cont’)

36. 12.5 MATLAB ’s Method
MATLAB includes three functions for solving non-stiff ODEs
ode 23, ode45
Implement a pair of explicit Runge-Kutta methods
second and third order and forth and fifth order
ode113
Fully variable step-size ODE solver based on the Adams-Bashforth-Moulton family of formulas of orders 1-12
syntax for function
[t, y] = ode23(‘F’, tspan, y0), where tspan=[t0 t_final]
‘F’: string contatining the name of an ODE file
F(t, y): must return a column vector of values
each row of solution array y corresponds to a time returned in column vector t
y0: initial conditions are given in the vector y0

37. Appendix A

38. 12.2 RUNGE-KUTTA METHOD Runge-Kutta방법의 유도 이라 할 때, (1)
여기서 실수 a, b, α, β 는 (1)를 Taylor 급수 전개하였을 때,
가능한 한 Taylor 방법과 같아 지도록 결정되어져야 할 상수이다.
이제, y(xn+1)을 투 근방에서 Taylor 급수 전개하면,
(2)
이다.

39. 12.2 RUNGE-KUTTA METHOD 한편, 2변수 함수의 Taylor전개를 이용하면, 이므로, k2를 (1)에 대입하면
을 얻는다.
이제, (1)식과 (2)식을 비교하여 h항과 h2항을 같도록 하려면
(3)

40. Truncation error 수치해 ui에 포함된 오차 ei 전체 절단오차
단계의 수가 (i+1)이라면, Xi+1에서 전체 절단오차는 각 단계 발생오차와 계산 단계 수를 곱한 것이라 할 수 있다.
로 부터 를 얻을 수 있으므로 각 단계 발생오차와 계산 단계 수를 곱하면 전체 절단 오차는 이다. →first order method