MATLAB 입문
CHAPTER 8
Numerical Calculus and Differential Equations
ACSL, POSTECH
1
Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

The Euler Method

Consider the equation
dy
 r (t ) y
dt

(8.5-1)
From the definition of derivative,
dy
y(t  t )  y(t )
 lim
dt t 0
t

r(t) : a known function
( t is small enough)
dy y (t  t )  y (t )

dt
t
(8.5-2)
Use (8.5-2) to replace (8.5-1) by the following approximation:
y(t  t )  y (t )
 r (t ) y (t )
t
y (t  t )  y (t )  r (t ) y (t )t
(8.5-3)
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

Assume that the right side of (8.5-1) remains constant over the time interval
(t , t  t ) . Then equation (8.5-3) can be written in more convenient form as
follows:
y(tk 1 )  y(tk )  r (tk ) y(tk )t
(8.5-4) , where
tk 1  tk  t
t : step size
.

The Euler method for the general first-order equation y  f (t , y) is
y(tk 1 )  y(tk )  tf [tk , y(tk )]

(8.5-5)
The accuracy of the Euler method can be improved by using a smaller step size.
However, very small step sizes require longer runtimes and can result in a large
accumulated error because of round-off effects.
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Euler method for the free response of
dy/dt = –10y, y(0) = 2
r= -10;
deltaT = 0.01 ;
% step size
t=[0:deltaT:0.5];
% our vector of time values
y(1)=2;
% initial value at t=0
for k=1: length(t)-1
% for every value in t vector
y(k+1) = y(k) + r*y(k)*deltaT;
end
y_true = 2*exp(-10*t);
plot(t,y,'o',t,y_true), xlabel('t'), ylabel('y');
4
Euler method solution for the free response of dy/dt  –10y,
y(0)  2. Figure 8.5–1
8-19
Euler method solution of dy/dt  sin t, y(0)  0. Figure 8.5–2
8-20
More? See pages 490-492.
Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

The Predictor-Corrector Method

The Euler method can have a serious deficiency in problems where the variables
are rapidly changing because the method assumes the variables are constant
over time interval t .
dy
 f (t , y ) (8.5-7)
dt
y(tk 1 )  y(tk )  tf [tk , y(tk )]

(8.5-8)
Suppose instead we use the average of the right side of (8.5-7) on the interval
(tk , tk 1 ) .
y (t k 1 )  y (t k ) 
t
( f k  f k 1 )
2
(8.5-9) , where
f k  f [tk , y(tk )]
(8.5-10)
7
Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

Let
h  t, yk  y (tk ), yk 1  y (tk 1 )
Euler predictor:
yk 1  yk  hf (tk , yk )
h
y

y

[ f (tk , yk )  f (tk 1 , yk 1 )]
Trapezoidal corrector: k 1
k
2
(8.5-11)
(8.5-12)

This algorithm is sometimes called the modified Euler method. However, note
that any algorithm can be tried as a predictor or a corrector.
 For purposes of comparison with the Runge-Kutta methods, we can express the
modifided Euler method as
g1  hf (t k , yk )
g 2  hf (t k  h, yk  g1 )
yk 1  yk 
1
( g1  g 2 )
2
(8.5-13) g1 is deltaY given by f(tk,yk)
(8.5-14) g2 is deltaY given by f(tk+1,yk+1)
(8.5-15)
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Modified Euler solution of
dy/dt = –10y, y(0) = 2
r= -10; deltaT = 0.01 ;
% same step size as before
y(1)=2;
t=[0:deltaT:0.5];
% our vector of time values
for k=1:length(t)-1
g1 = deltaT*r*y(k);
% estimate of deltaY at t(k),y(k)
g2 = deltaT*r*(y(k) + g1); % est of deltaY at t(k+1),y(k+1)
y(k+1) = y(k) + 0.5*(g1 + g2);
end
y_true = 2*exp(-10*t);
plot(t,y,'o',t,y_true), xlabel('t'), ylabel('y');
9
Modified Euler solution of dy/dt  –10y, y(0)  2. Figure 8.5–3
8-21
Modified Euler solution of dy/dt  sin t, y(0)  0. Figure 8.5–4
8-22
More? See pages 493-496.
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Numerical Methods for Differential
Equations

Runge-Kutta Methods

The second-order Runge-Kutta methods:
yk 1  yk  w1 g1  w2 g 2
g1  hf (tk , yk )
g 2  hf (tk  h, yk  hf k )

(8.5-17) , where w1 , w2 : constant weighting factors
(8.5-18)
(8.5-19)
To duplicate the Taylor series through the h2 term, these coefficients must
satisfy the following:
w1  w2  1
1
2
1
w2  
2
w1 
(8.5-19)
(8.5-19)
(8.5-19)
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

The fourth-order Runge-Kutta methods:
yk 1  yk  w1 g1  w2 g2  w3 g3  w4 g4
(8.5-23)
g1  hf (t k , yk )
g 2  hf (t k  h, yk  1 g1 )
g 3  hf [t k   2 h, yk   2 g 2  ( 2   2 ) g1 ]
g 4  hf [t k   3h, yk   3 g 2   3 g 3  ( 3   3   3 ) g1 ]
(8.5-24)
w1  w4  1 6
w2  w3  1 3
1   2  1 2
2  1 2
 3  3  1
3  0
Apply Simpson’s rule for integration
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Numerical Methods for Differential
Equations

MATLAB ODE Solvers ode23 and ode45

MATLAB provides functions, called solvers, that implement Runge-Kutta methods
with variable step size.
 The ode23 function uses a combination of second- and third-order Runge-Kutta
methods, whereas ode45 uses a combination of fourth- and fifth-order methods.
<Table 8.5-1> ODE solvers
Solver name
ode23
ode45
ode113
ode23s
ode23t
ode23tb
ode15s
Description
Nonstiff, low-order solver.
Nonstiff, medium-order solver.
Nonstiff, variable-order solver.
Stiff, low-order solver.
Moderately stiff, trapezoidal-rule solver.
Stiff, low-order solver.
Stiff, variable-order solver.
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Stiff?

Stiff Differential Equations

A stiff differential equation is one whose response changes rapidly over a time
scale that is short compared to the time scale over which we are interested in the
solution.
 A small step size is needed to solve for the rapid changes, but many steps are
needed to obtain the solution over the longer time interval, and thus a large error
might accumulate.
 The four solvers specifically designed to handle stiff equations: ode15s (a
variable-order method), ode23s (a low-order method), ode23tb (another loworder method), ode23t (a trapezoidal method).
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations

Solver Syntax
<Table 8.5-2> Basic syntax of ODE solvers
Command
Description
[t, y] = ode23( ‘ydot’, tspan, y0)
Solves the vector differential equation y  f (t , y) specified
.
in function file ydot, whose inputs must be t and y and
whose output must be a column vector representing dy dt
.
; that is, y  f (t , y) . The number of rows in this column
vector must equal the order of the equation. The vector
tspan contains the starting and ending values of the
independent variable t, and optionally, any intermediate
values of t where the solution is desired. The vector y0
contains y (t0 ) . The function file must have two input
arguments t and y even for equations where f (t , y ) is not
a function of t. The syntax is identical for the other
solvers.
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Lets try these on the problem we've been
solving
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Numerical Methods for Differential
Equations
<example> Response of an RC Circuit
R
+
v
c
y
2
1.8
.
y  10 y
(RC=0.1s, v(0)=0V, y(0)=2V)
numerical solution
analytical solution
1.6
1.4
Capacitor Voltage
dy
RC
 y  v(t )
dt
1.2
1
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
Time(s)
0.25
0.3
0.35
0.4
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Numerical Methods for Differential
Equations

Effect of Step Size

The spacing used by ode23 is smaller than that used by ode45 because ode45
has less truncation error than ode23 and thus can use a larger step size.
 ode23 is sometimes more useful for plotting the solution because it often gives a
smoother curve.

Numerical Methods and Linear Equations

It is sometimes more convenient to use a numerical method to find the solution.
 Examples of such situations are when the forcing function is a complicated
function or when the order of the differential equation is higher than two.

Use of Global Parameters

The global x y z command allows all functions and files using that command to
share the values of the variables x, y, and z.
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Extension to Higher-Order Equations

The Cauchy form or the state-variable form

Consider the second-order equation
..
.
5 y  7 y  4 y  f (t )
..
y

If
1
4
7 .
f (t )  y  y
5
5
5
x1  y
(8.6-1)
(8.6-2)
.
,
x2  y , then
.
x1  x2
1
4 7
x2  f (t )  x1  x2
5
5
5
.
The Cauchy form
or state-variable form
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Extension to Higher-Order Equations
.
<example> Solve (8.6-1) for 0  t  6 with the initial conditions y(0)  3, y(0)  9 .
( Suppose that f (t )  sin( t ) and use ode45.)
%Define the function
function xdot=example1(t,x)
xdot(1)=x(2);
xdot(2)=(1/5) * (sin(t) - 4*x(1) - 7*x(2));
xdot = [xdot(1) ; xdot(2)];
The time interval of interest
%then, use it:
[t, x] = ode45('example1',[0,6],[3,9]);
plot(t,x);
The initial condition for the vector x.
10
8
6
4
2
0
-2
.
xdot(1)
xdot(2)
x(1)
x(2)
x.1
x2
x1
x2
-4
0
1
2
3
4
5
6
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Extension to Higher-Order Equations

Matrix Methods
< a mass and spring with viscous surface friction >
y
..
.
m y  c y  ky  f (t )
(8.6-6)
.
f(t)
x1  x2
k
( Letting
m
.
x2 
c
   0
 x.1    k
 x   m
 2
.
, x2  y )
1
k
c
f (t )  x1  x2
m
m
m
(Matrix form)
.
x1  y
.
(Compact form)
1  x   0 
c   1    1  f (t )
   x2   
m
m
    f (t )
 0
 k
 m
1 
c
 
m
(8.6-7)
0
1
 m 
x 
   1
 x2 
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
25
The following is extra
credit (2 pts)
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Extension to Higher-Order Equations

Characteristic Roots from the eig Function
.
x1  3x1  x2
(8.6-8)
x2   x1  7x2
(8.6-8)
.

Substituting
x1 (t )  A1e st , x2 (t )  A2e st
sA1e st  3 A1e st  A2e st
sA2e st   A1e st  7 A2e st

Cancel the
e st
A nonzero solution will exist for A1 and A2 if and only
if the deteminant is zero
terms
( s  3) A1  A2  0
A1  ( s  7) A2  0

s  3 1
 s 2  10s  22  0
1
s7
Its roots are s = -6.7321 and s = -3.2679.
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
Extension to Higher-Order Equations

MATLAB provides the eig function to compute the characteristic roots.
 Its syntax is eig(A)
<example> The matrix A for the equations (8.6-8) and (8.6-9) is
 3 1 
  1  7



To find the time constants, which are the negative reciprocals of the real parts
of the roots, you type tau = -1./real (r). The time constants are 0.1485 and
0.3060.
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Acsl, Postech
MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Extension to Higher-Order Equations

Programming Detailed Forcing Functions
< An armature-controlled dc motor>
R
▪ Apply Kirchhoff’s voltage low and Newton’s low
L
di
  Ri  K e w  v(t )
dt
dw
I
 KT i  cw
dt
L
i
cw
+
v
I
Kew
i
-
T  KT i
Letting
w
x1  i, x2  w
:
motor’s current ,
(8.6-11)
: rotational velocity
( matrix form )
 R
 .  
 x.1    L
 x   KT
 2
 I
L: inductance, R: resistance, I: inertia, KT : torque constant,
c: viscous damping constant,
w
(8.6-10)
Ke
Ke 
1
L   x1     v(t )
c   x2   L 
 
0
I 

: back emf constant,
v(t ) : applied voltage
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MATLAB 입문 : Chapter 8. Numerical Calculus and Differential Equations
Acsl, Postech
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