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MatLab Programming Algorithms to Solve

Differential Equations

MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

x1 x2 x3

x

y(x)1

23 5

4

Adapted from Figure 16.1.2. In Numerical Recipes in C:The Art of Scientific Computing.

2nd Ed. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.

Cambridge, UK: Cambridge University Press, 1992. p. 711. ISBN: 9780521431088. Figure by MIT OCW.

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.

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Revisit the task of recovering the motion of a

dynamical system from its equation of motion

Consider the simplest 1st order system:

0=+ kxxb&

What does this system corresponds to?

The solution of this system can of course be

obtained analytically but also simply numerically

by a single integrationCite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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How did we solve this class of problems?We use a very simple straight forward approach ofdoing numerical integration:

0 1 2 3 4 5 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

x(t)

y(t)

Projectile trajectory with x0=0, y0=0, vx0=5, vy0=5

X=0; y=0; vx=5;vy=5;t=0;dt=0.1

Y=

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The General Numerical Problem of Solving

Ordinary Differential Equations (ODEs)

),,',,( )1()( tyyyfy nn L=

Note that y does not have to be a scaler

but can be a vector as in the case formultiple degrees of freedom systems

),,( 21 myyyy L=

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Converting higher order differential equation to

a system of first order differential equation

Consider probably the most important case:

)()(')('' thytgytfy ++=

This can be readily converted to a system

of first order differential equations

)()()('

'

'

122

21

12

thytgytfy

yy

yyyy

++=

=

==


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General equivalence between higher order differential

equation and a system of first order equations

),,,,('

;,;';

),,',,(

12

)1()2(121

)1()(

tyyyfy

yyyyyyyy

tyyyfy

nn

nn

nn

nn

L

L

L

=

====

=

The problem of solving all higher order ordinary

differential equation is thus reduced to solving a systemof linear differential equations


S l i li fi t d diff ti l ti

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Solving linear first order differential equationby Euler Method

nityyyfty nii LL 1),,,,()(' 21 ==

In general, the system of equations look like:

Euler Method says:

ni

ttjtjytjyftjytjy niii

L

L

1

))1(),)1((,),)1((())1(()( 1

=

+=

This equation can be solved if we have theinitial conditions:

0202101 )0(,,)0(,)0( nn yyyyyy === L


f ?

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What is the accuracy of the Euler Method?

Euler method is equivalent to taking the

1st order Taylor series expansion for yi(t);

it is unsymmetric and uses only the derivative

information at the start of the time step

ni

tOttjtjytjyftjytjy niii

L

L

1

)())1(),)1((,),)1((())1(()( 21

=

++=

Correction is only one less order then the correction term.

How do we get better accuracy?


S h ti ll th diff b t

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Schematically, the differences between

Euler and 2nd order Runge-Kutta are fairly clear

Euler Method

2nd order

Runge-Kutta

From Numerical Recipe in CCite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

x1 x3x2

x

y(x) 1

2

x1 x3x2

x

y(x)

4

5312

Adapted from Figure 16.1.1. InNumerical Recipes in

: The Art of Scientific Computing.

nd Ed. W. H. Press, S. A. Teukolsky, W. T. Vetterling,nd B. P. Flannery. Cambridge, UK: Cambridge

niversity Press, 1992. p. 711. ISBN: 9780521431088.

igure by MIT OCW.

C

2a

U

F

Adapted from Figure 16.1.1. InNumerical Recipes inC: The Art of Scientific Computing.

2nd Ed. W. H. Press, S. A. Teukolsky, W. T. Vetterling

and B. P. Flannery. Cambridge, UK: Cambridge

niversity Press, 1992. p. 711. ISBN: 9780521431088.

Figure by MIT OCW.

U

A ki ODE l R K tt M th d

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A working ODE solver Runge-Kutta Method

ni

ktjytjy

ttjktjyktjyfk

ttjtjytjyfk

iii

nnii

nii

L

L

L

1

2))1(()(

))2/1(),2/1)1((,,2/1))1(((2

))1(),)1((,),)1(((1

11

1

=

+=

++=

=

Estimate where the mid-point for yi is first. Then,

Evaluate the slope at the mid-point to estimate

the next value of yi.

Because of symmetry, this method is good to

O(t3)

This is called the 2nd order Runge-Kutta method.Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

Hi h O d R K tt M th d

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Higher Order Runge-Kutta Method

Just like Simpson method can be extended to higher

order estimate, Runge-Kutta also has straightforwardHigher order analog. The most commonly used

one is the 4th order Runge-Kutta method

ni

tOkkkktjytjy

ttjktjyktjyfk

ttjktjyktjyfk

ttjktjyktjyfk

ttjtjytjyfk

iiiiii

nnii

nnii

nnii

nii

L

L

L

L

L

1

)(6/43/33/26/1))1(()(

)),3)1((,3))1(((4

))2/1(),2/2)1((,,2/2))1(((3

))2/1(),2/1)1((,,2/1))1(((2

))1(),)1((,),)1(((1

5

11

11

11

1

=

+++++=

++=

++=

++=

=

Runge-Kutta methods are implemented in MatLabas ODE23 and ODE45 functionsCite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

U i M tL b t l t f diff ti l ti

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Using MatLab to solve a system of differential equations

Consider solving the following system of ODE:


y'1= y2y3

y'2= -y1y3

y'3= -0.51y1y2

y1(0) = 0

y2(0) = 1

y3(0) = 1

Adapted from MATLAB Help Sections. Figure by MIT OCW.


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(1) First define the system of ODEs as a function:function dy = system(t,y)

dy = zeros(3,1); % a column vector

dy(1) = y(2) * y(3);

dy(2) = -y(1) * y(3);

dy(3) = -0.51 * y(1) * y(2);

(2) Call ODE45 or ODE23 using the function handle

[T,Y] = ode45(@system,[0 12],[0 1 1]);

(3) Plot result

plot(T,Y(:,1),'-',T,Y(:,2),'-.',T,Y(:,3),'.')Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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