MATLAB ODE

Using ODEs in MATLAB

Prepared By: Nahla Al Amoodi17th September 2007

ODEs

The initial value problem for an ordinary

differential equation involves finding a

function y(t) that satisfies:

With the initial condition y(t0)=y0

A numerical solution generates a sequence of values for the

independent variable, and a corresponding sequence of values for

the dependent variable, such that each yn approximates the solution

at tn.

Euler’s Method

Works mainly for simple first order differential

equations.

Consider the following equation:

and y(0) given

This differential can be replaced with its

Newton quotient:

),( yxfdx

dy

Euler’s Method

),()()(

yxfh

xyhxy

)1(),()()( yxhfxyhxy

),(1 iiii yxhfyy

Rearranging:

Developing an iterative scheme:

Where )( ii xyy

Euler’s Method

such that at the

beginning of step i,

Suppose that the ode is to be integrated over the

interval from x=a to x=b. This interval is divided

into m steps of length h,

h

abm

Where )( ii xyy

)0()1( 1 yyandhixi

Euler’s Method

The main disadvantage of this method

is that it is dependant on the step size

m. Errors are greatly reduced if the

step size is increased.

Example: Bacteria Growth

A colony of coliform bacteria is

multiplying at the rate of r=0.8. If the

initial count shows 1000 colonies,

determine how many colonies exists in

10 hours.

Solution

Defining:

N as the colony size.

r as the growth rate.

t as the time.

The rate equation can be given as:

1000)0( NrNdt

dN

Numerical solution by applying Euler’s

Algorithm:

Analytical solution; an approximate of the

exact solution:

iii rNhNN 1

rteNtN )0()(

Solving by Euler’s method in

MATLAB

The following code can be used to solve using Euler’s method and to compare with the exact solution for error observation:

h=0.5; %defining the interval length

r=0.8;

a=0;

b=10;

m=(b-a)/h; %estimating the number of steps

N(1)=1000; %N(0)

Euler’s code (cont)

t=a:h:b; %defining the interval with the step h

for i=1:m

N(i+1)=N(i)+r*h*N(i); %applying Euler's estimation

end

Nex=N(1)*exp(r*t); %calculating the exact answer from the analytical solution

disp([t' N' Nex'])

plot(t,N),xlabel('Hr'),ylabel('Bacteria'), title('Bacteria Growth')

Notice the huge error between the

numerical and the analytical solution.

The error can be reduced by reducing

h of the numerical solution.

ODE built-in solvers in

MATLAB

ODE built in solvers depend on the

type and complexity of the system

being solved and the accuracy

needed.

The frequently used solvers are the

ode23 and ode45 referring to

second/third order and fourth/fifth

order truncations of the tailor series

respectively.

To use these solvers a number of easy

steps must be clearly defined to MATLAB:

1. Create a function in an m file to define the

right hand side of the equation to be

solved.

2. Determine the interval length for the

independent variable or tspan vector

3. Enter the initial conditions n0.

4. Call the solver to obtain the solution by

typing the following command

[t,y]=ode23(@functionName, tspan, n0)

The left hand side of the above expression

is the output argument containing two

vectors; t the independent variable and y the

dependant variable. Other solvers use

similar syntax.

Example: Bacteria Growth-

Solving a first order equation

Define the function in an m file called

bacrate.m

function y=f(t,N)

y=0.8*N;

In another m-file or in command

window type the following:

a=0;

b=10;

n0=1000; %N(0)

tspan=[a b]; % defining the interval of integration

[t,N]=ode23(@bacrate, tspan, n0)

Nex=n0*exp(0.8*t); %calculating the exact answer from the analytical solution

disp([t' N' Nex'])

plot(t,N,'*'),xlabel('Hr'),ylabel('Bacteria')

, title('Bacteria Growth')

hold on

plot(t,Nex,'r'),hold off

Second order equations

The first step in solving a second (or

higher) order ordinary differential

equation in MATLAB is to write the

equation as a first order system.

Suppose y1 (x), y1’ (x), & y1

’’ (x) are

functions of x and y.

Setting these equations into a first

order system and defining initial

conditions:

y(x)= y1(x) y(0)=a

y’(x)=y2 (x), and y’(0)=b

y’’(x)=y2’ (x)

The second step is to enter the

derivative equations as column vectors

in MATLAB, then call the solver to

obtain the solution.

Example

Solve the following second order ODE

using ode45 solver:

At y(0) = 2 and t at the range

y'(0) = 4 0 to 5

Then plot t versus y.

teyyy ''' 3

Step 1-Set to a 1st order ODE

Set y=y1

y’=y2

y’’=y2’

This results in the following system:

Rearranging:

teyyy

yy

12

'

2

2

'

1

3

2

'

112

'

2 3 yyyyey t

Step 2: Create m-files

In an m-file enter the derivative

equations as a column vector and

save it:

function dy=secondode(t,y)

dy=[y(2);exp(t)-3*y(2)+y(1)];

In the command window call the

solver:

[t,y]=ode45('secondode',[0,5],[2,4]); %

having tspan=[0 5] and initial

conditions y(0)=2, y’(0)=4

plot(t,y)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70

EXERCISES

A fluid of constant density starts to flow

into an empty and infinitely large tank

at 8 L/s. A valve regulates the outlet

flow to a constant 4L/s. Derive and

solve the differential equation

describing this process over a 100

second interval.

Solving systems of ODEs

Systems of ODEs can be entered in an

m-file using the same procedure of 2nd

order ODE, however the initial values

are entered as a row vector.

System of 1st oder ODEs will be

discussed in this tutorial.

Solving systems of ODEs

Solve the following system:

At x(0)=-8, y(0)=8, z(0)=27, then plot z

versus x.

3

224

3

8

1010

xyzdt

dz

xzydt

dy

yxdt

dx

Solution

Set x as g(1)

y as g(2)

z as g(3)

Then enter the ODEs in an m-file:

function dg= lorenz(t,g)

dg=[-10*g(1) + 10*g(2); -g(2) - g(1)*g(3); -

8/3*g(3) + g(1)*g(2) – 224/3];

In the command window type:

>>tspan=[0,50];

>>g0=[-8;8;27];

>>[t,g]=ode45(@lorenz,tspan,g0);

>>plot(g(:,1),g(:,3))

-20 -15 -10 -5 0 5 10 15 20-30

-20

-10

0

10

20

30Z versus g

g

z

Stiffness

ODEs methods discussed use a step

value h over a given interval to

estimate a numerical solution for the

ode.

For some odes even such solvers

would require a very small

(infinitesimal) step value to converge.

Such equations are called “stiff”.

MATLAB has built-in solvers to deal

with stiff odes such as ode15s,

ode23s.

For a system of odes stiffness may

also occur if one or more of the

equations is converging too fast while

another equation (or more) is

converging too slowly.

Passing Additional Parameters

Letter parameters can be written to

generalize the model represented by

odes. Consider the system:

having

x(0)=105

y(0)=8

p=0.4, q=0.04

r=0.02, s=2

syrxydt

dy

qxypxdt

dx

Create an m-file by entering the ODEs:

function dx=ques(t,x,p,q,r,s)

dx=[p*x(1)-q*x(1)*x(2); r*x(1)*x(2)-

s*x(2)];

In another mfile or in the command

window type the following

p=0.4;q=0.04;r=0.02;s=2;

tspan=[0 10];

X0=[105;8];

[t,x]=ode23(@ques,tspan,X0,[],p,q,r,s);

plot(t,x)

xlabel('t')

ylabel('x,y')

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

t

x,y

Note that the term [] have been used.

The reason being the more general

format of calling a solver:

[t,x]=solver(handle, tspan, initial

conditions, options, additional

parameters)

When using additional parameters, the

above format must be used and

‘options’ can’t be omitted.

Using ‘[]’ will tell MATLAB to hold the

place for the options parameter, but

since nothing has been defined; the

default MATLAB options will be used

to obtain the solution.