Numerical Solution of Ordinary Diff Eqns
• Intro to numerical solution of Ordinary Differential
Equations (ODEs):
• RungeKutta methods
• MATLAB ODE solvers (sec 10.4)
• Solving systems of firstorder ODEs w/ MATLAB
omment on Lab 14 Part VI
•
Comment on Lab 14, Part VI
• Examples of systems of ODEs
•2
nd
order ODEs as a system
of 1
st
order ODEs
y
•
PredatorPrey problem
•
Boundary Value Problems
4/14/2009
1
MASE201 Spring 2009
Solving a
system of ODEs
he same methods used to solve a single ODE can
•
The same methods used to solve a single ODE can
be used to solve a system of ODEs
• There is one independent variable and many (
n
)
dependent variables with one ODE per dependent
variable (denoted
j
):
he numerical solution of the system can be stated
n
j
j
j
y
y
y
y
t
f
dt
dy
,
,
,
,
,
,
2
1
•
The numerical solution of the system can be stated
as:
:
that
such
)
(
find
,
to
1
For
t
y
n
j
j
(0)
conditions
initial
the
with
)
,
,
...
,
,
(
o
j
j
2
1
y
y
t
y
y
y
f
dt
dy
n
j
4/14/2009
2
MASE201 Spring 2009
]
,
0
[
domain
the
over
final
t
Solving a system of ODEs in MATLAB
he built
MATLAB functions can solve a system of
•
The builtin MATLAB functions can solve a system of
ODEs. The syntax is the same as for a single
ODE,except that
yo
becomes a vector
, rather than a
scalar and odefun must return a column vector
with
the values of
f
1
,
f
2
etc.
•
a function m
e:
In a function mfile:
function dyvecdt = odesysfun(x,y)
k=10; m=1;
dyvecdt(1,1)= yvec(2);
%velocity is dx/dt
dyvecdt(2,1)= k/m*yvec(1);
%acceleration is dv/dt
•
n the MATLAB command line:
On the MATLAB command line:
>> [t,y]=ode45(@odesysfun,[0 10],[0 3]);
% Note that t is a column vector of some length n and y is an n x 2
4/14/2009
3
% matrix. Each column contains the values of one dependent variable
% (x and v respectively)
MASE201 Spring 2009
Initial Value Problems and Boundary Value Problems
hen solving an ODE arbitrary constants
• When solving an ODE, arbitrary constants
appear in the solution.
or a second
rder ODE or higher it is
• For a secondorder ODE or higher, it is
important to distinguish between initial value
nd boundary value problems
and boundary value problems
– When the independent variable is time, the ODE
is called an initial value problem
and we need to
specify
y
(0), d
y
/d
t
(0)
etc.
– When the independent variable is position (e.g.)
e
DE is called a
oundary value
roblem
nd
the ODE is called a boundary value problem
and
we can specify some combination
the value of
the dependent variable or its derivatives at
ifferent values of
4/14/2009
4
different values of
x
MASE201 Spring 2009
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentReducing a secondorder ODE
second
rder ODE can be reduced to a system of
A secondorder ODE can be reduced to a system of
two first order ODEs as shown for the example below:
2
v
dt
dx
kx
dt
x
d
m
Let
0
2
x
m
k
dt
dv
or
kx
dt
dv
m
0
Then
Write the functions
f
1
and
f
2
that are functions of the
independent variable
t
and the dependent variables
x
and
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Boundary value problem, BVP, MASE201 Spring

Click to edit the document details