2. CISE301_Topic8L1 2
Objectives of Topic 8
Solve Ordinary Differential Equations
(ODEs).
Appreciate the importance of numerical
methods in solving ODEs.
Assess the reliability of the different
techniques.
Select the appropriate method for any
particular problem.
3. CISE301_Topic8L1 3
Outline of Topic 8
Lesson 1: Introduction to ODEs
Lesson 2: Taylor series methods
Lesson 3: Midpoint and Heun’s method
Lessons 4-5: Runge-Kutta methods
Lesson 6: Solving systems of ODEs
Lesson 7: Multiple step Methods
Lesson 8-9: Boundary value Problems
5. CISE301_Topic8L1 5
Learning Objectives of Lesson 1
Recall basic definitions of ODEs:
Order
Linearity
Initial conditions
Solution
Classify ODEs based on:
Order, linearity, and conditions.
Classify the solution methods.
6. CISE301_Topic8L1 6
Partial Derivatives
u is a function of
more than one
independent variable
Ordinary Derivatives
v is a function of one
independent variable
Derivatives
Derivatives
dt
dv
y
u
7. CISE301_Topic8L1 7
Partial Differential Equations
involve one or more
partial derivatives of
unknown functions
Ordinary Differential Equations
involve one or more
Ordinary derivatives of
unknown functions
Differential Equations
Differential
Equations
1
6
2
2
tv
dt
v
d 0
2
2
2
2
x
u
y
u
8. CISE301_Topic8L1 8
Ordinary Differential Equations
)
cos(
)
(
2
)
(
5
)
(
)
(
)
(
:
2
2
t
t
x
dt
t
dx
dt
t
x
d
e
t
v
dt
t
dv
Examples
t
Ordinary Differential Equations (ODEs) involve one or
more ordinary derivatives of unknown functions with
respect to one independent variable
x(t): unknown function
t: independent variable
9. CISE301_Topic8L1 9
Example of ODE:
Model of Falling Parachutist
The velocity of a falling
parachutist is given by:
velocity
v
t
coefficien
drag
c
mass
M
v
M
c
t
d
v
d
:
:
:
8
.
9
13. CISE301_Topic8L1 13
Order of a Differential Equation
1
)
(
2
)
(
)
(
)
cos(
)
(
2
)
(
5
)
(
)
(
)
(
:
4
3
2
2
2
2
t
x
dt
t
dx
dt
t
x
d
t
t
x
dt
t
dx
dt
t
x
d
e
t
x
dt
t
dx
Examples
t
The order of an ordinary differential equation is the order
of the highest order derivative.
Second order ODE
First order ODE
Second order ODE
14. CISE301_Topic8L1 14
A solution to a differential equation is a function that
satisfies the equation.
Solution of a Differential Equation
0
)
(
)
(
:
t
x
dt
t
dx
Example
0
)
(
)
(
)
(
:
Proof
)
(
t
t
t
t
e
e
t
x
dt
t
dx
e
dt
t
dx
e
t
x
Solution
17. CISE301_Topic8L1 17
Solutions of Ordinary Differential
Equations
0
)
(
4
)
(
ODE
the
to
solution
a
is
)
2
cos(
)
(
2
2
t
x
dt
t
x
d
t
t
x
Is it unique?
solutions.
are
constant)
real
a
is
(where
)
2
cos(
)
(
form
the
of
functions
All
c
c
t
t
x
18. CISE301_Topic8L1 18
Uniqueness of a Solution
b
x
a
x
t
x
dt
t
x
d
)
0
(
)
0
(
0
)
(
4
)
(
2
2
In order to uniquely specify a solution to an nth
order differential equation we need n conditions.
Second order ODE
Two conditions are
needed to uniquely
specify the solution
19. CISE301_Topic8L1 19
Auxiliary Conditions
Boundary Conditions
The conditions are not at
one point of the
independent variable
Initial Conditions
All conditions are at one
point of the independent
variable
Auxiliary Conditions
20. CISE301_Topic8L1 20
Boundary-Value and
Initial value Problems
Boundary-Value Problems
The auxiliary conditions are
not at one point of the
independent variable
More difficult to solve than
initial value problems
5
.
1
)
2
(
,
1
)
0
(
2 2
x
x
e
x
x
x t
Initial-Value Problems
The auxiliary conditions
are at one point of the
independent
variable
5
.
2
)
0
(
,
1
)
0
(
2 2
x
x
e
x
x
x t
same different
21. CISE301_Topic8L1 21
Classification of ODEs
ODEs can be classified in different ways:
Order
First order ODE
Second order ODE
Nth order ODE
Linearity
Linear ODE
Nonlinear ODE
Auxiliary conditions
Initial value problems
Boundary value problems
23. CISE301_Topic8L1 23
Numerical Solutions
Numerical methods are used to obtain a
graph or a table of the unknown function.
Most of the Numerical methods used to
solve ODEs are based directly (or
indirectly) on the truncated Taylor series
expansion.
24. CISE301_Topic8L1 24
Classification of the Methods
Numerical Methods
for Solving ODE
Multiple-Step Methods
Estimates of the solution
at a particular step are
based on information on
more than one step
Single-Step Methods
Estimates of the solution
at a particular step are
entirely based on
information on the
previous step
