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MECH201 Engineering Analysis
(2009)
p.
11
The central difference scheme is more accurate than the other twos as it is of
order
Δ
x
2
.
See figures 23.1 to 23.3 for higher order representation of derivative of a function.
The higher order or accuracy can be achieved by using more points in the
neighborhood of
x
i
.
Note that
1
i
i
x
x
x
±
=
± Δ
.
For second derivative of f(x), consider two additional points on each side of x
i
:
1
1/ 2
1/ 2
1
1
1
;
;
;
&
2
2
i
i
i
i
i
i
i
i
i
x
x
x
x
x
x
x
x
x
x
x
x
x


+
+
=
 Δ
=

Δ
=
+
Δ
=
+ Δ
Using central difference scheme:
1
1/ 2
i
i
i
f
f
df
dx
x



=
Δ
and
1
1/ 2
i
i
i
f
f
df
dx
x
+
+

=
Δ
2
1
1
2
i
i
i
df
df
d f
dx
dx
dx
x
+


=
Δ
1
1
2
1
1
2
2
2
i
i
i
i
i
i
i
i
f
f
f
f
f
f
f
d f
x
x
dx
x
x
+

+





+
Δ
Δ
=
=
Δ
Δ
Numerical differentiation of experimental data can easily lead to error if the data
is not smooth.
As slight deviation of the data point from a smooth curve will
render large error in the differentiated results.
See Figure 23.5 for graphical
explanation.
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(2009)
p.
12
4
SOLUTION OF ORDINARY DIFFERENTIAL
EQUATION.
4.1
The Ordinary Differential Equation
Pages 697705
The ordinary differential equation (ODE) is commonly found in the governing
equations of many engineering system.
For example, the damped simple harmonic motion equation:
2
2
0
d x
dx
m
c
kx
dt
dt
+
+
=
It can be in the form of a system of equations.
In the case of a 2D projectile we
have two equations:
2
2
d y
m
g
dt
dx
m
V
dt
= 
=
The word "Ordinary" that precedes the "Differential Equation" Implies that the
differentiations
are
all
"total
differentiation"
In
contrast
to
"partial
differentiation" which will be the topic In latter sections.
We shall see that the
methods of solving ordinary differential equations Is different to that used to
solve partial differential equations.
The aim of this section is to consider four commonly used methods for solving
ODEs.
These are: the Euler method, the Huen method, the midpoint method
and the RungKutta method.
4.2
The Euler Method
Section 25.1
Consider an ordinary differential equation:
( )
,
dy
f y x
dx
=
and the value of
y
at
x = 0
is given, that is
y(0) = y
0
.
To find the value of
y
at
x = x
1
where
x
1
x =
Δ
x
We may use the Taylor’s series to obtain the value of
y(
x)
form
y(0):
( ) ( )
2
2
2
0
0
1
0
....
2!
dy
d y
y
x
y
x
x
dx
dx
Δ
=
+
Δ +
Δ
+
that is:
( ) ( ) ( )
2
0
0
1
0
0,
....
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This note was uploaded on 10/27/2009 for the course MECH 201 taught by Professor Weihuali during the Three '09 term at University of Wollongong, Australia.
 Three '09
 WeihuaLi

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