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Introduction to Ordinary Differential Equations: Taylor Series and Euler Method, Slides of Mathematical Methods for Numerical Analysis and Optimization
An introduction to ordinary differential equations (odes) with a focus on taylor series and the euler method. It covers the expansion of a function using taylor series, the concept of an initial value problem, and the derivation of the euler method. The document also includes examples of applying the euler method to find approximate solutions to odes.
Typology: Slides
2011/2012
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Download Introduction to Ordinary Differential Equations: Taylor Series and Euler Method and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Ordinary
DifferentialEquations
Ordinary
DifferentialEquations
TAYLOR’S
SERIES
METHOD
We considered aninitial value problemdescribed by
0
0
(^
dy
f t y
y t
y
dt
Noting that
f
is an implicit
function of
y
, we have
( ,
)
x^
y
y
f t y
f^
f dy
y
f^
ff
x
y dx
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Similarly
2
2
2
2
(^
)^
(^
)
2
(^
)
3
3
(^
2
)
3(
)(
)
(^
)
xx^
xy^
xy^
yy^
y^
x^
y
xx^
xy^
yy^
y^
x^
y
IV
xxx
xxx
xyy
y^
xx^
xy^
yy
x^
y^
xy^
yy
y^
x^
y
y^
f^
ff
f^
f^
ff
f^
f^
ff
f^
ff
f^
f^
f^
f^
ff
y^
f^
ff
f^
f
f^
f^
ff
f^
f
f^
ff
ff
ff
f^
f^
ff
^
^
^
^
^
^
^
^
^
^
^
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Euler method is one of theoldest numerical methodsused for integrating theordinary differential equations.Though this method is notused in practice, itsunderstanding will help us togain insight into nature ofpredictor-corrector method
Consider the differentialequation of first order withthe initial condition
y
( t
) = 0
y
. 0
( ,
)
dy
f t y
dt
Here we use a property that in asmall interval, a curve is nearlya straight line. Thus at (
t^0
,^
y
), 0
we approximate the curve by atangent at that point.Therefore,
0
0
0
0
0
(^ ,
)
0
(^
,^
)
t^ y
y^
y
dy
f t
y
dt
t^ t
^
^
^
^
^
^
That is,
0
0
0
0
(^
)^
(^
,^
y
y
t^
t^
f t
y
Hence, the value of ycorresponding to
t
=
t
1
is
given by
1
0
1
0
0
0
(
)
(^
,^
)
y
y
t^
t^
f t
y
Thus, we obtain ingeneral, the solution ofthe given differentialequation in the form of arecurrence relation
1
(
,^
)
m
m
m
m
y
y
hf t
y
Geometrically, this methodhas a very simple meaning.The desired function curveis approximated by apolygon train, where thedirection of each part isdetermined by the value ofthe function
f
(
t, y
) at its
starting point.
SolutionSince the number of steps arefive, we shall proceed in stepsof (0.1)/5 = 0.02.Therefore, taking step size h
= 0.02, we shall compute the value of
y
at
t^
= 0.02, 0.04, 0.06, 0.08 and 0.
Thus
1
0
0
0
(^
,^
),
y^
y^
hf t
y
^
^
where
0
0 1,^
0
y^
t ^
1
1
0
1
1
0
y^
^
^
Therefore,
2
1
1
1
3
2
2
2
4
3
3
3
5
4
4
4
(^
)^
(^
,^
)^
(^
,^
)^
(^
,^
)^
y^
y^
hf t
y
y^
y^
hf t
y
y^
y^
hf t
y
y^
y^
hf t
y
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Similarly,