Some Applications of Generalized Taylor Series

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Mathematical Models for Engineering Science
Some Applications of Generalized Taylor
Series
M. Iltina, I. Iltins
In the work [1] it is proved that such series converges if all
(k ) (
Abraham Temkin (1919 – 2007). He proved that convolution of two
functions is to be put into series whose members are multiplication of
derivatives of one function and integrals of other function. The article
provides a numerical approach of calculating convolution, by
applying generalized Taylor series. Calculations show that
generalized Taylor series may be effectively applied for solving this
and probably other problems.
ϕ (0 ) (t ) = δ (t − a ) ,
Keywords - Taylor series, convolution, Laplace transform.
I. INTRODUCTION
(1)
a
can be put forward in the following series:
∞
∑ ξ (k ) (a ) ⋅ ϕ (− k −1) (t ) ,
II. FUNCTION EXPANDING IN GENERALIZED
TAYLOR SERIES
(2)
k =0
where
ξ (k ) (a )
is a derivative of function
Sequence of functions
recurrent formula:
Algorithms according to which a function is expanded in
generalized Taylor series is provided in formula (1) and (2). At
first, one must find Laplace transform of expandable function,
then it must be divided into factors, original of each factor
must be found and one of these factors is function ϕ(0)(t), the
second is function ξ(0)(t). Briefly, it can be written as follows:
ξ (t ) in point a.
ϕ (− k −1) (t ) may be obtained with a
F (s ) = L[ f (t )] ,
F (s ) = F1 (s ) ⋅ F2 (s ) ,
ϕ (0 ) (t ) = L−1[ F (s )] , ξ (0 ) (t ) = L−1[F (s )].
t
ϕ (− k ) (t ) = ∫ ϕ (− k +1) (τ )dτ , k=1, 2, 3,...
1
(3)
a
2
Example
Let us expand function f(t)=t⋅sint into generalized Taylor
series:
L[ f (t )] =
Manuscript received September 23, 2010.
Marija Iltina is prof. assistant, Dr.sc.ing. at Riga Technical University,
Faculty of Computer Science and Information Technology, 1/ 4 Meza Street,
Riga, LV-1048, Latvia. (phone: +371-67089528, e-mail: [email protected]).
Ilmars Iltins is asoc.professor, Dr.sc.ing. at Riga Technical University
Faculty of Computer Science and Information Technology, 1/ 4 Meza Street,
Riga, LV-1048, Latvia. (phone: +371-67089528, e-mail: [email protected]).
ISBN: 978-960-474-252-3
(4)
A.Temkin applied generalized Taylor series in his work [2]
in order to separate influence of initial conditions and external
impact on non-stationery temperature field of solid body.
Besides, this work included methods for solving various
thermal conductivity inverse problems based on application of
generalized Taylor series in solving thermal conductivity
equation. These series have also been applied for solving
different problems in later papers [3], [4]. In general,
application possibilities of discussed series in solving different
problems are not research enough yet.
t
f (t ) =
)
δ - Dirac delta function.
There is demonstrated in paper [1] how a function f(t), defined
by a convolution integral
f (t ) = ∫ ϕ (0 ) (τ ) ⋅ ξ (0 ) (a + t − τ )dτ ,
(− k ) (
)
a and integrals ϕ
t are limited.
derivatives ξ
A. Temkin named these series as generalized Taylor series
because Taylor series of function f(t) in environment of point a
is a special case of these series when
Abstract - Notion „generalized Taylor series” was introduced by
F2 (s ) =
101
(s
2s
2
1
2
)
+1
s +1
.
2
,
F1 (s ) =
2s
s2 + 1
,
Mathematical Models for Engineering Science
Further results of numerical modeling will be provided that are
based on aforesaid example. A table was established from
It follows hereof that
ϕ (0 ) (t ) = L−1[F2 (s )] = sin t ,
ξ (0 ) (t ) = L−1 [F (t )] = 2 cos t .
function
1
ϕ (0 ) (t )
with interval
t ∈ [0;2π ] step
π
10
.
The table is as follows:
{0.,0.309017,0.587785,0.809017,0.951057
,1.,0.951057,0.809017,0.587785,0.309017
,0.,-0.309017,-0.587785,-0.809017,0.951057,-1.,-0.951057,-0.809017,0.587785,-0.309017,0.}
ξ (0 ) (t )
It is obvious that functions
and
could
have been also chosen vice versa.
Series partial sums are as follows after their simplification,
defining that a=0:
S 0 (t ) = 2 − 2 cos t ,
Having applied this table and generalized
Taylor
series,
values
of
function
S 2 (t ) = 4 − t − 4 cos t ,
2
t
f (t ) = ∫ ϕ (t )ξ (t − τ )dτ
4
t
− 6 cos t ,
12
4
t6
2 t
S 6 (t ) = 8 − 3t + −
− 8 cos t .
6 360
Figure 1 shows difference S 6 (t ) − f (t ) .
S 4 (t ) = 6 − 2t 2 +
ϕ (t ) = sin t
will
0
t ∈ [0;2π ]
interval
and step
be
π
10
found
at
. Assuming that
ξ(t)=2cost and ϕ(0)(t)=sint, result must be close to
corresponding
f(t)=tsint ≈
1
∞
values
( )
(
∑ (2 cos t ) (sin t )
k
of
− k −1)
function
= S N (t ) .
k =0
0.5
1
2
3
4
5
6
The following tables show the result of integration of the
previous table. As upper boundary of integration is variable,
result of integration is a table of the same size like the given
one. After three integrations one obtains:
-0.5
-1
-1.5
-2
{0,0.00119768,0.00741751,0.0326452,0.09
94274,0.234486,0.468298,0.831067,1.3500
7,2.04694,2.93558,4.02075,5.29766,6.752
41,8.36333,10.1031,11.9416,13.8485,15.7
965,17.764,19.7371}
-2.5
-3
Figure 1
Difference between the sixth partial sum of function
series and function
After five integrations one obtains:
Calculations with various functions show that total tendency
is that for approximation of a function with its partial sum at
higher t values one should use a larger number of partial sum
addends.
{0,0.0000295517,0.000212683,0.00133013,
0.00597051,0.0209306,0.059846,0.146013,
0.31549,0.619681,1.12749,1.92667,3.1243
,4.84627,7.23601,10.4523,14.6665,20.059
9,26.8204,35.1401,45.2131}
8
After nine integrations one obtains:
III. CALCULATION OF CONVOLUTION IF ONE OF THE
FUNCTIONS INCLUDED THEREIN IS SET BY A TABLE
0, 1.79912 ΄ 10- 8 , 1.41076 ΄ 10- 7 , 1.20605 ΄ 10- 6 , 7.91401 ΄ 10- 6 ,
0.0000441096, 0.000208458, 0.000840931, 0.00293814,
0.00904963, 0.02501, 0.063012, 0.146694, 0.319097, 0.654536,
1.2756, 2.37671, 4.25568, 7.35513, 12.3153, 20.0401
Some of the functions included in a convolution may be set
with a table, for instance, if it is acquired in the result of
measurements. In such a case one requires to approximate the
table that causes hardly assessable influence of approximation
error onto result of convolution. In this case one can use
generalized Taylor series for calculating convolution, by
choosing a table as function ϕ and choosing an analytically set
function as function ξ. It is possible to integrate the table, by
applying standard programs built-in mathematics software
whereas derivation never cases any problems.
ISBN: 978-960-474-252-3
<
As series form as linear combination of these tables whose
coefficients are 2, -2 and 0, one sees that partial sum cannot
change significantly at
102
 π
t ∈ 0;  .
 2
Mathematical Models for Engineering Science
The below table shows difference between values of partial
sum
S 9 (t )
and step
π
10
and function f(t)=tsint at interval
t ∈ [0;2π ]
:
{0.,-0.00233769,-0.00174997,0.000660865,-0.000470007,0.0000182753,0.000325384,0.000610813,0.
00104655,0.00208954,0.00498201,0.012796
3,0.0325983,0.0795091,0.183841,0.402957
,0.840041,1.67258,3.19399,5.87241,10.43
13}
It is evident that this difference at
t=
π
is -0.00233769, but difference is 0.00208954, at
10
9π
t=
, thus approximately equal following a module. This
10
difference increases at higher t values. Therefore one may
consider that the ninth partial sum approximates the given
function at interval
 9π 
t ∈ 0; 
 10 
with maximum possible
accuracy and the figures given in the last table that comply
with t values
 9π 
t ∈ 0; 
 10 
are inevitable errors whose cause
is replacement of a function by its table, but it is possible to
approximate values of a function that comply with values of
argument higher than
9π
10
better with generalized Taylor
series, by taking a larger number of partial sum elements.
IV. CONCLUSIONS
That far generalized Taylor series has been applied less for
solving applied problems. The discussed example shows that it
is applicable for interpretation and processing of real
experiment data. One can forecast assuredly that many other
problems still exist for solving which these series might be of
efficient use.
REFERENCES
[1]
[2]
[3]
[4]
А. Г. Темкин. Обобщенный ряд Тейлора и теорема умножения
изображений. В кн. Сборник научных трудов. Куйбышев,
Куйбышевский индустриальный институт, 1956.
А. Г. Темкин. Обратные методы теплопроводности. M., Энергия,
1973, 464 c.
A. Temkin, J. Gerhards, I. Iltins. The Temperature Field of Cable
Insulation. Latvian Journal of Physics and Technical Sciences. 2005, Nr
2, p 12-26.
A. Temkin, J. Gerhards, I. Iltins. Decomposition of Potential and
Temperature Fields for a Complex – Shaped Body: Simplest NonLinearity Case. Latvian Journal of Physics and Technical Sciences.
2003, Nr 2, p 17-33.
ISBN: 978-960-474-252-3
103
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