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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