TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu Departmet of Busiess Admiistratio, Fethiye Faculty of Maagemet, Mugla Uiversity, Mugla, 48000, Turkey Received: 2 February 2015, Revised: 23 March 2015, Accepted: 2 May 2015 Published olie: 2 September 2015 Abstract: I this study, we preset a practical matrix method to fid a approximate solutio of higher order liear differece equatio with costat coefficiets uder the iitial-boudary coditios i terms of Taylor polyomials To obtai this goal, we first preset time scale extesio of previous polyomial approach, the restrict the formula to the Itegers with h step This method coverts the differece equatio to a matrix equatio, which may be cosidered as a system of liear algebraic equatios Keywords: Time Scales Calculus, Differece Equatios, Matrix Method 1 Itroductio The theory of time scales, which has recetly received a lot of attetio, was itroduced by Hilger 1] i order to uify cotiuous ad discrete aalysis This theory is appealig because it provides a useful tool for modelig dyamical processes Time scale calculus has bee created i order to uify the study of differetial ad differece equatios, so called dyamic equatios 2, 3] The cocept of dyamic equatios has motivated a huge size of research work i recet years 4, 5, 6, 7] Sice time scale calculus has mai purposes as uificatio, extesio ad discretizatio 8], it allows us to use the theory to solve a differece equatio by the methods are used to solve differetial equatios Several umerical methods were used such as the successive approximatios, Adomia decompositio, Chebyshev ad Taylor collocatio, Haar Wavelet, Tau ad Walsh series etc 9] 17] Sice the begiig of the 1994, Taylor ad Chebyshev matrix methods have also bee used by Sezer et al to solve liear differetial, Fredholm itegral ad Fredholm itegro differetial-differece equatios Sice time scale calculus has uificatio purpose, it allows us to cosider so called dyamic equatios as the uificatio of differetial ad differece equatios Also, the Taylor matrix method has bee used to fid the approximate solutios of differetial, itegral ad itegro-differetial equatios i recet years I this paper we preset the time scale aalogue of this matrix method i terms of delta derivatives ad restrict this method to Itegers with h step to solve higher order differece equatios with costat coefficiets This approach is based o the matrix relatios betwee Taylor polyomials ad their derivatives which are modified to Correspodig author e-mail: veyselfuathatipoglu@muedutr c 2015 BISKA Bilisim Techology
130 Veysel Fuat Hatipoglu: Taylor polyomial solutio of differece equatio solve the -th order liear ODE with costat coefficiets uder the iitial-boudary coditios P k y (k) (x) = g(x) (1) 1 a ik y k (a) + b ik y k (b) = λ i, i = 1,2,, 1 a ik, b ik, ad λ i are costats I the method of Taylor polyomial solutio to differetial equatios; it is assumed that the solutio ca be costructed by the Taylor polyomial form y(x) = =0 y (x c), y = y() (c),,a c b! such that the Taylor coefficiets to be determied are y for = 1,2,, I this paper, we study a matrix method to fid a approximate solutio of higher order liear differece equatio with costat coefficiets uder the iitial-boudary coditios i terms of Taylor polyomials via the time scales calculus extesio of the method I Sectio 2, we give the extesio of the matrix method for geeral time scales The solutio to a geeralized dyamic equatio of higher order with costat coefficiets We also study the accuracy of the error I Sectio 3, we cosider the dyamic equatio as a differece equatio with costat coefficiet by choosig the geeralized time scale as a particular oe, that is hz The, we obtai the matrix relatios for the preset method I sectio 4, we give our coclusios about the method 2 Time scales calculus extesio of taylor polyomial approach Let us cosider the time scales aalogue of Eq 1; ie -th order dyamic equatio with costat coefficiets: P k y k (t) = g(t) (2) k deotes k-th order delta derivative of y : T R, T is i a arbitrary time scale, ad t T κ I 2], Agarwal et al preseted Taylor polyomial for arbitrary time scales as follows: y(t) t T κ, c T for a arbitrary time scale; ad h k (t,c) ca be defied recursively as h k (t,c)y k (c), (3) c h 0 (t,c) = 1, h k+1 (t,c) = h k (τ,c) τ t c 2015 BISKA Bilisim Techology
TMSCI 3, o 3, 129-135 (2015) / wwwtmscicom 131 If we rewrite the equatio (3) i the matrix form, we may obtai the matrix relatio y(t) = H(t)Y, (4) ] H(t) = h 0 (t,c) h 1 (t,c) h (t,c) (5) ad ] T Y = y(c) y (c) y k (c) (6) O the other had, it is straightforward that the relatio betwee the matrix H(t) ad its delta derivative H (t) is H (t) = H(t)B, (7) 0 1 0 0 0 0 1 0 B = 0 0 0 0 1 0 0 0 0 0 By the meas of the matrix equatio (7), it follows that H (t) = H(t)B H 2 (t) = H (t)b = H(t)B 2 H k (t) = H(t)B k By usig the equatios (4), (7), ad delta derivative relatios of H(t), we obtai the recurrece relatios y k (t) = H k (t)y = H(t)B k Y, k = 1,2,, (8) Substitutig the equatio (8) to (2) leads us to the matrix relatio ow let us cosider the Taylor expasio of g(t): g(t) P k H(t)B k Y = g(t) (9) It is also possible to cosider this expasio as the matrix form h k (t,c)g k (c) g(t) = H(t)G, (10) c 2015 BISKA Bilisim Techology
132 Veysel Fuat Hatipoglu: Taylor polyomial solutio of differece equatio ] T G = g(c) g (c) g k (c) We are ow able to costruct the fudametal matrix equatio correspodig to the equatio (2) Substitutig the matrix relatio (10) ito the equatio (2), the simplifyig lead us to fudametal matrix equatio: { P k B k } Y = G (11) Briefly, equatio (11) ca be writte i the form of a augmeted matrix as W : G], (12) W = ω pq ] = P k B k, p,q = 1,2,, ow let us cosider matrix represetatio of the iitial boudary coditios We ca obtai the correspodig matrix form for the coditios This forms leads us to followig matrix equatios: m 1 a k y k (a) + b k y k (b) = λ i, i = 1,2,,m 1 m 1 (a ik H(a) + b ik H(a))B k Y = λ i ], i = 1,2,,m 1 We ca also cosider this equatio i the form of U i Y = λ i, i = 1,2,,m 1, (13) m 1 U i = (a ik H(a) + b ik H(a))B k, i = 1,2,,m 1 To obtai the solutio of equatio (2) uder the iitial-boudary coditios, if we replace the row matrices (13) by the last m rows of the matrix (12), the we have the ew augmeted matrix ω 00 ω 01 ω 0 : g 0 ω 10 ω 11 ω 1 : g 1 W : G] = ω m,0 ω m,1 ω m, : g m U 00 U 01 U 0 : λ 0 U 10 U 11 U 1 : λ 1 U m 1,0 U m 1,1 U m 1, : λ If rakw = rakw : G] = + 1, the Y = ( W ) 1 G, (14) c 2015 BISKA Bilisim Techology
TMSCI 3, o 3, 129-135 (2015) / wwwtmscicom 133 ad the matrix Y is uiquely determied This solutio is give by the Taylor polyomial solutio 21 Accuracy of the solutio We ca easily check the accuracy of this solutio method Sice the trucated Taylor series is a approximated solutio of the equatio (2), whe the fuctio y (t) ad its delta derivatives are substituted i equatio (2), the resultig equatio must be satisfied approximately, that is, for ξ q a,b] ad q = 1,2, Error(ξ q ) 10 ξ q { } If max 10 ξ q = 10 ξ for a positive iteger ξ is prescribed, the the trucatio limit is icreased, util the Error(ξ q ) at each of the poits becomes smaller tha the prescribed 10 ξ O the other had, the error ca be estimated by the fuctio E (t) = P k y k (t) g(t) For the sufficietly large eough, if E is approachig to zero the the error decreases 3 Taylor polyomial solutio of higher order differece equatio I this sectio, we preset approximate solutio of higher order differece equatio For this purpose, we assume the time scale is itegers with h step The, delta derivative of y(t) becomes h y(t); ie h-differece of y(t) We ca also cosider discrete aalogue of Taylor expasio as follows: y(t) ( t c k ) k h y(c) For the sake of simplicity, we assume h = 1, ad the correspodig differece equatio with costat coefficiet is P k k y(t) = g(t) (15) It is possible to costruct the iteger aalogues of the previous method The matrix relatio of the y(t) stills same with H(t) = 1 t c ( t c) ( 2 t c ) ] (16) ad ] T Y = y(c) y(c) 2 y(c) y(c) (17) Recurrece relatios of delta derivative of y(t) which is preseted i previous sectio also becomes k y(t) = k H(t)B k Y, k = 1,2,, (18) c 2015 BISKA Bilisim Techology
134 Veysel Fuat Hatipoglu: Taylor polyomial solutio of differece equatio By the equatios (15 18) ad the discrete Taylor expasio of the fuctio g(t) which is g(t) ( t c k ) k g(c), we obtai that the fudametal matrix equatio is equal to the equatio (11) 4 Coclusios I this study, we preset a ew techique to solve a differece equatio with costat coefficiets by usig the extesio property of the time scales calculus The most importat advatage of this method is that the result is obtaied i terms of arbitrary time scales Therefore, to obtai approximate solutios of various types of differece equatios with costat coefficiets oe may just eed to chage the matrix H(t) rather tha to chage whole computatio procedure Also, it is possible to apply this method to other equatios with costat coefficiets sice the method is fully dyamic Refereces 1] SHilger, Aalysis o measure chai a uified approach to cotious ad discrete calculus, Results i Mathematics, vol 18, 1-1, 1999, pp 18-56 2] M Boher, A Peterso Dyamic Equatios o Time Scales: A Itroductio with Applicatios, Birkhauser, Bosto, Mass, USA 2001 3] M Boher, A Peterso Advaces i Dyamic Equatios o Time Scales, Birkhauser, Bosto, Mass, USA 2003 4] FM Atici, G Sh Guseiov O Grees fuctios ad positive solutios for boudary value problems o time scales, Joural of Computatioal ad Applied Mathematics, vol 141, o 1-2, 2002, 75-99 5] SP Atmaca ormal ad osculatig plaes of delta-regular curves, Abstract ad Applied Aalysis Volume 2010, Article ID 923916, pp 8 doi:101155/2010/923916 6] V F Hatipoǧlu, D Uçar, ad Z F Koçak, ψ -Expoetial Stability of oliear Impulsive Dyamic Equatios o Time Scales,Abstract ad Applied Aalysis, vol 2013, Article ID 103894, 5 pages, 2013 doi:101155/2013/103894 7] SP Atmaca, Ö Akgüller, Surfaces o time scales ad their metric properties, Advaces i Differece Equatios, 2013:170 8] R Agarwal, M Boher, D Oarega, A Peterso Dyamic equatios o time scales: a survey, Joural of Computatioal ad Applied Mathematics, 141, 2002, 1-26 9] S Yalçıbaş ad M Sezer The approximate solutio of high-order liear Volterra-Fredholm itegro-differetial equatios i terms of Taylor polyomials, Appl Math Comput, 2000, 112, 291-308 10] M Gülsu ad M Sezer A method for the approximate solutio of the high-order liear differece equatios i terms of Taylor polyomials, It J Comput Math 82 (5), 2005, 629-642 11] A Karamete ad M Sezer A Taylor collocatio method for the solutio of liear itegro-differetial equatios, It J Comput Math 79 (9), 2002, 987-1000 12] B Bülbül, M Gülsu, M Sezer A ew Taylor collocatio method for oliear Fredholm-Volterra itegro-differetial equatios umer Methods Partial Diff Eq, 26, 5, 2010, 1006-1020 13] Kurt ad M Sezer Polyomial solutio of high-order liear Fredholm iteqro-differetial equatios with costat coefficiets, J Frakli Is 345, 2008, 839-850 14] RP Kawal, KC Liu A Taylor expasio approach for solvig itegral equatios, It J Math Educ Sci Techol, 20(3), 1989, 411-414 c 2015 BISKA Bilisim Techology
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