A novel approach to processing fractal dynamical systems using the Yang-Fourier transforms

Transcription

1 From the SelectedWorks of Xiao-Ju Yag March A ovel approach to procesg fractal damical sstems ug the Yag-Fourier trasforms Yag Xiaoju Available at:

2 Advaces i Electrical Egieerig Sstems (AEES) 35 Vol. No. 3 ISSN X Copright World Sciece Publisher Uited States A Novel Approach to Procesg Fractal Damical Sstems Ug the Yag-Fourier Trasforms Xiao-Ju Yag Meg-Ke Liao 3 Jia-Nig Wag * Departmet of Mathematics ad Mechaics Chia Uiversit of Miig ad Techolog Xuzhou Campus Xuzhou Jiagsu 8 P. R. Chia College of Water Coservac Shihezhi Uiversit Shihezhi Xijiag 833 P. R. Chia 3 School of Mathematics ad Sstems Sciece Sheag Normal Uiversit Sheag Liaoig 34 P. R. Chia dagiaoju@63.com Abstract I the preset paper local fractioal cotiuous o-differetiable fuctios i fractal space are ivestigated ad the cotrol method for procesg damic sstems i fractal space are proposed ug the Yag- Fourier trasform based o the local fractioal calculus. Two illustrative paradigms for cotrol problems i fractal space are give to elaborate the accurac ad reliable results. Kewords No-differetiable fuctios Fractal damical sstems Yag-Fourier trasforms Fractal space Local fractioal calculus. Itroductio As is well-kow Fourier aalsis is oe of the most frequetl used tools i cotrol procesg. A sstem is defied as a fuctio or a equatio. However some damical sstems fuctios are fractal curves which are everwhere cotiuous but owhere differetiable [-9]. As a result we caot emplo the classical Fourier aalsis which requires that the defied fuctios should be differetiable to describe the damical states i fractal space. Recetl local fractioal calculus (fractal calculus) which was dealig with fractal fuctios had bee proposed ad developed i[-]. For these merits local fractioal calculus was successfull applied i the fractal elasticit [3 the fractal release equatio [7] fractal wave equatio [9] local fractioal Laplace equatios[] fractal boudar problems [] fractal sigals[47] the geeralized Newto iteratio method[] the Yag- Laplace trasforms [9 5] the local fractioal Fourier aalsis [4] the Yag-Fourier trasforms[87] the discrete Fourier trasform [8] the fast Yag-Fourier trasforms i fractal space [3] the local fractioal Melli trasform i fractal space [9] the local fractioal Stieltjes trasform i fractal space [6] the local fractioal Z trasform i fractal space [] the local fractioal short time trasform [56] ad the local fractioal wavelet trasform [56]. I this paper we appl the Yag-Fourier trasform to deal with the fractal damical sstems with local fractioal derivative. The orgaizatio of this paper is as follows. I sectio the prelimiar results are preseted. The fractal damical sstems are ivestigated i sectio 3. Two eamples of fractal damical sstems are studied i sectio 4. Coclusios are i sectio 5.. Prelimiaries.. Local fractioal cotiuit Defiitio If there is [5-9] f f (.) with for ad. Now f is called local fractioal cotiuous at deote b lim f f. The f is called local fractioal cotiuous o the iterval a b deoted b f C a b. (.) Defiitio A fuctio f is called a odifferetiable fuctio of epoet which satisf Hölder fuctio of epoet the for X such that [6-7] f f C Defiitio 3 A fuctio f. (.3) is called to be cotiuous of order or shortl cotiuous whe we have the followig relatio [6-7] f f o. (.4) Remark.. Compared with (.4) (.) is stadard defiitio of local fractioal cotiuit. Here (.3) is uified local fractioal cotiuit... Local Fractioal Derivatives ad Itegrals

3 Xiao-Ju Yag et al. AEES Vol. No. 3 pp Defiitio 4 Settig f C a b derivative of f of order at d f f f f lim d local fractioal is defied [5-9] ( where f f f f For a a b there eists f D f deoted b. f D a b. Local fractioal derivative of high order is writte i the form k times k f D... D f (.6) ad local fractioal partial derivative of high order k times k f... f..5) (.7) Defiitio 5 Settig f C a b itegral of f of order i the iterval defied [6-9] a b I f lim where t t t b a f t jn dt j t j f t local fractioal t j j j j t ma t t t j... t j t j j... N the iterval deoted b Remark.. If have N a b is (.8) ad t a t b is a partitio of a b. For a a b I f a f I a b. f D a b or f C a b..3. Special Fuctios i Fractal Space there eists I a b we Defiitio 6 The Mittag-Leffler fuctio i fractal space is defied b [56] E : k k k ad. (.9) Defiitio 7 The e fuctio i fractal space is b the epressio [56] (k) k : ad. k k (.) Defiitio 8 The coe fuctio i fractal space is give [56] k k : k ad. k The followig rules hold [ ]: E E E E E E E i E i E i E i i ta E i E i i E i E i ta ta ta ta ta ta ta ta ta ta ta ta (.)

4 Xiao-Ju Yag et al. AEES Vol. No. 3 pp m m m m m m E i i k where k where k where. Remark. 3. i is fractal imagiar uit for more details see [5 6]..4. Yag-Fourier Trasform i Fractal Space Defiitio 9 Suppose that f C the F Yag-Fourier trasform deted b F f f is writte i the form [8 7] F f f F E i f d (.) where the latter coverges. Ad of course a sufficiet coditio for covergece is f E i d f d K. Defiitio If F f f F formula is writte i the form [8 7] f F F f Suppose that F f f F E i f d its iversio F F ad F g g the followig formulas are valid [8 7]: (.3). (.4) F F f the F af bg af bg (.5) If If lim F (.6) F f i f lim f f... f the F. (.7) F f i f 3. Fractal Damical Sstems I this sectio we start with the damical sstems of differetial equatios from fractioal derivative to local fractioal derivative. At first we itroduce some of the most importat methods of solvig liear fractioal differetial equatios with fractioal derivative: t t i m i A i B i mi i i t i t. (3.)

5 Xiao-Ju Yag et al. AEES Vol. No. 3 pp where both t t ad t t represet a fractioal derivative[-3] ad both t ad t cotiuous. Moreover we will itroduce some of the most importat methods of solvig liear local fractioal differetial equatios with local fractioal derivative: t t are i m i A i B i mi i i t i t (3.) where both t t ad fractioal derivative ad both t t t ad represet a local t are local fractioal cotiuous. Mostl we will cosider a iitialvalue problem composed of the equatio (3.) ad appropriate iitial coditios i fractal space which deped o a costructio of all operators k t for k... m... f t k 4. Two Illustrative Eamples. (3.3) I this sectio we give the usual cases. Suppose we cosider fractal sstems described b the fractioal differetial equatios with local fractioal derivative. Eample. A fractal sstem is described b the differetial equatio with local fractioal derivative t t a a t b b t t t. ( 4.) Takig the Yag-Fourier trasform we have F F i a a i b b. (4.) Therefore the trasform fuctio is writte i the from F i b b F i a a. (4.3) Eample. A fractal sstem is described b the differetial equatio with local fractioal derivative t t t a a a t b b t t t t. (4.4) Take the Yag-Fourier trasform F F i a i a a i b b the the trasform fuctio is give F i b b F a i a a 5. Coclusios (4.5). (4.6) Local fractioal derivative ad itegrals are revealed as oe of useful tools to deal with everwhere cotiuous but owhere differetiable fuctios i fractal areas ragig from fudametal sciece to egieerig. I this paper we show that the fractal damical sstems are derived from local fractioal derivative which are proposed ug the Yag-Fourier trasform based o the local fractioal calculus ad give two paradigms for cotrol problems to elaborate accurac ad reliable results. Our method ca be applied to other academic fields [6-33]. Ackowledgemets This work is grateful for the fiace supports of the Natioal Natural Sciece Foudatio of Chia (Grat No ). Refereces [] K. M. Kolwakar A. D. Gagal Fractioal differetiabilit of owhere differetiable fuctios ad dimesios Chaos. 6 (4) (996) [] F. B. Adda J. Cresso About o-differetiable fuctios Joural of Mathematical Aalsis ad Applicatios 63 () [3] A. Carpiteri B. Chiaia P. Coretti Static-kiematic dualit ad the priciple of virtual work i the mechaics of fractal media Computer Methods i Applied Mechaics ad Egieerig 9(-) () [4] Y. Che Y. Ya K. Zhag O the local fractioal derivative Joural of Mathematical Aalsis ad Applicatios 36()7-33. [5] X. J. Yag Local Fractioal Fuctioal Aalsis ad Its Applicatios. Asia Academic publisher Limited Hog Kog. [6] X. J. Yag Local fractioal itegral trasforms Progress i Noliear Sciece 4 () -5. [7] X. J. Yag Applicatios of local fractioal calculus to egieerig i fractal time-space: Local fractioal differetial equatios with local fractioal derivative. ArXiv:6.3v [math-ph]. [8] W. P. Zhog F. Gao X. M. She Applicatios of Yag-Fourier trasform to local fractioal equatios with local fractioal derivative ad local fractioal itegral Advaced Materials Research 46 () [9] W. P. Zhog F. Gao Applicatio of the Yag-Laplace trasforms to solutio to oliear fractioal wave equatio with fractioal derivative I: The 3rd Iteratioal Coferece o Computer Techolog ad Developmet ASME ASME Chedu pp.9-3. [] X. J. Yag A ew viewpoit to the discrete approimatio discrete Yag-Fourier trasforms of discrete-time fractal sigal. 7.6v [math-ph]. [] K. S. Miller B. Ross A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios Wile New York 993. [] I. Podlub Fractioal Differetial Equatios Academic Press New York 999. [3] R. Hilfer Applicatios of Fractioal Calculus i Phsics World Scietific Sigapore. [4] X. J. Yag Local fractioal Fourier aalsis Advaces i Mechaical Egieerig ad its Applicatios() () -6. [5] S.M. Guo L. Q. Mei Y. Li Y.F. Su The improved fractioal sub-equatio method ad its applicatios to the space time fractioal differetial equatios i fluid mechaics Phsics Letters A. 376(4) () [6] G.S. Che The local fractioal Stieltjes trasform i fractal space Advaces i Itelliget Trasportatio Sstems () () 9-3. [7] X. J. Yag M. K. Liao J. W. Che A ovel approach to procesg fractal sigals ug the Yag-Fourier trasforms Procedia Eg 9 ()

6 Xiao-Ju Yag et al. AEES Vol. No. 3 pp [8] X.J. Yag The discrete Yag-Fourier trasforms i fractal space Advaces i Electrical Egieerig Sstems () () [9] G. S. Che Local fractioal Melli trasform i fractal space Advaces i Electrical Egieerig Sstems () () [] X. J. Yag Epressio of geeralized Newto iteratio method via geeralized local fractioal Talor series Advaces i Computer Sciece ad its Applicatios () () [] X. J. Yag Local fractioal partial differetial equatios with fractal boudar problems Advaces i Computatioal Mathematics ad its Applicatios () () [] X. J. Yag Local fractioal Z trasform i fractal space Advaces i Digital Multimedia () () 96-. [3] X. J. Yag Fast Yag-Fourier trasforms i fractal space Advaces i Itelliget Trasportatio Sstems () () 5-8. [4] X. J. Yag Geeralized Samplig Theorem for Fractal Sigals Advaces i Digital Multimedia () ()88-9. [5] X. J. Yag A short itroductio to Yag-Laplace Trasforms i fractal space Advaces i Iformatio Techolog ad Maagemet ()() [6] Yudog Zhag Shuihua Wag Yuakai Huo Lea Wu Feature Etractio of Brai MRI b Statioar Wavelet Trasform ad its Applicatios Joural of Biological Sstems 8(s) () 5-3. [7] Yudog Zhag Lea WU A Fast Documet Image Deoig Method based o Packed Biar format ad Source Word Accumulatio Joural of Covergece Iformatio Techolog 6() () [8] Yudog Zhag Lea Wu Shuihua Wag Bacterial foragig optimizatio based eural etwork for short-term load forecastig Joural of Computatioal Iformatio Sstems 6(7) () [9] Yudog Zhag Lea Wu A Novel Geetic At Colo Algorithm Joural of Covergece Iformatio Techolog 7() () [3] Yudog Zhag Yuakai Huo Qig Zhu Shuihua Wag Lea Wu Polmorphic BCO for Protei Foldig Model Joural of Computatioal Iformatio Sstems 6(6) () [3] Yudog Zhag Lea Wu Fast Documet Image Biarizatio based o a improved adaptive Otsu s method ad destiatio word accumulatio Joural of Computatioal Iformatio Sstems 7(6) () [3] Yudog Zhag Lea Wu Recursive Structure Elemet Decompositio Ug Migratio Fitess Scalig Geetic Algorithm Spriger Berli / Heidelberg pp [33] Yudog Zhag Lea Wu Bacterial chemotais optimizatio for protei foldig model i: 5th Iteratioal Coferece o Natural Computatio ICNC 9 August August 6 9 IEEE Computer Societ Tiajia Chia 9 pp Vitae Yag Xiao-Ju was bor i 98. He worked as a scietist ad egieer i CUMT. His research iterest icludes Fractal mathematics (Geometr applied mathematics ad fuctioal aalsis) fractal Mechaics (fractal elasticit ad fractal fracture mechaics fractal rock mechaics ad fractioal cotiuous mechaics i fractal media) fractioal calculus ad its applicatios fractioal differetial equatio local fractioal itegral equatio local fractioal differetial equatio local fractioal itegral trasforms local fractioal short-time aalsis ad wavelet aalsis local fractioal calculus ad its applicatios ad local fractioal fuctioal aalsis ad its applicatios. Liao Meg-Ke is a lecturer i Egieerig Mechaics i Shihezi Uiversit. Mai research iterest is rock ad soil mechaics. Wag Jia-Nig is a master studet at Sheag Normal Uiversit. Her research iterest is cotrol of saturated sstems.