Applications of Two Dimensional Fractional Mellin Transform

Transcription

1 Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Volume 2 Issue 9 September 2014 PP ISSN X (Prit) & ISSN (Olie) Applicatios of Two Dimesioal Fractioal Melli Trasform V. D. Sharma Departmet of Mathematics Arts Commerce ad Sciece College Kira Nagar Amravati(M.S.) Idia. vdsharma@hotmail.co.i P. B. Deshmukh Departmet of Mathematics IBSS College of Egieerig Amravati (M.S.) Idia. Abstract: The Melli trasform is a basic tool for aalyzig the behavior of may importat fuctios i mathematics ad mathematical physics. Melli trasform has may applicatios such as algorithms correlators avigatio vowel recogitio cryptographic scheme quatum calculus radar classificatio of ships electromagetic stress distributio agriculture medical stream statistics probability sigal processig optics ad patter recogitio. Keywords: Fractioal Melli Trasform Geeralize fuctio Melli Trasform Testig Fuctio space 1. INTRODUCTION The Fourier trasform is ivariat i modulus to traslatios i frequecy but ot to dilatios. Therefore the Fourier trasform is o loger the appropriate trasform to chage the represetatio space of these sigals. It has to be replaced by a ew a ew trasform the Melli trasform which is ivariat i modulus to dilatios ad decomposes the sigal o a basis of hyperbolic sigals. Historically Riema (1876) first recogized the Melli trasform. I the last few decades the iterest of the scietific commuity towards the fractioal calculus experieced a exceptioal boost so that its applicatios ca ow be foud i a great variety of atural scieces. Two-dimesioal Melli trasform ca covert the auditory images of vowel souds from vocal tracts with differet sizes ito a ivariat Melli image (MI) ad thereby facilitate the extractio ad separatio of the size ad shape iformatio associated with a give vowel type. I sigal processig terms the MI of a soud is the Melli trasform of a stabilised wavelet trasform of the soud. Toshio Irio a Roy D. Patterso discussed i their article that the MI provides a good model of auditory vowel ormalizatio ad that this provides a good framework for auditory processig from cochlea to cortex [1]. Robert Frotczak Raier Schobel preseted i their work that how to modify the approach to value America call optios o divided-payig stocks. He proposed a ew itegral equatio to determie the price of a America call optio ad its free boudary usig modified Melli trasforms ad also show how to derive the pricig formula for perpetual America call optios usig the ew framework [2]. We ca compute the price of a optio o a basket of stocks usig Melli trasforms i several variables [3]. Salvatore Butera Mario Di Paola discussed i their work that usig Melli trasform solutio of the multiorder differetial equatios are calculated [4]. They also provide i their article that the solutio of a multi-order multi-degree-of-freedom fractioal differetial equatio is addressed by usig the Melli itegral trasform [5]. The two dimesioal fractioal Melli trasform with parameter θ of f ( x deoted by FRMT { f ( x } performs a liear operatio give by the itegral trasform ARC Page 794

2 V. D. Sharma & P. B. Deshmukh FRMT { f ( x } F ( u f ( x K ( x y u dxdy (1) where the kerel - K iu 2iv i [ u v log x log y 2 ] si si ta ( x y u x y e 0. (2) 2 Where (3) This paper will discussed some applicatios two dimesioal fractioal Melli trasform ad defied the testig fuctio space ad two dimesioal fractioal Melli trasform. 2. PRELIMINARIES 2.1 The Test Fuctio Space E A ifiitely differetiable complex valued fuctio x y compact set K S where I S b o S a { x : xr x a a 0} S b { y : y R y b b 0} R belogs to E R I K R if for each q E q [ ( x ] sup Dx y ( x xi yk Thus E( R ) will deote the space of all ( x E( R ) with compact support cotaied i S & S b. Note that the space E is complete ad therefore a Frechet space. Moreover we say that f ( x is a fractioal Melli trasformable if it is a member of 2.2 Two Dimesioal Fractioal Melli Trasform (FRMT) * E the dual of E. * The two dimesioal fractioal Melli trasform of f ( x E ( R ) ca be defied by FRMT{ f ( x } F ( u f ( x K ( x y u K ( x y u x iu 1 si 2 2 y 2iv 1 si e i [ u v log x log y ] ta Right had side of equatio (4) has a meaig as the applicatio of K ( x y u E. * f ( x E to It ca be exteded to the complex space as a etire fuctio give by FRMT{ f ( x } F ( p k) f ( x K ( x y p k) The right had side is meaigful because for each p k C K ( x y p k) E 3. APPLICATIONS 3.1 Prove that as a fuctio of xy. Puttig Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 795

3 Applicatios of Two Dimesioal Fractioal Melli Trasform Where 3.2 Prove that 3.3 Prove that Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 796

4 V. D. Sharma & P. B. Deshmukh Puttig Where Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 797

5 Applicatios of Two Dimesioal Fractioal Melli Trasform 3.4 Prove that Puttig Where Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 798

6 V. D. Sharma & P. B. Deshmukh Hece proved 4. CONCLUSION I this paper we have defied Distributioal two dimesioal fractioal Melli trasform with compact support. Some applicatios for two dimesioal fractioal Melli trasform is proved. REFERENCES [1] Toshio Irio a* Roy D. Patterso Segregatig iformatio about the size ad shape of the vocal tract usig a time-domai auditory model: The stabilized wavelet-melli trasform Speech Commuicatio 36 (2002) [2] Robert Frotczak Raier Schöbel O modified Melli trasforms Gauss_Laguerre quadrature ad the valuatio of America call optios Joural of Computatioal ad Applied Mathematics 234 (2010) [3] R. Paii ad R. P. Srivastav Optio Pricig with Melli Trasforms Mathematical ad Computer Modellig 40 (2004) [4] Salvatore Butera Mario Di Paola Fractioal differetial equatios solved by usig Melli trasform Commu Noliear Sci Numer Simulat 19 (2014) [5] Salvatore Butera Mario Di Paola Melli trasform approach for the solutio of coupled systems of fractioal differetial equatios Commu Noliear Sci Numer Simulat (2014) AUTHORS BIOGRAPHY V. D. Sharma is curretly workig as a Assistat professor i the departmet of Mathematics Arts Commerce ad Sciece College Kira Nagar Amravati (M.S.) Idia. She has got 18 years of teachig ad research experiece. She has obtaied her Ph.D. degree i 2007 from SGB Amravati Uiversity Amravati. Her field of iterest is Itegral Trasforms. Six research studets are workig uder her supervisio. She has published more tha 50 research articles.. P. B. Deshmukh is a Assistat professor i the departmet of Mathematics IBSS College of Egieerig Amravati (M.S.) Idia. She has obtaied her master degree i 2006 from Sat Gadage Baba Amravati Uiversity Amravati. She has got 7 years of teachig experiece. She has 6 research articles i jourals to her credit. Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 799