Application of Fractional Calculus Operators to Related Areas

Transcription

1 Gen. Math. Notes, Vol. 7, No., Novembe 2, pp ISSN ; Copyight ICSRS Publication, 2 Available fee online at Application of Factional Calculus Opeatos to Related Aeas ishan Shama Depatment of Mathematics, NRI Institute of Technology and Management, Baaghata, Net to S.G. Motos, Jhansi Road, Gwalio-474, India Addess: B-3, ishna Pui, Taaganj, Lashka, Gwalio (M.P.-474, India dkishan77@yahoo.com (Received: 6-7-/ Accepted: 4-- Abstact In this pape a new function called as -function, which is an etension of the genealization of the Mittag-Leffle function[,] and its genealized fom intoduced by Pabhaka[2], is intoduced and studied by the autho in tems of some special functions and deived the elations that eists between the - function and the opeatos of Riemann-Liouville factional integals and deivatives. eywods: Factional calculus, Riemann- Liouville factional integals and deivatives. Intoduction Factional Calculus is a field of applied mathematics that deals with deivatives and integals of abitay odes. Duing the last thee decades Factional Calculus has been applied to almost evey field of Mathematics like Special Functions etc., Science, Engineeing and Technology. Many applications of Factional Calculus

2 34 ishan Shama can be found in Tubulence and Fluid Dynamics, Stochastic Dynamical System, Plasma Physics and Contolled Themonuclea Fusion, Non-linea Contol Theoy, Image Pocessing, Non-linea Biological Systems and Astophysics. The Mittag-Leffle function has gained impotance and populaity duing the last one decade due mainly to its applications in the solution of factional-ode diffeential, integal and diffeence euations aising in cetain poblems of mathematical, physical, biological and engineeing sciences. This function is intoduced and studied by MittagLeffle[,] in tems of the powe seies ( Eα, + ( α > α (. A genealization of this seies in the following fom, ( Eα β, α ( α, β > (.2 has been studied by seveal authos notably by Mittag-Leffle[,],Wiman[3], Agawal[5], Humbet and Agawal[8] and Dzbashjan[,2,3]. It is shown in [5] that the function defined by (. and (.2 ae both entie functions of ode ρ and type σ. A detailed account of the basic popeties of these two functions ae given in the thid volume of Bateman manuscipt poject[4] and an account of thei vaious popeties can be found in [2,2]. The multiinde Mittag-Leffle function is defined by iyakova[9] by means of the powe seies E ( ρi, ( µ i ϕ z j + m j µ ρj (.3 whee m > is an intege, ρj and µ j ae abitay eal numbes. The multiinde Mittag-Leffle function is an entie function and also gives its asymptotic, estimate, ode and type see iyakova[9]. A genealization of (. and (.2 was intoduced by Pabhaka [2] in tems of the seies epesentation γ α, β E ( γ n! α, ( α, β, γ C, Re( α >

3 Application of Factional Calculus Opeatos 35 whee (γ n is Pochamme s symbol defined by (.4 ( γ γ ( γ +...(( γ + ( n, n N, γ. n It is an entie function of ode ρ [Re( α]. An inteesting genealization of (.2 is ecently intoduced by ilbas and Saigo[6] in tems of a special entie function of the fom whee,, Eα m l c, Γ[ α( im + l + ] c Γ[ α ( im + l + + i ] c (.5 and an empty poduct is to be intepeted as unity. Cetain popeties of this function associated with factional integals and deivatives[2]. The pesent pape is oganized as follows; In section 2, we give the definition of the -function and its elation with anothe special functions, namely genealization of the Mittag-Leffle function[] and its genealized fom intoduced by Pabhaka[2] and othe special functions. In section 3, elations that eists between the -function and the opeatos of Riemann-Liouville factional calculus ae deived. 2 The -Function The -function intoduced by the autho is defined as follows: p ( a,, ap; b,..., b; p ( a...( ap ( b...( b ( γ! α ( 2. whee α, β, γ C, Re( α > and ( a j and ( b j ae the Pochamme symbols. The seies(2. is defined when none of the paametes bj s, j,2,...,, is a negative intege o zeo. If any numeato paamete a j is a negative intege o zeo, then the seies teminates to a polynomial in. Fom the atio test it is evident that the seies is convegent fo all if p > +. When p + and

4 36 ishan Shama p aj bj, the seies can convege in some cases. Let γ j j. It can be shown that when p + the seies is absolutely convegent fo if ( R ( γ <, conditionally convegent fo - if R ( γ < and divegent fo if R (γ. Special cases: (i When thee is no uppe and lowe paamete, we get ( γ ( ; ; E! α γ α, β ( 2.2 which educes to the genealization of the Mittag-Leffle function[] and its genealized fom intoduced by Pabhaka[2]. (ii If we put γ in (2.2, we get α, β ; ( ; ; E α α, β E α, β ( 2.3 which is the genealized Mittag-Leffle function[]. (iii If we take β in (2.3, we get α,; ( ; ; E α + α, E α, E α which is the Mittag-Leffle function[]. (iv If we take α in (2.4, we get α,; ( ; ; α E +, E which is the Eponential function[4] denoted by e., E ( 2.4 ( 2.5

5 Application of Factional Calculus Opeatos 37 3 Relations with Riemann-Liouville Factional Calculus Opeatos In this section we deive elations between the -function and the opeatos of Riemann-Liouville Factional Calculus. Theoem 3. Let > ; α, β, γ C (Re( α > and I be the opeato of Riemann-Liouville factional integal then thee holds the elation: I p + α β γ, ; ( p + a +,, ap, ; b,..., b, + ; ( 3. Poof. Following Section 2 of the book by Samko, ilbas and Maichev[2], the factional Riemann-Liouville(R-L integal opeato(fo lowe limit a w.. t. vaiable is given by I f ( t f ( t dt ( Γ By vitue of (3.2 and (2., we obtain ( a...( a I p ( b...( b ( γ t! α α p ( t dt ( 3.2 ( 3.3 Intechanging the ode of integation and evaluating the inne integal with the help of Beta function, it gives I p ( a...( ap ( ( γ + ( b...( b ( +! α + p + + ( a,, ap, ; b,..., b, + ; ( 3.4 The intechange of the ode of integation and summation is pemissible unde the conditions stated along with the theoem due to convegence of the integals involved in this pocess. This shows that a Riemann-Liouville factional integal of the -function is again the -function with indices p+, +. This completes the poof of the theoem (3..

6 38 ishan Shama Theoem 3.2 Let > ; α, β, γ C (Re( α > and D be the opeato of Riemann-Liouville factional deivative then thee holds the elation: ( D p ( p + a,, ap, ; b,..., b, ; + ( 3.5 Poof. Following Section 2 of the book by Samko, ilbas and Maichev[2], the factional Riemann-Liouville(R-L integal opeato(fo lowe limit a w.. t. vaiable is given by d D f ( d Γ ( t f ( t dt whee [ ] +. ( 3.6 Fom (2. and (3.6 it follows that D p d d ( a...( a ( b...( b ( γ t! α n p ( t dt ( 3.7 Intechanging the ode of integation and evaluating the inne integal with the help of Beta function, it gives D p ( a...( ap ( ( b...( b ( ( γ! α p + + ( a,, ap, ; b,..., b, ; ( 3.8 This shows that a Riemann-Liouville factional deivative of the -function is again the -function with indices p+, +. This completes the poof of the theoem ( Conclusion It is epected that some of the esults deived in this suvey may find applications in the solution of cetain factional ode diffeential and integal euations aising poblems of physical sciences and engineeing aeas.

7 Application of Factional Calculus Opeatos 39 Acknowledgements The autho is vey thankful to the efeees fo giving seveal valuable suggestions in the pesentation of the pape. Refeences [] M.M. Dzbashjan, On the integal epesentation and uniueness of some classes of entie functions (in Russian, Dokl. AN SSSR, 85( ( [2] M.M. Dzbashjan, On the integal tansfomations geneated by the genealized Mittag-Leffle function (in Russian, Izv. AN Am. SSR, 3(3 ( [3] M.M. Dzbashjan, Integal Tansfoms and Repesentations of Functions I n the Comple Domain (in Russian, Nauka, Moscow, (966. [4] A. Edelyi, W. Magnus, F. Obehettinge and F.G. Ticomi, Highe Tanscendental Functions (Vol. 3, McGaw-Hill, New Yok-Toonto- London, (955. [5] R. Goenflo, A.A. ilbas and S.V. Rosogin, On the genealized Mittag- Leffle type functions, Integal Tansfoms and Special Functions, 7(3-4(998, [6] R. Goenflo and F. Mainadi, The Mittag-Leffle type function in the Riemann-Liouville factional calculus, In: A.A. ilbas, (ed. Bounday value poblems, special functions and factional calculus, Poc. Int. Conf. Minsk, Belaussian State Univesity, Minsk., (996, [7] R. Goenflo and F. Mainadi, Factional calculus: integal and diffeential euations of factional ode, In: A. Capintei and F. Mainadi (Eds., Factals and Factional Calculus in Continuum Mechanics, Spinge, Wien, (997. [8] P. Humbet and R.P. Agawal, Su la function de Mittag-Leffle et uelues unes de ses. genealizations, Bull Sci. Math., (77(2 (953, [9] V.S. iyakova, Multiple (multi inde Mittag-Leffle functions and elations to genealized factional calculus, J. Comput. Appl. Math., 8 (2, [] G.M. Mittag-Leffle, Su la nuovelle function E α (, C. R. Acad. Sci. Pais, (37(2 (93, [] G.M. Mittag-Leffle, Su la epesentation analytiue de une banche unifome une function monogene, Acta. Math., 29(95, -8. [2] S.G. Samko, A. ilbas and O. Maichev, Factional Integals and Deivatives, Theoy and Applications, Godon and Beach Sci. Publ., New Yok et alibi, (99. [3] A. Wiman, Ube die nullsteliun de fuctionen E α (, Acta Math., 29(95, [4] E.D. Rainville, Special Functions, Chelsea Publishing Company, Bon, New Yok, (96.

8 4 ishan Shama [5] R.P. Agawal, A popos d une note M. Piee Humbet, C. R. Acad. Sc. Pais, 236(953, [6] A.A. ilbas and M. Saigo, Factional integals and deivatives of Mittag- Leffle type function (Russian English summay, Doklady Akad. Nauk Belausi, 39(4 (995, [7] A.A. ilbas, Factional calculus of genealized Wight function, Fac. Calc. Appl. Anal., 8(25, [8] A.M. Mathai and R.. Saena, The H-function with Applications in Statistics and othe Disciplines, John Wiley and Sons, Inc., New Yok, (978. [9] M. Shama and R. Jain, A note on a genealized M-seies as a special function of factional calculus, Fact. Calc. Appl. Anal., 2(4 (29, [2] T.R. Pabhaka,A Singula Integal Euation with a Genealized Mittag- Leffle Function in the enel, Yokohama Math. J., 9(97, 7-5.