Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

• Authors: Virginia Kiryakova , Yuri Luchko
• Languages of publication: EN

Abstracts

EN

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ
1,…,δ
m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

Keywords

EN

fractional calculus; operators of Riemann-Liouville and Caputo type; Erdélyi-Kober operators; special functions; integral transforms; Cauchy problems

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1314-1336

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

Virginia Kiryakova

• Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bontchev” Str., Block 8, 1113, Sofia, Bulgaria

author

Yuri Luchko

• Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, D - 13353, Berlin, Germany

References

• [1] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000) http://dx.doi.org/10.1142/9789812817747[Crossref]
• [2] V.V. Uchaikin, Method of fractional derivatives (Artishok, Ul’janovsk, 2008), in Russian
• [3] R.R. Nigmatullin, D. Baleanu, Fract. Calc. Appl. Anal. 15, 718 (2012)
• [4] R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, Singapore, 2011) http://dx.doi.org/10.1142/8072[Crossref]
• [5] J.L.A. Dubbeldam, A. Milchev, V.G. Rostiashvili, T.A. Vilgis, Phys. Rev. E 76, 010801 (2007) http://dx.doi.org/10.1103/PhysRevE.76.010801[Crossref]
• [6] A. Freed, K. Diethelm, and Yu. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus (NASA’s Glenn Research Center, Ohio, 2002)
• [7] R. Klages, G. Radons, and I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications (Wiley-VCH, Weinheim, 2008) http://dx.doi.org/10.1002/9783527622979[Crossref]
• [8] Yu. Luchko, Forum der Berliner mathematischen Gesellschaft 19, 53 (2011)
• [9] Yu. Luchko, A. Punzi, Int. J. Geom. 1, 257 (2011) http://dx.doi.org/10.1007/s13137-010-0012-8[Crossref]
• [10] R.L. Magin, Crit. Rev. Biomed. Eng. 32, 195 (2004) http://dx.doi.org/10.1615/CritRevBiomedEng.v32.i34.10[Crossref]
• [11] F. Mainardi, Chaos. Soliton. Fract. 7, 1461 (1996) http://dx.doi.org/10.1016/0960-0779(95)00125-5[Crossref]
• [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press & World Sci., London - Singapore, 2010) http://dx.doi.org/10.1142/9781848163300[Crossref]
• [13] R. Metzler and J. Klafter, J. Phys. A. Math. Gen. 37, R161 (2004) http://dx.doi.org/10.1088/0305-4470/37/31/R01[Crossref]
• [14] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon & Breach. Sci. Publ., London-N. York, 1993)
• [15] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering 198 (Academic Press, N. York-Boston, 1999)
• [16] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
• [17] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific, Singapore, 2012)
• [18] M. Caputo, Geophys. J. R. Astr. Soc. 13, 529 (1967) http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.x[Crossref]
• [19] Ch. Li, D. Qian, YQ Chen, Discrete Dynamics in Nature and Society 2011, 562494 (2011)
• [20] I. Dimovski, Compt. Rend. Acad. Bulg. Sci. 19, 1111 (1966)
• [21] V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes in Math. 301(Longman Sci. Tech. & J. Wiley, Harlow-N. York, 1994)
• [22] S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Ser.: Mathematics and Its Applications 287 (Kluwer Acad. Publ., Dordrecht-Boston-London, 1994) http://dx.doi.org/10.1007/978-94-011-1196-6[Crossref]
• [23] A.P. Prudnikov, Yu. Brychkov, O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions (Gordon and Breach Sci. Publ., N. York-London-Paris-Tokyo, etc., 1990)
• [24] H.M. Srivastava, R.G. Buschman, Theory and Applications of Convolution Integral Equations (Kluwer Acad. Publ., Dordrecht-Boston-London, 1992) http://dx.doi.org/10.1007/978-94-015-8092-2[Crossref]
• [25] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vols. 1, 2, 3 (McGraw-Hill, New York, 1953)
• [26] V. Kiryakova, Recent Advances in Appl. Mathematics’ 96 (Proc. Intern. Workshop, Kuwait University, 1996) 281
• [27] V. Kiryakova, Fract. Calc. Appl. Anal. 2, 445 (1999)
• [28] Yu. Luchko, Fract. Calc. Appl. Anal. 2, 463 (1999)
• [29] V. Kiryakova, J. Comput. Appl. Math. 118, 214 (2000)
• [30] Yu. Luchko, S.B. Yakubovich, Math. Balkanica 7, 119 (1993)
• [31] E. Buckwar, Yu. Luchko, J. Math. Anal. Appl. 227, 81 (1998) http://dx.doi.org/10.1006/jmaa.1998.6078[Crossref]
• [32] R. Gorenflo, Yu. Luchko, F. Mainardi, Fract. Calc. Appl. Anal. 2, 383 (1999)
• [33] R. Gorenflo, Yu. Luchko, F. Mainardi, J. Comput. Appl. Math. 11, 175 (2000) http://dx.doi.org/10.1016/S0377-0427(00)00288-0[Crossref]
• [34] Yu. Luchko, R. Gorenflo, Fract. Calc. Appl. Anal. 1, 63 (1998)
• [35] V. Kiryakova, Comput. Math. Appl. 59, 1885 (2010) http://dx.doi.org/10.1016/j.camwa.2009.08.025[Crossref]
• [36] V. Kiryakova, Yu. Luchko, American Institute of Physics-Conf. Proc. 1301, 597 (2010)
• [37] M. Dzrbashjan, Izv. Akad. Nauk Armen. SSR 13, 21 (1960), In Russian
• [38] I.N. Sneddon, Lecture Notes in Math. 457, Fractional Calculus and Its Applications (Proc. Internat. Conf. Held in New Haven, Springer, N. York, 1975) 37 http://dx.doi.org/10.1007/BFb0067097[Crossref]
• [39] O.P. Agrawal, Fract. Calc. Anal. Appl. 15, 700 (2012)
• [40] S.L. Kalla, V.S. Kiryakova, Math. Japonica 35, 1 (1990)
• [41] V. Kiryakova, American Institute of Physics-Conf. Proc. 1410, 247 (2011)
• [42] Yu. Luchko, Fract. Calc. Appl. Anal. 7, 339 (2004)
• [43] Yu. Luchko, J.J. Trujillo, Fract. Calc. Appl. Anal. 10, 249 (2007)
• [44] G. Pagnini, Fract. Calc. Appl. Anal. 15, 117 (2012)
• [45] G. Pagnini, A. Mura, F. Mainardi, Intern. J. of Stochastic Analysis 2012, 427383 (2012)
• [46] L. Rodriguez-Germa, J.J. Trujillo, L. Vazquez, M. Pilar-Velasco, Fract. Calc. Appl. Anal. 11, 431 (2008)
• [47] Yu. Luchko, Integr. Transforms Spec. Funct. 19, 809 (2008) http://dx.doi.org/10.1080/10652460802091617[Crossref]
• [48] Yu. Luchko, V. Kiryakova, Fract. Calc. Appl. Anal. 16, 405 (2013)
• [49] H. Kober, Quart. J. Math. Oxford ll, 193 (1940)
• [50] V. Kiryakova, Fract. Calc. Appl. Anal. 11, 201 (2008)
• [51] J. Mikusinski, Cz. Ryll-Nardzewski, Studia Math. 12, 51 (1951)
• [52] I. Dimovski, V. Kiryakova, Numer. Funct. Anal. Opt. 21, 121 (2000) http://dx.doi.org/10.1080/01630560008816944[Crossref]
• [53] Yu. Luchko, V. Kiryakova, C.R. Acad. Bulg. Sci. 53, 17 (2000)
• [54] Yu. Luchko, V. Kiryakova, Algebraic Analysis and Related Topics, Banach Center Publ. 53 (Proc. Conf., Warsaw, 2000) 155
• [55] I. Ali, V. Kiryakova, S. Kalla, J. Math. Anal. Appl. 269, 172 (2002) http://dx.doi.org/10.1016/S0022-247X(02)00012-4[Crossref]
• [56] K. Furati, Fract. Calc. Appl. Anal. 16, 171 (2013)
• [57] M.A. Al-Bassam, Yu. Luchko, Journal of Fractional Calculus 7, 69 (1995)
• [58] S.B. Hadid, Yu. Luchko, Panamerican Mathematical Journal 6, 57 (1996)
• [59] Yu. Luchko, H.M. Srivastava, Comput. Math. Appl. 29, 73 (1995) http://dx.doi.org/10.1016/0898-1221(95)00031-S[Crossref]
• [60] I. Dimovski, V. Kiryakova, Proc. Conf. Complex Anal. and Appl., Varna’ 81, 148 (1984)
• [61] F. Al-Musallam, V. Kiryakova, Vu Kim Tuan, Rocky Mountain J. Math. 32, 409 (2002) http://dx.doi.org/10.1216/rmjm/1030539678[Crossref]
• [62] M. Saigo, Mathematical Reports Kyushu Univ. 11, 135 (1978)
• [63] A. Rao, M. Garg, S.L. Kalla, Kuwait J. Sci. Engg. 37, 15 (2010)

Identifiers

DOI

10.2478/s11534-013-0217-1