On two classes of operators of generalized fractional integro-differentiation

S. M. Sitnik ; E. L. Shishkina

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Physics, Tome 198 (2021), pp. 109-122.

Résumé

The paper is a survey of two special classes of generalized fractional integro-differentiation operators studied earlier in detail by the authors; all results are presented without proofs. A list of various fractional integro-differentiation operators with brief historical background is given. Further, we discuss main results of the theory of Bushman–Erdélyi operators. This class includes the classical Riemann–Liouville and Erdélyi–Kober operators, the Mehler–Fock transformation, transformation operators for the singular Bessel operator, and others. Also, we present main results for operators of fractional Bessel integro-differentiation.

Détail

Keywords: fractional integro-differentiation, Riemann–Liouville operator, Erdélyi–Kober operator, transformation operator, Buschman–Erdélyi operator, fractional power of Bessel operators.

Bibliographie

[1] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1973

[2] Gerasimov A. N., “Obobschenie lineinykh zakonov deformirovaniya i ego primenenie k zadacham vnutrennego treniya”, Prikl. mat. mekh., 12:3 (1948), 251–260 | Zbl

[3] Dzhrbashyan M. M., “Obobschennyi operator Rimana—Liuvillya i nekotorye ego primeneniya”, Dokl. AN SSSR., 177:4 (1967), 767–770 | Zbl

[4] Dzhrbashyan M. M., “Obobschennyi operator Rimana—Liuvillya i nekotorye ego primeneniya”, Izv. AN SSSR. Ser. mat., 32:5 (1968), 1075–1111 | Zbl

[5] Dzhrbashyan M. M., “Teoriya faktorizatsii funktsii, meromorfnykh v kruge”, Mat. sb., 79 (121):4 (8) (1969), 517–615 | Zbl

[6] Dzhrbashyan M. M., “Kraevaya zadacha dlya operatora drobnogo poryadka tipa Shturma—Liuvillya”, Izv. AN Arm. SSR, 5:2 (1970), 71–96 | Zbl

[7] Dzhrbashyan M. M., “Teoriya faktorizatsii i granichnykh svoistv funktsii, meromorfnykh v kruge”, Usp. mat. nauk, 28:4 (172) (1973), 3–14 | MR | Zbl

[8] Dzhrbashyan M. M., “Ob odnom beskonechnom proizvedenii”, Dokl. AN SSSR, 240:3 (1978), 518–521 | MR | Zbl

[9] Dzhrbashyan M. M., “Bazisnost biortogonalnykh sistem, porozhdennykh kraevymi zadachami dlya differentsialnykh operatorov drobnogo poryadka”, Dokl. AN SSSR, 261:5 (1981), 1054–1058 | MR | Zbl

[10] Dzhrbashyan M. M., Nersesyan A. B., “O primenenii nekotorykh integro-differentsialnykh operatorov”, Dokl. AN SSSR, 121:2 (1958), 210–213 | MR | Zbl

[11] Dzhrbashyan M. M., Nersesyan A. B., “Razlozheniya po spetsialnym biortogonalnym sistemam i kraevye zadachi dlya differentsialnykh uravnenii drobnogo poryadka”, Dokl. AN SSSR, 132:4 (1960), 747–750 | Zbl

[12] Dzhrbashyan M. M., Nersesyan A. B., “Razlozheniya po nekotorym biortogonalnym sistemam i kraevye zadachi dlya differentsialnykh uravnenii drobnogo poryadka”, Tr. Mosk. mat. o-va, 10, 1961, 89–179 | Zbl

[13] Dzhrbashyan M. M., Nersesyan A. B., “Drobnye proizvodnye i zadachi Koshi dlya differentsialnykh uravnenii drobnogo poryadka”, Izv. Akad. nauk Arm. SSR, 3:1 (1968), 3–28 | MR

[14] Dzhrbashyan M. M., Rafaelyan S. G., “Interpolyatsionnye teoremy i razlozheniya po sistemam tipa Fure”, Dokl. AN SSSR, 285:4 (1985), 782–787 | MR | Zbl

[15] Dzhrbashyan M. M., Saakyan B. A., “Klassy formul i razlozheniya tipa Teilora—Maklorena, assotsiirovannye s differentsialnymi operatorami drobnogo poryadka”, Izv. AN SSSR. Ser. mat., 39:1 (1975), 69–122 | Zbl

[16] Katrakhov V. V., Sitnik S. M., “Kompozitsionnyi metod postroeniya $B$-ellipticheskikh, $B$-giperbolicheskikh i $B$-parabolicheskikh operatorov preobrazovaniya”, Dokl. RAN, 337:3 (1994), 307–311 | Zbl

[17] Katrakhov V. V., Sitnik S. M., “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Sovr. mat. Fundam. napr., 64, no. 2, 2018, 211–426 | MR

[18] Kilbas A. A., Teoriya i prilozheniya differentsialnykh uravnenii drobnogo poryadka. Kurs lektsii, Voronezh, 2009

[19] Meshkov S. I., Vyazkouprugie svoistva metallov, Metallurgiya, M., 1974

[20] Nakhushev A. M., Uravneniya matematicheskoi biologii, Vysshaya shkola, M, 1995

[21] Nakhushev A. M., Elementy drobnogo ischisleniya i ikh primenenie, Izd-vo KBNTs RAN, Nalchik, 2000

[22] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003

[23] Novozhenova O. G., Biografiya i nauchnye trudy Alekseya Nikiforovicha Gerasimova. O lineinykh operatorakh, uprugo-vyazkosti, elevteroze i drobnykh proizvodnykh, Pero, M., 2018

[24] Pskhu A. V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005

[25] Pskhu A. V., Kraevye zadachi dlya differentsialnykh uravnenii s chastnymi proizvodnymi drobnogo i kontinualnogo poryadka, Izd-vo KBNTs RAN, Nalchik, 2005

[26] Rabotnov Yu. N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977

[27] Repin O. A., Kraevye zadachi so smescheniem dlya uravnenii giperbolicheskogo i smeshannogo tipov, Samara, 1992

[28] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987

[29] Sitnik S. M., “Faktorizatsiya i otsenki norm v vesovykh lebegovykh prostranstvakh operatorov Bushmana—Erdeii”, Dokl. AN SSSR, 320:6 (1991), 1326–1330 | Zbl

[30] Sitnik S. M., “Drobnoe integro-differentsirovanie dlya differentsialnogo operatora Besselya”, Mat. Mezhdunar. simp. «Uravneniya smeshannogo tipa i rodstvennye voprosy analiza i informatiki» (Nalchik–Elbrus, 22-26 maya 2004 g.), 2004, 163–167

[31] Sitnik S. M., “Operatory preobrazovaniya i ikh prilozheniya”, Issledovaniya po sovremennomu analizu i matematicheskomu modelirovaniyu, eds. Korobeinik Yu. F., Kusraev A. G., Vladikavkaz, 2008, 226–293

[32] Sitnik S. M., “O yavnykh realizatsiyakh drobnykh stepenei differentsialnogo operatora Besselya i ikh prilozheniyakh k differentsialnym uravneniyam”, Dokl. Adyg. (Cherkes.) Mezhdunar. akad. nauk, 12:2 (2010), 69–75

[33] Sitnik S. M., “Operatory preobrazovaniya Bushmana—Erdeii, ikh klassifikatsiya, osnovnye svoistva i prilozheniya”, Nauch. ved. Belgorod. gos. un-ta. Mat. Fiz., 11 (208):39 (2015), 60–76

[34] Sitnik S. M., “Obzor osnovnykh svoistv operatorov preobrazovaniya Bushmana—Erdeii”, Chelyabinsk. fiz.-mat. zh., 1:4 (2016), 63–93 | MR | Zbl

[35] Sitnik S. M., Shishkina E. L., “Ob odnom tozhdestve dlya iterirovannogo vesovogo sfericheskogo srednego i ego prilozheniyakh”, Sib. elektron. mat. izv., 13 (2016), 849–860 | MR | Zbl

[36] Sitnik S. M., Shishkina E. L., Metod operatorov preobrazovaniya dlya differentsialnykh uravnenii s operatorami Besselya, Fizmatlit, M., 2019

[37] Shishkina E. L., “O svoistvakh odnogo usrednyayuschego yadra v vesovom klasse Lebega”, Nauch. ved. Belgorod. gos. un-ta. Mat. Fiz., 6:42 (2016), 12–19

[38] Shishkina E. L., “Vesovye obobschennye funktsii, otvechayuschie kvadratichnoi forme s kompleksnymi koeffitsientami”, Chelyabinsk. fiz.-mat. zh., 2:1 (2017), 88–98 | MR | Zbl

[39] Shostak R. Ya., “Aleksei Vasilevich Letnikov”, Ist.-mat. issled., 1952, no. 5, 167–240

[40] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1983

[41] Garra R., Orsingher E., “Random flights related to the Euler–Poisson–Darboux equation”, Markov Process. Related Fields., 22 (2016), 87–110 | MR | Zbl

[42] Garra R., Orsingher E., Polito F., “Fractional Klein–Gordon equations and related stochastic processes”, J. Stat. Phys., 155 (2014), 777–809 | DOI | MR | Zbl

[43] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, North-Holland, Amsterdam, 2006 | Zbl

[44] Kiryakova V., “All the special functions are fractional differintegrals of elementary functions”, J. Phys. A: Math. Gen., 30:14 (1997), 5085–5103 | DOI | Zbl

[45] Koornwinder T. H., “Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators”, SIGMA, 11:074 (2015) | Zbl

[46] Luchko Yu., “Fractional Integrals and Derivatives: “True” versus “False””, Mathematics, 2019, Special Issue

[47] Handbook of Fractional Calculus with Applications, de Gruyter, Berlin, 2019

[48] McBride A. C., “Fractional powers of a class of ordinary differential operators”, Proc. London Math. Soc., 3:45 (1982), 519–546 | DOI | Zbl

[49] Miller K. S., Ross B., An introduction to fractional calculus and fractional differential equations, Wiley, New York, 1993 | Zbl

[50] Podlubny I., Fractional differential equations, Academic Press, San Diego–New York–London, 1999 | Zbl

[51] Rossikhin Yu. A., “Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids”, Appl. Mech. Rev., 63:1 (2010), 12

[52] Rossikhin Yu. A., Shitikova M. V., “Comparative analysis of viscoelastic models involving fractional derivatives of different orders”, Fract. Calc. Appl. Anal., 10:2 (2007), 111–121 | MR | Zbl

[53] Rossikhin Yu. A., Shitikova M. V., “Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator”, Fract. Calc. Appl. Anal., 17:3 (2014), 674–683 | DOI | MR | Zbl

[54] Saigo M., “A remark on integral operators involving the Gauss hypergeometric functions”, Math. Rep. Kyushu Univ., 11:2 (1977/78), 135–143 | MR

[55] Samko S., Kilbas A., Marichev O., Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, Switzerland, 1993

[56] Shishkina E. L., “On weighted generalized functions associated with quadratic forms”, Issues Anal., 5 (23):2 (2016), 52–68 | DOI | MR | Zbl

[57] Shishkina E. L., “On the boundedness of hyperbolic Riesz B-potential”, Lithuan. Math. J., 56:4 (2016), 540–551 | DOI | Zbl

[58] Shishkina E. L., Sitnik S. M., “On fractional powers of Bessel operators”, J. Inequal. Spec. Funct., 8:1 (2017), 49–67 | MR

[59] Shishkina E. L., Sitnik S. M., “General form of the Euler–Poisson–Darboux equation and application of the transmutation method”, Electron. J. Differ. Equations., 2017, no. 177, 1–20

[60] Shishkina E. L., Sitnik S. M., “A fractional equation with left-sided fractional Bessel derivatives of Gerasimov–Caputo type”, Mathematics, 7:12 (2019), 21 | DOI

[61] Sitnik S. M., Transmutations and applications: A survey, arXiv: 1012.3741

[62] Sitnik S. M., “Buschman–Erdélyi transmutations, classification and applications”, Analytic Methods of Analysis and Differential Equations, Cambridge Scientific Publishers, Cottenham, 2013, 171–201

[63] Sitnik S. M., “A short survey of recent results on Buschman–Erdélyi transmutations”, J. Inequal. Spec. Funct., 8:1 (2017), 140–157 | MR

[64] Sprinkhuizen-Kuyper I. G., “A fractional integral operator corresponding to negative powers of a certain second-order differential operator”, J. Math. Anal. Appl., 72:2 (1979), 674–702 | DOI | MR | Zbl