“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

TMSF’ 2017 BOOK OF ABSTRACTS

Sofia, 2017 Institute of Mathematics and Informatics Bulgarian Academy of Sciences Website: http://www.math.bas.bg/∼tmsf/2017/ Organizing Committee: Emilia Bazhlekova and Jordanka Paneva-Konovska (Co-Chairs), Yulian Tsankov, Donka Pashkouleva, Georgi Dimkov, Nikolay Ikonomov, Ivan Bazhlekov (Local Members), Miglena Koleva (Ruse University), Djur- djica Takaci (Serbia), Biljana Jolevska-Tuneska (Macedonia), Nicoleta Breaz (Romania) Scientific Program Committee: Virginia Kiryakova, Stepan Tersian (Co-Chairs), Blagovest Sendov, Ivan Dimovski, Nedyu Popivanov, Tsvyatko Rangelov (Bulgaria); Teodor Atanackovic, Stevan Pilipovic, Arpad Takaci, Predrag Rajkovic (Serbia); Nikola Tuneski (Macedonia), Daniel Breaz (Romania), Yuri Luchko (Germany, FCAA), Igor Podlubny (Slovak R., FCAA), J. Ten- reiro Machado (Portugal, FCAA)

This Conference is dedicated to: • The 70th anniversary of Institute of Mathematics and Informatics – Bulgarian Academy of Sciences, created 27 October 1947 • The 20th volume of the specialized international journal “Fractional Calculus and Applied Analysis”, https://www.degruyter.com/view/j/fca • The 65th anniversary of its Ed.-in-Chief, Virginia Kiryakova

Acknowledgements: Thanks are due to the Institute of Mathematics and Informatics (IMI), http://math.bas.bg/index.php/en/ – Bulgarian Academy of Sciences (BAS), as a host of Conference providing its facilities and support. This Conference is organized by Department “Analysis, Geometry and Topology” at IMI and under the auspices of the bilateral agreements (2017– 2019) between Bulgarian Academy of Sciences and Serbian and Macedonian Academies of Sciences and Arts. Some of the reported research results are in frames of the working programs of the corresponding bilateral projects as well as of research projects with National Science Fund of Bulgaria, related to the TMSF 2017 topics (Grant DFNI-I 02/9 and Grant DFNI-I 02/12). 1

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

HISTORY OF “TMSF” International Meetings in Bulgaria Website: http://www.math.bas.bg/∼tmsf/

• The 1st International Workshop “TMSF, Sofia’ 94” took place near Sofia (the resort town of Bankya, 20 km far from Sofia), 12-17 August 1994. Attended by 46 mathematicians and 6 accompanying persons from 15 countries. • The 2nd International Workshop “TMSF, Varna’ 96” took place in Varna (the Black Sea resort “Golden Sands”), 23-30 August 1996. Attended by 73 participants and several accompanying persons from 19 countries. • The 3rd International Workshop “TMSF, AUBG’ 99” was in the town of Blagoevgrad (100km south from Sofia), 13-20 August 1999, with the kind assistance and co-organization of the American University in Bulgaria (AUBG), with 47 participants from 16 countries. • The 4th International Workshop “TMSF, Borovets’ 2003” took place in the frames of the 1st MASSEE (Mathematical Society of South-Eastern Europe) Congress, in Borovets (famous winter resort in the Rila mountain), 15-21 September 2003, with 40 participants from 13 countries. • The 5th International meeting “TMSF”, 27-31 August 2010, was organized as an International Symposium “Geometric Function Theory and Applications’ 2010” held in Sofia at IMI-BAS, and attracted 50 participants and 7 accompanying persons from 12 countries. Details at: http://www.math.bas.bg/∼tmsf/gfta2010/. • The 6th International conference “TMSF 2011” was held in Sofia, 20-23 October 2011, in IMI-BAS, with 62 participants from 19 countries. It was dedicated to the 80th Anniversary of Prof. Peter Rusev. For all details, Book of Abstracts and Proceedings of TMSF2011, photos, etc., visit http://www.math.bas.bg/∼tmsf/2011/. • The 7th International Minisymposium “TMSF 2014” was held in Sofia, 7-10 July 2014, in frames of International conference “Mathematics Days in Sofia’ 2014”, hosted by IMI-BAS. It was dedicated to the 80th An- niversary of Prof. Ivan Dimovski, 30 of all 280 participants were especially as TMSF participants. Details: http://www.math.bas.bg/∼tmsf/2014/. • And this is the 8th event of the series of “TMSF”. 2

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017 FRACT CALC APPL ANAL – 20 YEARS Vol. 1 (1998) – Vol. 20 (2017) Website: https://www.degruyter.com/view/j/fca The “FCAA” journal, abbreviated as “Fract. Calc. Appl. Anal.”, Print ISSN 1311-0454, Electronic ISSN 1314-2224, is a specialized in- ternational journal for theory and applications of an important branch of Mathematical Analysis (Calculus), where the differentiations and integra- tions can be of arbitrary non-integer order. A peer-reviewed journal with high standards guaranteed by the Editorial Board’s list and carefully se- lected external referees and proven by the recently achieved high values of: Thomson Reuters Impact Factor (JIF)= 2.974 (2013), 2.245 (2014), 2.246 (2015), 2.034 (2016), 5-year Impact Factor: 2.359; Scopus Impact Rang (SJR) = 2.106 (2013), 1.433 (2014), 1.602 (2015), CiteScore 2016: 2.18; these have launched “FCAA” to top 10 places in the TR ranking lists for Maths and Appl. Maths. Abstracted / Indexed in: ISI Science Citation Index Expanded and Journal Citation Reports/Science Edition (Thomson Reuters); Sco- pus (Elsevier); SCImago (SJR); WorldCat (OCLC); Mathematical Reviews (MathSciNet); Zentralblatt Math.; Summon (Serials Solutions / ProQuest); Primo Central (ExLibris); Celdes; JournalTOCs; Google Scholar; etc. Published with the kind assistance and cooperation of: Institute of Mathematics and Informatics – Bulgarian Academy of Sciences.

Established in 1998, with Founding Publisher: (Vol. 1–Vol. 13, till 2010) Institute of Mathematics and Informatics – Bulgarian Academy of Sciences. Contents and abstracts of all volumes, back volumes 2005–2010, and journal’s details are available at Editor’s websites: http://www.math.bas.bg/∼fcaa , http://www.diogenes.bg/fcaa. Co-published in the period 2011–2014, Vol. 14–Vol. 17, by: Versita = De Gruyter Open, Warsaw and Springer-Verlag, Wien, Website (former): http://http://degruyteropen.com/serial/fcaa/ and Online Electronic version then was at SpringerLink. Current Publisher – since 2015, Vol. 18: Walter de Gruyter GmbH, Berlin / Boston. Publisher’s website: http://www.degruyter.com/view/j/fca, with all journal’s details and online electronic version (Vols. 14–17, free). 3

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

INSTITUTE OF MATHEMATICS AND INFORMATICS – Bulgarian Academy of Sciences Website: http://math.bas.bg/index.php/en/ One of the events to which this Conference is dedicated, is the 70th an- niversary of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI – BAS). IMI-BAS was created shortly after the end of World War II by the ef- forts and high professionalism of a generation of Bulgarian mathematicians whom we now thankfully style the pivots of Bulgarian mathematics. The date 27 October 1947 is rightly considered the birth date of IMI, when the Executive Council of the Academy confirmed the plan of the scientific activ- ity in 1947/1948. Then, the section devoted to the mathematical sciences included the work of three commissions: • Commission for demographic studies (Acad. Kiril Popov) • Commission for mathematical studies of the representative method in statistics (Acad. Nikola Obrechkoff) • Commission for financial mathematical research of state bonds and external bonds guaranteed by the state (Acad. Kiril Popov). The scientific plan contained also detailed individual plans by the mathematicians • Acad. Ivan Tzenov, • Acad. Ljubomir Tchakalov, • Acad. Nikola Obrechkoff, • Acad. Kiril Popov. Fully in line with the dynamics of the time and development of research in mathematical sciences, our institution changed its name several times. In 1949 the Mathematical Institute (MI) was established at the Physical and Mathematical Branch of the Academy and Acad. Ljubomir Tchakalov was appointed as its head. The MI was renamed Mathematical Institute with Computing Centre (MI with CC) in 1961 (when the 1st Computing Centre in Bulgaria was created), then as Institute of Mathematics and Me- chanics with Computing Centre (IMM with CC) in 1972, and Institute of Mathematics (IM) in 1994. The flourishing period 1971–1988 was the time of the United Centre for Science and Training in Mathematics and Mechan- ics, combining together the stuff (more than 500 members) and missions of both IMM with CC (Institute) and FMM (Faculty of Mathematics and Mechanics – Sofia University), with Director Acad. Ljubomir Iliev. Then it started the 3 levels of university education, the system which was accepted in whole Europe 20 years later. The current name, IMI, goes back to 1995. 4

Since its creation in 1947, IMI has been a leading Bulgarian centre for research and training of highly qualified specialists and exercising an effi- cient, widerange, consistent policy related to the fundamental trends in the development of mathematics, computer science and information technolo- gies, including also working and care for teachers and talented pupils. Mission of IMI: • Development of fundamental and applied research in mathematics and informatics in compliance with the national and European priorities, integration of IMI into the European Research Area; • Scientific research in the fields of mathematical structures, mathe- matical modeling and mathematical informatics and linguistics, enriching the theoretical foundations of mathematics and informatics and leading to innovative applications in other sciences, in information and communica- tion technologies, industry and society; • Application of mathematics and informatics in the national educa- tional programmes and educational processes at all levels in the country; • Establishing the Institute of Mathematics and Informatics as a lead- ing research centre in Bulgaria in the field of mathematics and informatics. http://math.bas.bg/index.php/en/en-about-mission-2/en-departments.

Department “Analysis, Geometry and Topology” (AGT) was created in March 2011 by merging the former departments Complex Analy- sis, Geometry and Topology, and Real and Functional Analysis, among the first scientific groups in Institute, since 1962 (Dept. of Advanced Analysis). The stuff (25 members) is currently working in two basic trends: the research projects “Transform Methods, Special Functions and Complex Ap- proximations” and “Several Complex Variables, Differential Geometry and Topology”. The AGT Department is one of the biggest and most active parts of the Institute; with authors of several monographs and yearly long lists of published scientific papers in prestigious journals and high impact citations; organizing several international conferences (almost each year); handling research projects under National Science Fund, bilateral academic agreements, European programs; with a Joint Seminar on AGT areas; par- ticipating in publishing of several international mathematical journals.

Details at: http://math.bas.bg/index.php/en/en-analysis-department, http://math.bas.bg/index.php/en/en-analysis-geometry-topology-staff, http://www.math.bas.bg/complan/seminar/SeminarAGT EN.html. 5

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

TMSF’ 2017 – ABSTRACTS

Main Topics in Scientific Program of TMSF 2017 include:

• “Fractional Calculus and Applied Analysis” (FCAA) - the topics of the journal “Fract. Calc. Appl. Anal.” • “Transform Methods and Special Functions”(TMSF) - topics as: Special Functions, Integral Transforms, Convolutional and Operational Calculus, Fractional and High Order Differential Equations, Numerical Methods, Generalized Functions, Complex Analysis, etc. • “Geometric Function Theory and Applications” (GFTA) • Applications, etc. 6

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

WAVE EQUATION WITH FRACTIONAL DERIVATIVES OF REAL AND COMPLEX ORDER

Teodor ATANACKOVIC´ Serbian Academy of Sciences and Arts Nikole Pasica 6, 21 000 - Novi Sad, SERBIA e-mail: [email protected]

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of real and complex order. Thus in the constitutive equations that we shall study the left Riemann-Liouville fractional derivative operator of real order α ∫ t α 1 d u(x, τ) 0Dt u(x, t) = α dτ, 0 < α < 1. Γ(1 − α) dt 0 (t − τ) and the following fractional operator of complex order ( ) 1 − D¯ α,β := ˆb Dα+iβ + ˆb Dα iβ , 0 t 2 1 0 t 2 0 t √ iβ −iβ where i = −1, ˆb1 = T , ˆb2 = T , and |ˆb1| = |ˆb2| (T is a constant having the dimension of time) are used. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed in detail.

This is joint work with Sanja Konjik, Marko Janev and Stevan Pilipovic.

Partially supported and in the frames of the project “Analytical and nu- merical methods for differential and integral equations ...” under bilateral agreement between SASA and BAS. MSC 2010: 26A33; 74D05 Key Words and Phrases: wave equations; fractional order deriva- tives 7

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SUBORDINATION APPROACH TO FRACTIONAL WAVE EQUATIONS Emilia BAZHLEKOVA §, Ivan BAZHLEKOV, Ivan GEORGIEV Institute of Mathematics and Informatics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., Block 8, Sofia – 1113, BULGARIA e-mails: [email protected], [email protected], [email protected] By means of the principle of subordination (Pr¨uss,1993) it is possible to construct solutions of linear time-fractional initial-boundary value problems from a known solution, e.g., of the corresponding first or second order equation, or of a simpler time-fractional one. We establish subordination results for a class of fractional evolution equations with Caputo time-derivatives, which orders are discretely or con- tinuously distributed over an interval [α, β], such that 1 < β ≤ 2 and β − α ≤ 1. The solution is explicitly represented in terms of the solution of the corresponding second-order problem and of a probability density function (p.d.f.), obtained from the fundamental solution of the spatially one-dimensional version of the considered equation. Explicit integral rep- resentation of the p.d.f. is derived and its properties are studied. Subor- dination to the corresponding single-term time-fractional equation of order β is also established. The obtained subordination representations are applied for understand- ing the regularity and asymptotic behaviour of the solution, as well as for the numerical computation of the solution in some particular cases. The analytical findings are supported by numerical work. Acknowledgements: The authors acknowledge the support by Grant DFNI-I 02/9 by Bulgarian National Science Fund, and Project “Analytical and numerical methods for differential and integral equations...” under bilateral agreement between BAS and SASA. MSC 2010: 26A33, 31B10, 35Q35, 35R11, 44A10 Key Words and Phrases: Caputo fractional derivative, time-fractional diffusion-wave equation, Bernstein function, solution operator, strongly continuous cosine family 8

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON THE SOLUTIONS OF A FRACTIONAL DIFFERENTIAL INCLUSION WITH RANDOM EFFECTS

Aurelian CERNEA Faculty of Mathematics and Computer Science – University of Bucharest Academiei 14, Bucharest - 010014, ROMANIA e-mail: [email protected]

In general, if the parameters appearing in a dynamical system are known from the statistical point of view, the approach used in the math- ematical models is the one of random differential equations or stochastic differential equations. We note that random differential equations gener- alize deterministic differential equations and appear in a large number of applications. We are concerned, mainly, with the following problem

(Dα,βx)(t, w) ∈ F (t, x(t, w), w) a.e. t ∈ [0,T ], w ∈ Ω, 1−γ (1) (I x)(t, w)|t=0 = φ(w), w ∈ Ω, where α ∈ (0, 1), β ∈ (0, 1], γ = α + β − αβ, T > 0, Ω is a measurable space, φ :Ω → R is a measurable function, F : [0,T ] × R × Ω → P(R) is a set-valued map, I1−γ is the left-sided Riemann-Liouville integral of order (1−γ) and Dα,β is the Hilfer fractional derivative of order α and type β. Our aim is to adapt suitably Filippov’s ideas in order to obtain the existence of solutions for problem (1). Recall that for a differential inclusion defined by a lipschitzian set-valued map with nonconvex values, Filippov’s theorem consists in proving the existence of a solution starting from a given “quasi” solution. Moreover, the result provides an estimate between the “quasi” solution and the solution obtained. In this way we improve some existence results for problem (1) already existing in the literature. MSC 2010: 26A33, 34A60 Key Words and Phrases: random fractional differential inclusion, Hilfer fractional derivative, selection 9

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

COMPLEX AUTO-WAVE SOLUTIONS IN FRACTIONAL REACTION-DIFFUSION SYSTEMS CLOSE TO BIFURCATION POINT Bohdan DATSKO 1,§, Vasyl GAFIYCHUK 2 1 Dept. of Math. and Appl. Phys. – Rzeszow University of Technoogy al. Powstancow Warszawy 8, Rzeszow – 35-959, POLAND e-mail: [email protected] 2 Inst. of Appl. Problems of Mech. and Math., NAS of Ukraine Naukova Str. 3b, Lviv – 79-060, UKRAINE e-mail: [email protected] In contrast to the integer-order dynamical systems, conditions of insta- bility in fractional dynamical systems are realized in a qualitatively different manner. As a result, dynamics of the fractional order systems can be much more complex in compare with the integer order systems and substantially depend on the orders of fractional derivatives. In our investigation we study complex auto-wave solutions in nonlinear fractional reaction-diffusion systems (FRDS). The main attention is paid to nonlinear dynamics at the parameters close to bifurcation point. Despite the fact that the homogeneous state is stable at the parameters lower than bifurcation ones, a variety of nonlinear solutions are realized in the subcrit- ical domain. Depending on the standard bifurcation parameters and the orders of fractional derivatives, new types of auto-wave solutions is revealed in such systems. For basic FRDS, through linear stability analysis and numerical simu- lations, conditions of existence and main properties of auto-wave solutions are studied. An overall picture of different types of nonlinear dynamics depending on orders of fractional derivatives are presented. The obtained results significantly enrich the nonlinear dynamics that we imagine to have in fractional reaction-diffusion systems at subcritical bifurcation. MSC 2010: 35K57, 35B36, 35C07, 34A99 Key Words and Phrases: fractional differential equations; fractional reaction-diffusion system; auto-wave solutions; instability; bifurcation 10

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

CHERCHEZ LA FEMME . . . Georgi DIMKOV Institute of Mathematics and Informatics Bulgarian Academy of Sciences “Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIA e-mail: [email protected]

A few days ago the Nobel Foundation and the Norwegian Nobel Com- mittee announced the names of the new Nobel Prize Laureates in Chem- istry, Literature, Peace, Physics and Physiology and Medicine. My God !!! And what about Mathematics ??? That science who helped and still is helping the development of all natural sciences, technical and engineering sciences and in last time social sciences as well. Good question. In the Testament of Alfred Nobel there are no reasons for the above selection. Thus we could make a lot of speculations: – Nobel wasn’t good in Maths; – Among the friends of Nobel there was no mathematician to advise him to include Mathematics in the Testament; – Among the friends of Nobel there were many mathematicians and he disliked their behavior; – Maths is not a direct producing science; – Finally, why not, Chercher la femme. Excellent idea! It gives many people the possibility to exercise and demonstrate their imagination. But who could be the offender? For many people there is only one answer – Mittag-Leffler: Swedish, mathematician, man about town, especially during his stay in Helsinki. Doubtless this is the reason of Alfred Nobel. Let us scrutinize the life of Magnus Gustav Mittag-Leffler and then decide where the truth is. MSC 2010: 01A55-01A60, 33E12, 30B40, 30D20 Key Words and Phrases: life of Magnus Gustav Mittag-Leffler; Mittag-Leffler function and fractional calculus 11

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

HYPER-BESSEL TRANSFORMATIONS

Ivan DIMOVSKI Institute of Mathematics and Informatics Bulgarian Academy of Sciences “Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIA e-mail: [email protected]

The idea of transmutation operators could be traced to the (wrong) paper [1] of J. Delsarte and J.L. Lions. An extension of the ideas of this paper can be seen in the posthumous paper of J. Delsarte [2]. Some 40 years ago the author proposed transmutation operators in an explicit form which “transmute” an arbitrary hyper-Bessel differential operator of a fixed integer order m ≥ 1 in each other hyper-Bessel operator of the same order, in particular, into (d/dt)m. In [3], this transmutation operator is presented in an integral form, using Meijer G-function. It helps also to represent in similar way, the fractional powers of the hyper-Bessel operators. Here we are to exhibit the simple principle behind this transmutation operator. MSC 2010: 26A33, 47B38 Key Words and Phrases: transmutation operator; hyper-Bessel op- erator; Meijer’s G-functions

References [1] J. Delsarte, J.L. Lions, Transmutations d’op´erateurs diff´erentiales dans le domaine complexe. Comment. Math. Helv. 32, No 2 (1957), 113–128. [2] J. Delsarte, Un principle g´en´eral de construction d’op´erateurs de transposition. In: Ouvres de Jean Delsarte, II, Editions du CNRS, 1971, 893–948. [3] I.H. Dimovski, V.S. Kiryakova, Transmutations, convolutions and fractional powers of Bessel type operators via Meijer’s G-function. In: Proc. “Complex Analysis and Applications 1983”, Sofia, 1985, 47–66. 12

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

NONLOCAL RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION INCLUSIONS IN BANACH SPACE

Tzanko DONCHEV 1,§, Mohamed ZIANE 2 1 Department of Mathematics University of Architecture and Civil Engineering 1 Hr. Smirnenski Str., Sofia – 1046, BULGARIA e-mail: [email protected] 2 Department of Mathematics, Ibn Khaldoun University Tiaret, ALGERIA e-mail: [email protected]

Let E be a Banach space and let A : D(A) → E be a densely defined linear operator A : D(A) → E. In this paper we study the existence of solution to the following nonlocal fractional evolution equation   L q ∈ ′ ( D0+ x)(t) = Ax(t) + fx(t), a.e. t I = (0, 1], f (t) ∈ F (t, x(t)), (1)  x xq(0) = g(xq(·))(= x0),

L q · where ( D0+ ) is the Riemann-Liouville fractional derivative of order q, 1−q 1−q 0

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

PERTURBED FRACTIONAL EIGENVALUE PROBLEMS

Maria F˘arc˘a¸seanu 1,2 1 Department of Mathematics – University of Craiova 13 A. I. Cuza, Craiova - 200585, ROMANIA e-mail: [email protected] 2 “Simion Stoilow” Institute of Mathematics of the Romanian Academy 21 Calea Grivit¸ei, Bucharest – 010702, ROMANIA

Let Ω ⊂ RN (N ≥ 2) be a bounded domain with Lipschitz boundary. s For each p ∈ (1, ∞) and s ∈ (0, 1) we denote by (−∆p) the fractional (s, p)-Laplacian operator. In this paper we study the existence of nontrivial s p−2 solutions for a perturbation of the eigenvalue problem (−∆p) u = λ|u| u, in Ω, u = 0, in RN \Ω, with a fractional (t, q)-Laplacian operator in the left-hand side of the equation, when t ∈ (0, 1) and q ∈ (1, ∞) are such that s − N/p = t − N/q. We show that nontrivial solutions for the perturbed eigenvalue problem exists if and only if parameter λ is strictly larger than the first eigenvalue of the (s, p)-Laplacian.

This is a joint work with Mihai Mih˘ailescuand Denisa Stancu-Dumitru.

MSC 2010: 35P30, 49J35, 47J30, 46E35 Key Words and Phrases: perturbed eigenvalue problem; non-local operator; variational methods; fractional Sobolev space 14

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

THE PRABHAKAR FUNCTION: THEORY AND APPLICATIONS Roberto Garrappa Department of Mathematics – University of Bari Via E. Orabona 4, Bari - 70126, Bari, ITALY e-mail: [email protected]

A three parameter Mittag-Leffler function ∞ 1 ∑ Γ(γ + k)zk Eγ (z) = α,β Γ(γ) k! Γ(αk + β) k=0 was introduced in 1971 by the Indian mathematician Tillk Raj Prabhakar and inherited his name. This Prabhakar function has been recently re- discovered and widely exploited, mainly for its applications in relaxation processes in anomalous dielectrics of Havriliak-Negami type. After preliminarily discussion of some important applications of the Prabhakar function and its main properties, some new results on the asymp- totic expansion for large arguments are illustrated. Fractional-order opera- tors of Prabhakar type are hence discussed and some results on the solution of differential equations with Prabhakar derivatives are shown. We conclude by illustrating some schemes for the numerical solution of differential equations with the fractional Prabhakar derivative. MSC 2010: 33E12, 26A33, 34E05 Key Words and Phrases: Prabhakar function; Mittag-Leffler func- tion; fractional calculus; asymptotic expansion References [1] R. Garrappa, On Gr¨unwald-Letnikov operators for fractional relax- ation in Havriliak-Negami models. Commun. Nonlinear Sci. Numer. Simul. 38 (2016), 178–191. [2] R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone functions, Fract. Calc. Appl. Anal. 19, No 5 (2016), 1105–1160. [3] R. Garra and R. Garrappa, The Prabhakar or three parameter Mittag–Leffler function: theory and application. Submitted. 15

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

OPERATIONAL METHOD FOR FRACTIONAL FOKKER-PLANCK EQUATION

Katarzyna GORSKA´ 1 Department of Mathematical Physics and Theoretical Astrophysics Institute of Nuclear Physics, Polish Academy of Sciences ul. Radzikowskiego 152, Krak´ow31-342, POLAND e-mail: [email protected]

We will present some results for fractional equations of Fokker-Planck type using the evolution operator method. Exact forms of one-sided L´evy stable distributions are employed to generate a set of self-reproducing so- lutions. Explicit cases are reported and studied for various fractional order of derivatives, with different initial conditions, and for different versions of Fokker-Planck operators.

MSC 2010: 26A33, 33E12, 35Q84 Key Words and Phrases: fractional calculus; special functions; inte- gral transforms and operational calculus; fractional Fokker-Planck equation 16

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

NUMERICAL COMPARISON OF ITERATIVE METHODS FOR THE SIMULTANEOUS DETERMINATION OF POLYNOMIAL COMPLEX ZEROS Roselaine NEVES MACHADO 1, Luiz GUERREIRO LOPES 1,§ 1 Federal Institute of Rio Grande do Sul, IFRS, Bento Gon¸calves Campus 95700-000 Bento Gon¸calves, RS, BRAZIL e-mail: [email protected] 2 Faculty of Exact Sciences and Engineering, University of Madeira Penteada Campus, 9000-390 Funchal, Madeira Is., PORTUGAL e-mail: [email protected]

There are lot of iterative methods for the simultaneous approximation of polynomial complex zeros, from the more classical numerical algorithms, such as the well-known Weierstrass-Durand-Dochev-Kerner method, the Maehly-Ehrlich-Aberth method, and the B¨orsch-Supan-Nouren method, to the more recent ones. However, relatively little has been done in order to numerically compare and evaluate these simultaneous zero-finding methods. Moreover, examples in literature may give a distorted picture of the efficiency of these methods and the quality of the approximations produced by them. In this work, an extensive computational analysis of the main known it- erative methods for the simultaneous approximation of polynomial complex zeros is performed. The simultaneous iterative methods considered in this study were implemented in Matlab/Octave, and evaluated and compared by using a very large set of test polynomials. Robustness, computational efficiency and accuracy of the generated approximations were used as basic evaluation criteria for these iterative numerical algorithms. The obtained results provide information on the relative merits and performance of the different simultaneous iterative methods studied. MSC 2010: 30C10, 65H04, 65H05, 65Y20 Key Words and Phrases: polynomial zeros; simultaneous iterative methods; computational efficiency; numerical accuracy 17

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

RECENT ADVANCES IN FRACTIONAL VISCOELASTICITY Andrea GIUSTI 1,2 1 Department of Physics & Astronomy University of Bologna and INFN Via Irnerio 46, Bologna – 40126, ITALY e-mail: [email protected] 2 Arnold Sommerfeld Center for Theoretical Physics Ludwig-Maximilians-Universit¨at Theresienstraße 37, M¨unchen – 80333, GERMANY The aim of this paper is to provide a summary of two recently proposed fractional order viscoelastic models. In particular, we deal with the so- called Bessel models and with a modification of the fractional Maxwell model involving the Prabhakar derivatives. The Bessel models arise naturally from a generalization of a model for fluid-filled elastic tubes, first proposed by Giusti and Mainardi in 2014. Interestingly, in the Laplace domain, both memory and material functions characterizing this class of theories are represented by ratios of modified Bessel functions of contiguous order ν > −1. Furthermore, we also discuss all the main viscoelastic features and asymptotic behavior of these models. Finally, by studying the propagation of transient waves in a Bessel medium, we provide the analytic expression for the response function of the material as we approach the wave-front, for different values of the parameter ν. Secondly, we also discuss a recently proposed linear viscoelastic model based on the so-called Prabhakar operators of fractional calculus. In par- ticular, we present a detailed description of the modified fractional Maxwell model, in which we replace the Caputo fractional derivative with the Prab- hakar one. Furthermore, we also show how to recover a formal equivalence between the new model and the known classical models of linear viscoelas- ticity by means of a suitable choice of the parameters in the Prabhakar derivative. MSC 2010: 26A33, 34A35, 44A10, 76A10, 33C10 Key Words and Phrases: fractional calculus; special functions; Bessel models; Prabhakar derivative 18

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

TOWARDS UNIFIED DESCRIPTION OF RELAXATION PROCESSES

Andrzej HORZELA Department of Mathematical Physics and Theoretical Astrophysics Institute of Nuclear Physics, Polish Academy of Sciences ul. Radzikowskiego 152, Krak´ow31-342, POLAND e-mail: [email protected]

We study functions related to the Havriliak-Negami dielectric relax- α −β ation pattern given in the frequency domain by ∼ [1 + (iωτ0) ] with τ0 being some characteristic time. For rational α = l/k < 1 and β > 0 we furnish exact and explicit expressions for response and relaxation func- tions in the time domain and suitable probability densities in their “dual” domain. These functions are expressed as finite sums of generalized hy- pergeometric functions, convenient to handle analytically and numerically. Moreover, they are related to the one-sided L´evystable distributions which self-similarity may be used to introduce two-variable densities and to show that they satisfy the integral evolution equations.

MSC 2010: 26A33, 46F12, 33E99, 49M20 Key Words and Phrases: fractional calculus; special functions; Prab- hakar function; Meijer G-function 19

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SOME STABILITY PROPERTIES WITH INITIAL TIME DIFFERENCE FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

Snezhana HRISTOVA 1,§, Ravi AGARWAL 2, Donal O’REGAN 3

1 University of Plovdiv “Paisii Hilendarski” Tzar Asen 24, Plovdiv – 4000, BULGARIA e-mail: [email protected] 2 Texas A&M University-Kingsville Kingsville, TX 78363, USA e-mail: [email protected] 3 School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway, IRELAND e-mail: [email protected]

Lipschitz stability with initial time difference and Mittag-Leffler stabil- ity with initial time difference for nonlinear nonautonomous Caputo frac- tional differential equation are defined and studied. The stability with ini- tial time difference allow us to compare the behavior of two solutions with different initial values as well as different initial times. The dependence of the Caputo fractional derivative on the initial time causes several problems with the study of the defined types of stability. The fractional order ex- tension of comparison principle via scalar fractional differential equations with a parameter is employed. Some sufficient conditions are obtained by the application of the Lyapunov like functions. The relation between both types of stability is discussed theoretically and it is illustrated on examples. MSC 2010: 34A08, 26A33, 34D20 Key Words and Phrases: Caputo fractional derivatives; Lipschitz stability; Mittag-Leffler stability; initial time difference Acknowledgements. This research was partially supported by Fund MU17-FMI-007, Fund Scientific Research, University of Plovdiv “Paisii Hilendarski”. 20

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

RESULTS ON THE EXPONENTIAL INTEGRAL

Biljana JOLEVSKA-TUNESKA Department of Mathematics and Physics Ss. Cyril and Methodius University in Skopje Faculty of Electrical Engineering and Informational Technologies Karpos 2 bb, 1000 Skopje, Republic of MACEDONIA e-mail: [email protected]

Results on the convolution product of the exponential integral and ex- ponential function are given. These results are found in a space of distri- butions. The convolution products gained in this work may be considered as a generalization of Chandrasekhar’s functions which are needed in the problem of diffuse reflections and transmission of radiation by an atmo- sphere. MSC 2010: 33F10, 46F10 Key Words and Phrases: exponential integral; exponential function; distribution; convolution Work in frames of bilateral agreement between MANU and BAS. 21

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON A HIGHER ORDER MULTI-TERM TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATION INVOLVING CAPUTO-FABRIZIO DERIVATIVE

Erkinjon KARIMOV 1,§, Sardor PIRNAFASOV1 1 Institute of Mathematics named after V.I. Romanovsky – Uzbekistan Academy of Sciences Durmon yuli str. 29, Tashkent – 100125, UZBEKISTAN e-mail: [email protected], [email protected]

In this talk we aim to show an algorithm how to reduce initial value problem (IVP) for multi-term fractional differential equation (DE) with Caputo-Fabrizio (CF) derivative to the IVP for integer order DE and using this result to prove a unique solvability of a boundary value problem (BVP) for partial differential equation (PDE) involving CF derivative on time- variable. Separation of variables leads us to the following IVP for fractional DE: ∑k  α+n λn CF D0t T (t) + µ T (t) = f(t),  n=0 (1) (i) T (0) = Ci, i = 0, 1, 2, ..., k. Here λn and µ are given real numbers, f(t) is a given function, k ∈ N0, ∫t α 1 ′ − α (t−s) D T (t) = T (s)e 1−α ds (2) CF 0t 1 − α 0 is a fractional derivative of order α (0 < α < 1) in Caputo-Fabrizio sense. Remark. We note that the used algorithm allows us to investigate fractional spectral problems{ such Dα+1 T (t) + µ T (t) = 0, CF 0t (3) T (0) = 0,T (1) = 0, reducing them to second order usual spectral problems. MSC 2010: 33E12 Key Words and Phrases: fractional derivative; initial and boundary value problems; integer and fractional order differential equations 22

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON A SYSTEM OF NONLINEAR NONLOCAL REACTION DIFFUSION EQUATIONS

Mokhtar KIRANE Department of Mathematics – University of La Rochelle Avenue M. Cr´epeau, 17000 – La Rochelle, FRANCE e-mail: [email protected]

The reaction diffusion system with anomalous diffusions and a balance law α/2 ut + (−∆) u = −f(u, v), (1) β/2 vt + (−∆) u = +f(u, v), (2) for RN , t > 0 and supplemented with the initial conditions

N u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ R , (3) where 0 < α, β ≤ 2 is considered. The existence of global solutions is proved in two situations: (i) a polynomial growth condition is imposed on the reaction term f when 0 < α ≤ β ≤ 2; (ii) no growth condition is imposed on the reaction term f when 0 < β ≤ α ≤ 2.

MSC 2010: 35K57, 35R11 Key Words and Phrases: reaction-diffusion systems; fractional Lapla- cian 23

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FRACTIONAL CALCULUS OPERATORS OF SPECIAL FUNCTIONS ? – THE RESULT IS WELL PREDICTABLE !

Virginia KIRYAKOVA Institute of Mathematics and Informatics Bulgarian Academy of Sciences “Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIA e-mail: [email protected]

Recently many authors are spending lot of time and efforts to evaluate various operators of fractional order integration and differentiation and their generalizations of classes of, or rather particular, special functions. Practically, these are exercises to calculate improper integrals of products of different special functions. The list of such works is rather long and yet growing daily. So, to illustrate our general approach, we limit ourselves to mention here only a few of them. Since there is a great variety of special functions, as well as of operators of fractional calculus, the mentioned job produces a huge flood of publications. Many of them use same formal and standard procedures, and besides, often the results sound not of practical use, with except to increase authors’ publication activities. In this work, first we point out on some few basic classical results and ideas descending yet from 80’s (as Bateman-Erd´elyi’stables of integral transforms, 1954; handbook by Askey, 1975; a survey by Lavoie-Osler- Tremblay, 1976) - with examples for Riemann-Liouville fractional integrals and derivatives of some basic elementary and hypergeometric functions. We need to emphasize also that the basic “Method of Calculation of Integrals of Special Functions” (as title of Marichev’s book in Russian of 1978, EN edition as a handbook of 1983) is the use of Mellin transform techniques. The approach presented here combines these classics with this author’s ideas and developments for many years (since 1990, [1]–[6]). We show how one can do the mentioned task at once, in the rather general case: for both operators of generalized fractional calculus and generalized hypergeometric functions. In this way, the greater part of the results in the mentioned publications are well predicted and fall just as rather special cases of the discussed general scheme. 24

As necessary preliminaries, we give a brief sketch on the operators of Generalized Fractional Calculus (GFC) whose detailed theory is presented in [2], as well as the definitions of the generalized hypergeometric functions (Fox, Meijer, Wright) in which scheme the basic classes of Special Functions (SF) are included. Then, general results for the images of the Wright generalized hypergeo- metric functions pΨq (therefore, also of pFq and all their special cases) under the generalized fractional (multi-order) integrals and derivatives are pro- vided. These operators of GFC, defined by means of H- and G-functions in the kernels, are also commutable (m-tuple) compositions of Erd´elyi-Kober operators, each one increasing by 1 the indices p and q of the mentioned SF. Then the GFC images of pΨq and pFq are expectedly, special functions of the type p+mΨq+m and p+mFq+m with suitable additional parameters. The mentioned 15 examples from works recently published by various authors show the effectiveness of the proposed general scheme to encompass at once such results. A work under the bilateral agreements of BAS with SANU and MANU. MSC 2010: 26A33, 33C60, 33E12, 44A20 Key Words and Phrases: generalized fractional integrals and deriva- tives; generalized hypergeometric functions; integrals of special functions References [1] V. Kiryakova, Poisson and Rodrigues type fractional differintegral for- mulas for the generalized hypergeometric functions pFq. Atti Sem. Mat. Fis. Univ. Modena 39 (1990), 311–322. [2] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman - J. Wiley, Harlow - N. York, 1994. [3] V. Kiryakova, All the special functions are fractional differintegrals of elementary functions. J. Phys. A: Math. & General 30, No 14 (1997), 5085–5103; doi:10.1088/0305-4470/30/14/019. [4] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and re- lations to generalized fractional calculus. J. Comput. Appl. Math. 118 (2000), 241–259; doi:10.1016/S0377-0427(00)00292-2. [5] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Computers and Math. with Appl. 59, No 3 (2010), 1128–1141; doi:10.1016/j.camwa.2009.05.014. [6] V. Kiryakova, Fractional calculus operators of special functions? – The result is well predictable! Chaos Solitons and Fractals 102 (2017), 2–15; doi:10.1016/j.chaos.2017.03.006. 25

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FRACTIONAL STURM-LIOUVILLE PROBLEM – EXACT AND NUMERICAL SOLUTIONS

Malgorzata KLIMEK 1,§, Tomasz BLASZCZYK 1, Mariusz CIESIELSKI 2 1 Institute of Mathematics – Czestochowa University of Technology Armii Krajowej 21, Czestochowa - 42-201, POLAND e-mail: [email protected], [email protected] 2 Institute of Computer and Information Sciences – Czestochowa University of Technology Dabrowskiego 73, Czestochowa - 42-201, POLAND e-mail: [email protected]

We consider the fractional differential equation (called the fractional Sturm-Liouville equation) ( ) C α C α Db− p (x) Da+ y (x) + q (x) y (x) = λw (x) y (x) , (1) subject to the homogeneous mixed boundary conditions

C α y (a) = 0, p(x) Da+ y (x) x=b = 0, (2) ( ] ∈ 1 ≥ where order α 2 , 1 , p, q, w are given functions such that: p(x) 0; q, C C w are continuous, w(x) > 0 and Da+ , Db− denote Caputo derivatives. This fractional Sturm-Liouville problem (FSLP) is a special case of FSLPs involving the left and right derivatives introduced in paper [1], in- teresting and meaningful as it leads to a countable system of orthogonal eigenfunctions similar to the FSLP with homogeneous Dirichlet conditions studied previously in article [2]. We transform the fractional differential problem to the equivalent integral one and apply the Hilbert-Schmidt op- erators theory. Then, we prove that under suitable assumptions such an eigenvalue problem has a purely discrete, real, unbounded spectrum and the associated continuous eigenfunctions form an orthogonal basis in the respective Hilbert space. 26

Next, we develop a numerical method of solving system (1)-(2). The introduced numerical scheme is based on its integral form and it leads to the set of eigenvalues and to an orthogonal system of approximate solutions. The experimental order of convergence (EOC), both for eigenvalues and eigenfunctions, is analyzed and we show that for the new scheme it is higher then the one characterizing results presented in [3]. MSC 2010: 26A33, 34A08, 34B09 Key Words and Phrases: fractional differential equations; fractional eigenvalue problem; discrete spectra; numerical analysis References

[1] M. Klimek, O.P. Agrawal, Fractional Sturm-Liouville problem, Com- put. Math. Appl. 66 (2013), 795–812.

[2] M. Klimek, T. Odzijewicz, A.B. Malinowska, Variational methods for the fractional Sturm-Liouville problem, J. Math. Anal. Appl. 416, No 1 (2014), 402–426.

[3] M. Ciesielski, M. Klimek, T. Blaszczyk, The fractional Sturm- Liouville problem - numerical approximation and application in frac- tional diffusion, J. Comput. Appl. Math. 317 (2017), 573–588. 27

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON THE ASYMPTOTICS OF SEQUENCES OF PADE´ APPROXIMANTS

Ralitza KOVACHEVA Institute of Mathematics and Informatics – Bulgarian Academy of Sciences Acad. Bonchev str. 8, 1113 – Sofia, BULGARIA e-mail: [email protected]

In the present talk, the asymptotics of sequences of Pad´eapproximants – classical and multipoint and closed to rows will be discussed. We show that under appropriate conditions on the approximated function f there { } → ∞ is a sequence of Pad´eapproximants πn,mn , mn = o(n) as n which behaves like the Taylor series of the meromorphic continuation of f.

MSC 2010: 41A20, 41A21, 30E10 Key Words and Phrases: complex analysis; domain of m-mero- morphy; Pad´eapproximants 28

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SYMMETRIC QUANTUM BERNSTEIN FUNCTIONS AND APPLICATIONS

Valmir B. KRASNIQI Department of Mathematics – University of Prishtina Prishtin¨e– 10000, Republic of KOSOVO e-mail: [email protected]

We introduce the symmetric quantum Bernstein functions, and give sufficient and necessary conditions for a function to belong to the class of symmetric quantum Bernstein functions. Also, we give sufficient and neces- sary conditions for a function to belong to the class of symmetric quantum completely monotonic functions. For some classes of functions we give results concerning symmetric quantum completely monotonic and - Bern- stein functions. The obtained results are symmetric quantum analogues of known results. MSC 2010: 05A30 Key Words and Phrases: symmetric quantum completely monotonic function; symmetric quantum Bernstein function 29

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

USING ORDINARY DIFFERENTIAL EQUATIONS TO SOLVE LINEAR FRACTIONAL INTEGRAL EQUATIONS

Daniel CAO LABORA 1,§, Rosana RODR´IGUEZ-LOPEZ´ 2

1,2 Department of Mathematical Analysis, Statistics and Optimization – University of Santiago de Compostela R´uaLope G´omezde Marzoa, Santiago de Compostela – 15782, SPAIN 1 e-mail: [email protected] , 2 e-mail: [email protected]

The main goal of this work will be to show a new method to solve some fractional order integral equations. The material is based on a re- cent manuscript submitted for publication, together with new ideas. In that sense, we thank for some advices from other mathematicians, like F. Chamizo. Although the original idea is a bit more general, we can think about the following: we consider a fractional order integral equation like

p1 pn q1 ··· qn c1I0+ x(t) + + cnI0+ x(t) = f(t), (1) where c ∈ C and pi ∈ Q+ ∪ {0} for every i ∈ {1, . . . , n}. Using exclusively i qi basic results about polynomials, we construct an integral operator that turns (1) into a similar equation, but with integer orders. If we have some hypotheses over f, we can rewrite the natural order integral equation as a linear ODE with constant coefficients, which can be easily solved. The procedure can be even used to solve some fractional differential equations endorsed with suitable initial conditions. For instance, we apply this philosophy to solve analytically Bagley-Torvik equation. Finally, some comments will be made about an ongoing computational implementation.

MSC 2010: Primary 34A08, 26A33; Secondary 45A05 Key Words and Phrases: fractional order integral and differential equation; linear problems; fractional operators; explicit solutions 30

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

DIFFUSION ENTROPY METHOD FOR ULTRASLOW DIFFUSION USING INVERSE MITTAG-LEFFLER FUNCTION

Yingjie LIANG 1,§, Wen CHEN 2 1,2 Institute of Soft Matter Mechanics, College of Mechanics and Materials Hohai University, Nanjing, Jiangsu 211100, CHINA 1 e-mail: [email protected] , 2 e-mail: [email protected]

Ultraslow diffusion has been observed in numerous complicated sys- tems. Its mean squared displacement (MSD) is not a power law function of time, but instead a logarithmic function, and in some cases grows even more slowly than the logarithmic rate. Recent investigation has used the structural derivative to describe ul- traslow diffusion dynamics by using the inverse Mittag-Leffler function as the structural function. In this study, the ultraslow diffusion process is ana- lyzed by using both the classical Shannon entropy and its general case with inverse Mittag-Leffler function. In addition, to describe the observation process with information loss in ultraslow diffusion, e.g., some defect in the observation process, two definitions of fractional entropy are proposed by using the inverse Mittag-Leffler function, in which the Pade approximation technique is employed to numerically estimate the diffusion entropy.

MSC 2010: 35Q82, 60G20, 33E20 Key Words and Phrases: ultraslow diffusion; diffusion entropy; in- verse Mittag-Leffler function; fractional entropy 31

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

MAXIMUM PRINCIPLES FOR THE FRACTIONAL DIFFERENTIAL EQUATIONS

Yuri LUCHKO Department of Mathematics, Physics, and Chemistry – Beuth Technical University of Applied Sciences Berlin Luxemburger Str. 10, Berlin – 13353, GERMANY e-mail: [email protected]

The maximum principles are a well known and widely applied tool in the theory of partial differential equations of elliptic and parabolic type. They have a clear and straightforward physical background and provide among other things important a priori information regarding solutions to the boundary or initial-boundary-value problems for the partial differential equation of elliptic or parabolic type, respectively. The fractional partial differential equations with the time-fractional derivative of order α between zero and one interpolate between the PDEs of elliptic and parabolic type. Thus it would be natural to expect that the maximum principle is valid for these equations, too. Because the fractional derivatives are non-local operators that do not vanish in the critical points of the functions (in contrast to the conventional first order derivative), the standard proof technique for the maximum principle does not work in the fractional case and thus the maximum principle for the fractional differen- tial equations remained unproved until very recently. It is worth mentioning that some arguments related to a kind of a maximum principle have been used for analysis of the fractional diffusion equation in the paper [1] by Kochubei. But the explicit form of the (weak) maximum principle for a time-fractional diffusion equation was formulated and proved for the first time in the paper [3] by Luchko. After its publication, the maximum prin- ciples for the fractional differential equations and their applications became a popular topic under very intensive development. One of the most recent research topics in the theory of the fractional differential equations are the general time-fractional diffusion equations, which generalize the single- and the multi-term time-fractional diffusion 32 equations as well as the time-fractional diffusion equation of the distributed order. In the recent paper [4], the maximum principle for the time-fractional diffusion equation with the general fractional derivative of the Caputo type was formulated and proved. The first part of the talk will be devoted to this result. In the publications mentioned above, the maximum principles for the time-fractional diffusion equations with the Caputo fractional derivative were formulated in the weak sense. A kind of a strong maximum principle for the single-term time-fractional diffusion equation was proved in the paper [2]. In the second part of this talk, a new result from [5] regarding the strong maximum principle for the weak solution of this equation from the fractional Sobolev space will be presented. The final part of the talk will be devoted to the very recent results ob- tained jointly with Masahiro Yamamoto regarding the maximum principle for the general space- and time-fractional evolution equation in the Hilbert space. A work partially supported by Bulgarian National Science Fund (Grant DFNI-I 02/9). MSC 2010: 26A33, 35A05, 35B30, 35C05, 35L05, 45K05, 60E99 Key Words and Phrases: time-fractional diffusion equations; time- and space-fractional partial differential equations; weak and strong maxi- mum principles; comparison principle References [1] A.N. Kochubei, Fractional-order diffusion. Differential Equations 26 (1990), 485–492. [2] Y. Liu, W. Rundell, M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fract. Calc. Appl. Anal. 19, No 4 (2016), 888–906; DOI: 10.1515/fca-2016-0048. [3] Yu. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223. [4] Yu. Luchko, M. Yamamoto, General time-fractional diffusion equa- tion: Some uniqueness and existence results for the initial-boundary- value problems. Fract. Calc. Appl. Anal. 19, No 3 (2016), 676–695; DOI: 10.1515/fca-2016-0036. [5] Yu. Luchko, M. Yamamoto, On the maximum principle for a time- fractional diffusion equation. Accepted in: Fract. Calc. Appl. Anal. 20, No 5 (2017); at https://www.degruyter.com/view/j/fca. 33

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FRACTIONAL CALCULUS: FUNDAMENTALS AND APPLICATIONS

Jos´eTenreiro MACHADO Department of Electrical Engineering – Institute of Engineering Polytechnic of Porto, Rua Dr. Ant´onioBernardino de Almeida, 431 Porto – 4249-015 Porto, PORTUGAL e-mail: [email protected]

Fractional Calculus (FC) started in year 1695 when Guillaume de l’Hˆo- pital (1661-1704) wrote a letter to Gottfried Leibniz (1646-1716) asking n 1 for the meaning of D y for n = 2 . FC emerged simultaneously with the standard differential calculus, but remained an obscure topic during sev- eral centuries. In spite of the contributions of important mathematicians such as Leonhard Euler (1707-1783), Joseph Liouville (1809-1882), Bern- hard Riemann (1826-1866) and many others, applied sciences were simply unaware of the mathematical concept. The first application is credited to Niels Abel (1802-1829) in the tautochrone problem. By the beginning of the twentieth century, the brothers Kenneth Cole (1900-1984) and Robert Cole (1914-1990) applied FC heuristic concepts in biology. Also, Olivier Heaviside (1850-1925) and Andrew Gemant (1895-1983) applied FC in the electrical and mechanical engineering, respectively. Surprisingly, these vi- sionary and important contributions were forgotten. Only during the eight- ies FC emerged associated with phenomena such as fractal and chaos and, consequently, in nonlinear dynamics. In the last years, FC became the “new” tool for the analysis of dy- namical systems, particularly in systems with long range memory effects, or power law behavior. The present day popularity of FC in the scientific arena, with a growing number of researchers and published papers, makes one forget that 20 years ago the topic was considered “exotic” and that a typical question was “FC? what is it useful for?” Nowadays, new ad- vances and directions of scientific progress are new definitions of operators, “fractionalization” of models and new applications. The proposal of new definitions of fractional-order operators, or the fractionalization of some 34 mathematical models, may represent dangerous adventures with possible misleading or even erroneous formulations. On the other hand, the in- depth study of some mathematical formulations, constitutes a solid basis, but its relative lack of ambition narrows considerably the scope of FC to a limited number of topics. New applications of FC require not only imagi- native ideas, but also motivates researchers to alleviate some mathematical properties required by a strict interpretation of FC. In this line of thought, possible new directions of progress in FC emerge in the fringe of classical science, or in the borders between several distinct areas. The present work starts with classical concepts and will present some innovative ideas involving fractional objects and complex systems. About FC it is often mentioned the Leibniz prophecy that FC “will lead to a paradox, from which one day useful consequences will be drawn”, but we should also have in mind Albert Einstein’s comment “Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution”. A work partially supported by Bulgarian National Science Fund (Grant DFNI-I 02/9).

MSC 2010: 26A33, 34A08, 60G22 Key Words and Phrases: fractional calculus; fractional derivatives; fractional dynamics 35

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON A GENERALIZED THREE-PARAMETER WRIGHT FUNCTION OF LE ROY TYPE Roberto GARRAPPA 1, Sergei ROGOSIN 2, Francesco MAINARDI 3,§ 1 Department of Mathematics – University of Bari via E. Orabona 4, Bari – I-70125, ITALY e-mail: [email protected] 2 Department of Economics – Belarusian State University Nezavisimosti ave 4, Minsk – BY-220030, BELARUS e-mail: [email protected] 3 Department of Physics and Astronomy – University of Bologna via Irnerio 46, Bologna – I-40126, ITALY e-mail: [email protected] Recently, S. Gerhold and R. Garra–F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family, ∞ ∑ zk F (γ)(z) = , z ∈ C, Re α > 0, β ∈ R, γ > 0. α,β [Γ(αk + β)]γ k=0 This entire function is a generalization of the function studied by E. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter γ) and provide its continuation to the case of a negative first parameter α. An asymptotic result is illustrated by numerical calculations. Discussion on possible further studies and open questions are also presented. Details will be available in the forthcoming paper by same authors and same title, accepted in: Fract. Calc. Appl. Anal. 20, No 5 (2017); at https://www.degruyter.com/view/j/fca. MSC 2010: 33E12, 30D10, 30F15, 35R11 Key Words and Phrases: special functions; Mittag-Leffler and Wright functions; integral representation; asymptotics; Laplace transforms 36

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

OPTIMAL CONTROL OF LINEAR SYSTEMS WITH FRACTIONAL DERIVATIVES

Ivan MATYCHYN 1,§, Viktoriia ONYSHCHENKO 2 1 Faculty of Mathematics and Computer Science – University of Warmia and Mazury in Olsztyn Sloneczna 54 Street, Olsztyn - 10-710, POLAND e-mail: [email protected] 2 State University of Telecommunications 7, Solomyanska Str., Kyiv – 03110, UKRAINE e-mail: [email protected]

A problem of time-optimal control of linear systems with fractional dynamics is examined using the technique of attainability sets and their support functions. The cases of Riemann–Liouville and Caputo fractional derivatives are discussed separately. A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of set-valued maps and represents a fractional version of Pontryagin’s maximum principle. Special emphasis is placed upon the problem of computing of the matrix Mittag-Leffler function (MMLF), which plays a key role in the proposed methods. A technique for computing MMLF using Jordan canonical form is discussed, which is implemented in the form of a Matlab routine. The theoretical results are supported by several examples. Optimal control functions, in particular of the “bang-bang” type, are obtained in the examples. MSC 2010: 26A33, 34A08, 49N05 Key Words and Phrases: fractional calculus; fractional differential equations; matrix Mittag-Leffler function; optimal control 37

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON THE CAUCHY PROBLEM FOR SOME FRACTIONAL NONLINEAR ULTRA-PARABOLIC EQUATIONS

Fatma AL-MUSALHI 1,§, Sebti KERBAL 2 1,2 Department of Mathematics and Statistics – Sultan Qaboos University Al-Khodh 123, Muscat - P.O. Box 36, OMAN 1 e-mail: [email protected] , 2 e-mail: [email protected]

Blowing-up solutions to fractional nonlinear ultra-parabolic equations and corresponding system are presented. The obtained results will con- tribute in the development of ultra-parabolic equations and enrich the ex- isting non-extensive literature on fractional nonlinear ultra-parabolic prob- lems and extend some recent work of classical multi-time parabolic equa- tion. Our method of proof relies on a suitable choice of a test function and the weak formulation approach of the sought for solutions. Moreover, necessary conditions for blow up is also obtained.

MSC 2010: 35A01, 26A33, 35K70 Key Words and Phrases: nonexistence; nonlinear ultra-parabolic equations; fractional space and time operators 38

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

DIFFERENTIAL AND INTEGRAL RELATIONS IN THE CLASS OF MULTI-INDEX MITTAG-LEFFLER FUNCTIONS Jordanka PANEVA-KONOVSKA 1,2 1 Faculty of Applied Mathematics and Informatics Technical University of Sofia 8 “Kliment Ohridski” Blv., Sofia 1000, BULGARIA 2 Institute of Mathematics and Informatics Bulgarian Academy of Sciences ”Acad. G. Bontchev” Str., Block 8, Sofia 1113, BULGARIA e-mail: [email protected] As it has been recently obtained by Oliveira at al., the n-th derivative of the (2-parametric) Mittag-Leffler function gives to a 3-parametric Mittag- Leffler function, introduced by Prabhakar (up to a constant). Following this analogy, the n-th derivative of the (2m) multi-index Mittag-Leffler functions [1] is obtained. It turns out that this integer order derivative is expressed by the (3m) Mittag-Leffler functions [2], also up to a constant. Further, some special cases of fractional order Riemann–Liouville and Erd´elyi–Kober integrals from the Mittag-Leffler functions are calculated and interesting relations are proved. Analogous relations connect the inte- grals of 2m-Mittag-Leffler functions and 3m-Mittag-Leffler functions. Finally, multiple Erd´elyi-Kober fractional integration operators ([1]) are shown to relate the 2m- and 3m-parametric Mittag-Leffler functions. A work under the bilateral agreements of BAS with SANU and MANU. MSC 2010: 26A33, 33E12 Key Words and Phrases: Mittag-Leffler functions; multi-index Mit- tag-Leffler functions; Riemann–Liouville and Erd´elyi–Kober fractional in- tegrals and derivatives References [1] V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and re- lations to generalized fractional calculus. J. Comput. Appl. Math. 118 (2000), 241–259; doi:10.1016/S0377-0427(00)00292-2. [2] J. Paneva-Konovska, Multi-index (3m-parametric) Mittag-Leffler functions and fractional calculus. Compt. rend. Acad. bulg. Sci. 64, No 8 (2011), 1089–1098; available at: http://www.proceedings.bas.bg/. 39

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

A GENERALIZATION OF CONVEX FUNCTIONS Donka PASHKOULEVA Institute of Mathematics and Informatics – Bulgarian Academy of Sciences Acad. G. Bonchev Str., Block 8, Sofia – 1113, BULGARIA e-mail: donka zh [email protected] ∑∞ k Let S denote the class of functions of the form f(z) = z + akz , k=2 which are analytic and univalent in the open unit disk E = {z : |z| < 1}. Let C denote the class of convex functions f(z) ∈ C if and only if for z ∈ E { } zf ′′(z) ℜ 1 + > 0. f ′(z)

For k ≥ 2, denote by Vk the class of normalized functions of bounded boundary rotation at most kπ. Thus g(z) ∈ Vk if and only if g(z) is analytic in E, g′(z) ≠ 0, g(0) = g′(0) − 1 = 0 and for z ∈ E ∫ 2π ′ ′ ℜ(zg (z)) ≤ ′ dθ kπ. 0 g (z) Let f(z) be analytic in E, f ′(0) ≠ 0 and normalized so that f(0) = 0, ′ f (0) = 1. Then for k ≥ 2, f(z) ∈ Tk if there exists a function g(z) ∈ Vk, such that for z ∈ E f ′(z) ℜ > 0. g′(z) Clearly T2 = K, the class of close-to-convex functions. Let f(z) be analytic in E and normalized so that f(0) = 0, f ′(0) = 1 ′ ̸ ∈ ∗ ≥ ∈ and f (z) = 0. Then f(z) Ck (k 2) if there exists a function g(z) Vk ∈ such that for z E ′ (zf ′(z)) ℜ > 0. g′(z) In this talk we present sharp results involving growth and distortion ∗ properties for the classes Vk and Ck . MSC 2010: 30C45 Key Words and Phrases: univalent functions; convex functions 40

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SPECIAL FUNCTIONS FOR QUANTUM OSCILLATORS IN INTEGER AND NON-INTEGER DIMENSIONS

Irina PETRESKA Ss. Cyril and Methodius University in Skopje Faculty of Natural Sciences and Mathematics, Institute of Physics Arhimedova 3, PO Box 162, 1000 Skopje, MACEDONIA e-mail: [email protected]

Partial differential equations in non-integer dimensions, as mathemati- cal models that lead to a closed-form solution, have already been success- fully applied to describe various physical objects and phenomena. In the present work we are in particular, interested in the Schr¨odingerequation in non-integer dimensions, that drags special attention, because of its applica- bility for modeling of, for example, excitonic properties of low-dimensional anisotropic systems, such as quantum dots, quantum wells and various semiconducting heterostructures. Time-independent Schr¨odingerequation is solved, using the modified Laplace operator in spherical coordinates, by employing the method of separation of variables. We further consider the cases of harmonic potential, as well as the Kratzer potential. The closed form solutions in terms of special functions for each of them are obtained. We show that in non-integer dimensions, Gegenbauer polynomials appear in a solution of the angular part, whilst for the radial part we obtain Laguerre polynomials. We further analyze the known cases in integer dimensions and comment on the relations between the Gegenbauer and Legendre polynomi- als, and their generalization with the Gauss’ hypergeometric function. The studied potential energy forms are suitable to model particles and quasi- particles, that act as quantum oscillators when confined in various porous and disordered media. MSC 2010: 35Q40, 81Q05, 33C45, 42C05 Key Words and Phrases: Schr¨odingerequation; special functions; partial differential equations; hypergeometric functions; Laguerre Legen- dre and Gegenbauer polynomials; quantum oscillator; harmonic oscillator; Kratzer potential 41

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

Rd ULTRADISTRIBUTIONS ON + AND THE LAGUERRE EXPANSIONS

Stevan PILIPOVIC´ Department of Mathematics and Informatics, University of Novi Sad Trg. “D. Obradovica” 4, Novi Sad - 21 000, SERBIA e-mail: [email protected]

The first part of the talk is devoted to the G-type spaces, i.e. the spaces α Rd ≥ Gα( +), α 1 and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justifica- tion of obtained results. The Laguerre type expansions of the elements in α Rd Gα( +) and their duals characterize these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological α Rd description and by the nuclearity of Gα( +) the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the G-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo- differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.

This is a joint work with Smiljana Jakˇsi´cand Bojan Prangoski.

Partially supported and in the frames of the project “Analytical and nu- merical methods for differential and integral equations ...” under bilateral agreement between SASA and BAS. MSC 2010: 46F05, 47G30, 47L50 Key Words and Phrases: ultradistributions; Gelfand-Shilov type spaces and dual spaces; pseudo-differential operators 42

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SOME ASPECTS OF MODELING OF REAL MATERIALS

Igor PODLUBNY Institute of Control and Informatization of Production Processes BERG Faculty Technical University of Kosice B. Nemcovej 3, 04200 Kosice, SLOVAKIA e-mail: [email protected]

Modeling of real materials (elastic, viscoelastic, porous, granular, etc.) will be considered from various viewpoints: modeling of the geometric structure of materials, modeling of their physical properties, some ap- proaches to classification of materials based on their structure and proper- ties, modeling of dynamical processes in materials based on their geometric and physical models, and some other aspects. Methods for identification of parameters of mathematical models of real materials will be discussed, as well as some approaches to performing experimental measurements for that purpose. The use of integer-order models, fractional-order models, numerical ap- proaches and models, computational methods, and other currently available approaches in relation to modeling of real materials will also be explored.

Acknowledgements: This work is partially supported by grants VEGA 1/0908/15, APVV-14-0892, and ARO WF911NF-15-1-0228.

MSC 2010: 26A33 Key Words and Phrases: fractional calculus; fractals; mathematical modeling; porous materials; granular materials; rheology 43

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SPECIAL FUNCTIONS IN STATISTICS Tibor K. POGANY´ Faculty of Maritime Studies – University of Rijeka Studentska 2, 51000 Rijeka, CROATIA e-mail: [email protected]

The first main aim is to give the real order moments for the rv ξ having three parameter exponentiated exponential Poisson distribution. We prove that the existing [1] series and integral form expressions for Eξν, ν ∈ N are valid for all ν > 1 − α, α > 0, [3]. Next, the moments for the gamma exponentiated exponential Weibull distribution are expressed in terms of the 3,1 1Ψ0- and Meijer G1,3-functions, [4]. Further, we expose moments for the four-parameter Marshall-Olkin exponential Weibull distribution, discussing by the way the modality issue, [4]. We end the exposition with the definite integral representation of the normalization constant Z(λ, ν) which defines the CoM–Poisson distribution, [5]. MSC 2010: 33CXX, 33E20, 60E05, 60E10, 62E15, 62F10 Key Words and Phrases: moments; hypergeometric type functions; exponentiated exponential Poisson; gammaexponentiated exponential Wei- bull; Marshall-Olkin exponential Weibull; Conway–Maxwell Poisson References [1] M.M. Risti´c,S.A. Nadarajah, A new lifetime distribution. J. Statist. Comput. Simul. 84, No 1 (2014), 135-150. [2] T.K. Pog´any, The exponentiated exponential Poisson distribution re- visited. Statistics 49, No 4 (2015), 918–929. [3] T.K. Pog´any, A. Saboor, The Gamma exponentiated exponential- Weibull distribution. Filomat 30, No 12 (2016), 3159–3170. [4] T.K. Pog´any, A. Saboor, S. Provost, The Marshall–Olkin exponential Weibull distribution. Hacettepe J. Math. Statist. 44, No 6 (2015), 1579–1594. [5] T.K. Pog´any, Integral form of the COM-Poisson renormalization con- stant. Stat. Probab. Letters 119 (2016), 144–145. 44

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

POHODHAEV IDENTITIES FOR SEMI-LINEAR MIXED TYPE ELLIPTIC-HYPERBOLIC EQUATIONS

Nedyu POPIVANOV Faculty of Mathematics and Informatics – Sofia University 5 James Bourchier blvd., Sofia – 1164, BULGARIA e-mail: [email protected]fia.bg

It is well known result of Pohozhaev (1965), that the homogeneous Dirichlet problem for semilinear elliptic equations in a bounded subset Ω of Rn, with n > 2, permits only the trivial solution if the domain is star- shaped, the solution is sufficiently regular, and the power of nonlinearity p > 2∗(n) := 2n/(n − 2), where the latter quantity is the critical exponent 1 p ∗ in the Sobolev embedding of H0 (Ω) into L (Ω) for p < 2 (n). To the opposite of this fact, in the case 2 < p < 2∗(n) there exist nontrivial solutions. In the last 50 years the Pohozhaev identities and results have been used and extended for a large class of elliptic problems. Let us mention now that in [1], [2] it has been shown that the nonexistence principle in supercritical case also holds for certain two dimensional problems for the mixed elliptic-hyperbolic Gellersted operator L (instead of ∆), with some appropriate boundary conditions. It is also valid for a large class of such problems even in higher dimensions [3]. In dimension 2, such operators have a long-standing connection with transonic fluid flow. Of course, the critical Sobolev embedding in this case is for a suitable weighted version of 1 p H0 (Ω) into L (Ω). As usual, in the BVP for such mixed elliptic-hyperbolic Gellersted operator L, the boundary data are given only on the proper subset of the boundary of Ω. To compensate the lack of a boundary condition on a part of boundary, a sharp Hardy-Sobolev inequality is used, as was first done in [1], [2] and later in [3], [4]. Some further results, already published or in progress, pre- pared jointly with colleagues from Italy and Norway will be also discussed.

MSC 2010: 35M10, 35M12, 35G30, 46E35 45

Key Words and Phrases: Pohodhaev identities; mixed type elliptic- hyperbolic equations; Sobolev embedding; Gellersted operator; boundary value problem References [1] D. Lupo, K. Payne, Critical exponents for semi-linear equations of mixed elliptic-hyperbolic types. Comm. Pure Appl. Math. 56 (2003), 403–424.

[2] D. Lupo, K. Payne, Conservation laws for equations of mixed elliptic- hyperbolic type. Duke Math. J. 127 (2005), 251–290.

[3] D. Lupo, K. Payne, N. Popivanov, Nonexistence of nontrivial solu- tions for supercritical equations of mixed elliptic-hyperbolic type. In: Progress in Non-Linear Differential Equations and Their Applications, Birkh¨auserVerlag, Basel, Vol. 66 (2005), 371–390.

[4] D. Lupo, K. Payne, N. Popivanov, On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type. Nonlinear Analysis, Series A: Theory, Methods & Appl. 108 (2014), 29–56. 46

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON THE COMPUTATION OF THE MATRIX MITTAG–LEFFLER FUNCTION WITH APPLICATIONS TO FRACTIONAL CALCULUS Roberto GARRAPPA 1, Marina POPOLIZIO 2,§ 1 Dipartimento di Matematica – Universit`adegli Studi di Bari Via E. Orabona n.4 – 70125 Bari, ITALY e-mail: [email protected] 2 Dipartimento di Matematica e Fisica “Ennio De Giorgi” Universit`adel Salento Via per Arnesano – 73100 Lecce, ITALY e-mail: [email protected] The important role played by the Mittag-Leffler (ML) function in frac- tional calculus is widely known. Furthermore, the ML function evaluated in matrix arguments has useful applications in studying theoretical properties of systems of fractional differential equations and in finding their solution. In this talk we introduce the ML function with matrix arguments and, after reviewing some of its main applications, we discuss the problem of its computation with the challenges it raises. Since the evaluation at matrix arguments may require the computation of derivatives of the ML function of possible high order, we discuss in detail this topic and we show some new formulas for the ML function derivatives. MSC 2010: 65F30, 26A33, 65F60 Key Words and Phrases: Mittag–Leffler function; special functions; matrix function; fractional calculus References [1] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53, No 3 (2015), 1350–1369. [2] R. Garrappa and M. Popolizio, Computing the matrix Mittag–Leffler function with applications to fractional calculus. Submitted. [3] R. Garrappa and M. Popolizio, On the use of matrix functions for frac- tional partial differential equations. Math. Comput. Simulation 81, No 5 (2011), 1045–1056. 47

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SINGULAR SOLUTIONS OF PROTTER PROBLEMS FOR THE WAVE EQUATION

Nedyu POPIVANOV 1, Todor POPOV 2,§ Faculty of Mathematics and Informatics – Sofia University 5 James Bourchier blvd., Sofia - 1164, BULGARIA e-mails: 1 [email protected]fia.bg , 2 [email protected]fia.bg

Boundary value problems introduced by M. H. Protter for the non- homogeneous wave equation are studied in a (3+1)-D domain, bounded by two characteristic cones and a non-characteristic ball. They could be considered as multidimensional analogues of the Darboux problem in the plane. In the frame of classical solvability the Protter problem is not Fred- holm, because it has an infinite-dimensional cokernel. Alternatively, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity at the vertex O of the boundary light cone. This singularity is isolated at the point O and does not propagate along the cone. We present some conditions on the smooth right-hand side functions that are sufficient for existence of a generalized solution and give some a priori estimates for its possible singularity.

MSC 2010: 35L05, 35L20, 35D30, 35C10, 35B40 Key Words and Phrases: wave equation; boundary value problem; generalized solution; singular solution; special functions References [1] N. Popivanov, T. Popov, A. Tesdall, Semi-Fredholm solvability in the framework of singular solutions for the (3+1)-D Protter-Morawetz problem. Abstract and Applied Analysis 2014 (2014), Article ID 260287, 19 p. 48

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

TIME-FRACTIONAL DIFFUSION WITH MASS ABSORPTION UNDER HARMONIC IMPACT

Yuriy POVSTENKO 1,§, Tamara KYRYLYCH 2 1 Institute of Mathematics and Computer Science – Jan D lugoszUniversity in Czestochowa Armii Krajowej 13/15, Czestochowa - 42-200, POLAND e-mail: [email protected] 2 Institute of Law, Administration and Management – Jan D lugoszUniversity in Czestochowa Zbierskiego 2/4, Czestochowa – 42-200, POLAND e-mail: [email protected]

The time-fractional diffusion-wave equation describes many important physical phenomena in different media. The book [1] systematically presents solutions of this equation in Cartesian, cylindrical and spherical coordinates under different kinds of boundary conditions. In this paper, we consider the time-fractional diffusion-wave equation with mass absorption and with the source term varying harmonically in time: ∂αc = a∆c − bc + Qδ(x) eiωt. (1) ∂tα The particular case of equation (1) with b = 0 was studied in [2]. MSC 2010: 26A33, 35K05, 45K05 Key Words and Phrases: fractional calculus; integral transforms; Mittag-Leffler function; differential equations; harmonic impact References [1] Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkh¨auser,New York (2015). [2] Y. Povstenko, Fractional heat conduction in a space with a source vary- ing harmonically in time and associated thermal stresses. J. Thermal Stresses 39, No 11 (2016), 1442–1450. 49

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

THE ITERATIVE METHODS IN FRACTIONAL q–CALCULUS Predrag RAJKOVIC´ 1,§ Miomir STANKOVIC´ 2, Sladjana MARINKOVIC´ 3 1 Faculty of Mechanical Engineering – University of Niˇs A. Medvedeva 14, Niˇs– 18 000, SERBIA e-mail: [email protected] 2 The Mathematical Institute of the SASA Kneza Mihaila 36, Belgrade – 11001, SERBIA 3 Faculty of Electronic Engineering – University of Niˇs A. Medvedeva 14, Niˇs– 18 000, SERBIA

Fractional calculus and q–calculus, as two special mathematical disci- plines, provide a lot of new operators and functions and have great influence in the science in XX century with a lot of applications. They able us to consider quite new equations. In that situation, we can try to apply well-known methods with more or less success. We chose to prepare a few modifications of the well known methods for numerical solving of such equations or the systems. Starting from q-Taylor formula for the functions of several variables and mean value theorems in q–calculus, we develop some new methods for solving systems of equations. We prove its convergence and give an estimation of the error. Especially, we include Newton’s, the Newton– Kantorovich and gradient method. The purpose is to adapt them to cases when the functions are given in the form of infinite products. The examples comprehend the infinite q–power products and prove that the methods are pretty suitable for them. They are very useful when the continuous function does not have fine smooth properties. MSC 2010: 33D60, 26A24, 65H10 Key Words and Phrases: fractional calculus; q-calculus; iterative methods 50

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

DYNAMIC RESPONSE OF A VISCOELASTIC ANISOTROPIC PLANE WITH CRACKS VIA FRACTIONAL DERIVATIVES AND BIEM Tsviatko RANGELOV 1,§, Petia DINEVA 2 1 Department of Differential Equations and Mathematical Physics – Institute of Mathematics and Informatics, Bulg. Acad. Sci. Acad. G. Bonchev str., Block 8, Sofia – 1113, BULGARIA e-mail: [email protected] 2 Department of Solid Mechanics – Institute of Mechanics Bulgarian Academy of Sciences Acad. G. Bonchev str., Block 4, Sofia – 1113, BULGARIA e-mail: [email protected] The aim of this study is to develop an efficient numerical technique using the non-hypersingular, traction boundary integral equation method for solving wave propagation problems in anisotropic, viscoelastic media containing cracks that is valid at the macro scale and, with additional modifications, at the nano-scale. Within the framework of continuum me- chanics, this modelling effort employs linear fracture mechanics, the frac- tional derivative concept for viscoelastic wave propagation and the surface elasticity model of Gurtin and Murdoch (1975) which leads to nonclassical boundary conditions at the nano-scale. Following validation of the model through comparison studies, extensive numerical simulations reveal the de- pendence of the stress intensity factor and of the stress concentration factor that develop in a material plane with cracks and inhomogeneities, respec- tively, on the type of anisotropy, on the viscoelastic parameters in the Zener rheological model, on the size effect and associated surface elasticity phe- nomena, on the type and characteristics of the imposed dynamic load and on the bulk properties of the surrounding material. MSC 2010: 35Q74, 74S15, 74H35 Key Words and Phrases: viscoelastic; fractional constitutive equa- tion; anisotropy; cracks; nano-cracks; surface elasticity; boundary elements; stress intensity; stress concentration Supported by Bulgarian National Science Fund (Grant DFNI-I 02/12). 51

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SOME APPLICATIONS OF FIXED POINT THEORY TO FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS

Daniel CAO LABORA 1 , Rosana RODR´IGUEZ-LOPEZ´ 2,§ 1,2 Department of Statistics, Mathematical Analysis, and Optimization – University of Santiago de Compostela Campus Vida, Santiago de Compostela – 15782, SPAIN 1 e-mail: [email protected], 2 e-mail: [email protected]

In the modelization of real processes, we often face the difficulty to express in an exact way the essential data and the effect of different factors that might be involved, due to the imprecision inherent to this information. Fuzzy mathematics, in general, and fuzzy differential equations, in par- ticular, have been an interesting approach to deal mathematically with the uncertainty present in real phenomena. To this purpose, different con- cepts for derivatives of fuzzy functions have been proposed, and several ap- proaches independent of these definitions have also been developed (Zadeh’s Extension Principle, differential inclusions, ...). On the other hand, the introduction of the concept of solution for frac- tional differential equations with uncertainty allows to consider problems for fuzzy differential equations involving arbitrary order fuzzy derivatives. In this context, the extension by Agarwal, Arshad, O’Regan, and Lupulescu of Schauder fixed point theorem to semilinear spaces has given interesting existence results when applied to the study of fuzzy fractional differential equations. We will focus our attention on the implications of fixed point theory in spaces without a vectorial structure for the properties of the solutions to fractional differential equations with uncertainty.

MSC 2010: 34G20, 34A07, 34A08, 34A12, 34A34 Key Words and Phrases: fixed point theory; fractional calculus; differential equations; fuzzy mathematics 52

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

UNIQUENESS OF THE REPRESENTATIONS BY SCHLOMILCH¨ SERIES Peter RUSEV Institute of Mathematics and Informatics Bulgarian Academy of Sciences ”Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIA e-mail: [email protected]

The series ∑∞ anJ0(nx), x, an ∈ R, n = 0, 1, 2,..., (∗) n=0 where J0 is the Bessel function of first kind with zero index, have been studied for the first time by O.X. Schl¨omilch [1] who gave sufficient condi- tions a real function defined on the interval [0, π] to be represented there by a series of the kind (∗). The representation (∗) is, in general, not unique. For example, it holds that [2, 7.10.3., (47)] ∞ 1 ∑ + (−1)nJ (nx) = 0, x ∈ (0, π). 2 0 n=1 Sufficient condition for uniqueness of the expansions (∗) is given in [3, 19.54] which is an analogue of the Riemann theorem for uniqueness of the trigonometric series. This communication is devoted to another criterion for uniqueness of Schl¨omilch’s series which is based on the uniqueness of the expansions in uniformly convergent Fourier series as well as on the Riemann-Liuville type operator for fractional integration, see e.g. [4, p. 103]. Necessary and sufficient conditions a complex function holomorphic in the strip {z ∈ C : |ℑz| < τ0 ≤ ∞} to be represented there by a series of Schl¨omilch are given in the paper [5]. Moreover, the uniqueness property also holds for such representations, [4, p. 104]. MSC 2010: 30B50, 33C10, 26A33 53

Key Words and Phrases: Schl¨omilch’s series; Bessel functions; rep- resentations of holomorphic functions; integration of fractional order References [1] O.X. Schl¨omilch, Uber¨ die Besselschen Funktionen. Zeitschr. Math. Phys., II (1857), 137–155. [2] H. Bateman, A. Erd´elyi, Higher Transcendental Functions, II. Mc- Graw Hill Co., 1953. [3] G.N. Watson, Treatise on the Theory of Bessel Functions. Oxford University Press, 1945. [4] P. Rusev, An Invitation to Bessel Functions. Prof. Marin Drinov Publ. House of Bulg. Acad. Sci., Sofia, 2016. [5] P. Rusev, Representation of holomorphic functions by Schl¨omilch series. Fract. Calc. Appl. Anal. 16, No 2 (2013), 431–435; DOI: 10.2478/s13540-013-0026-7. 54

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

INITIAL BOUNDARY VALUE PROBLEMS FOR A FRACTIONAL DIFFERENTIAL EQUATION WITH HYPER-BESSEL OPERATOR

Fatma AL-MUSALHI 1, Nasser AL-SALTI 1,§, Erkinjon KARIMOV 2 1 Department of Mathematics and Statistics, Sultan Qaboos University P.O. Box 36, PC 123 Al-Khoudh, OMAN e-mail: [email protected] (Fatma) e-mail: [email protected] (Nasser) 2 Institute of Mathematics named after V. I. Romanovsky Academy of Science of Republic of Uzbekistan Durmon yuli 29, Tashkent 100125, UZBEKISTAN e-mail: [email protected]

Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are consid- ered. Solutions to these problems are constructed based on appropriate eigenfunction expansion and results on existence and uniqueness are estab- lished. To solve the resultant equations, a solution to a non-homogeneous fractional differential equation with regularized Caputo-like counterpart hyper-Bessel operator is also presented.

MSC 2010: 35G16, 35R30, 34A08, 35C10 Key Words and Phrases: initial-boundary value problems; inverse problems; time-fractional differential equation; series solutions; hyper-Bessel operator 55

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FROM CONTINUOUS TIME RANDOM WALKS TO FRACTIONAL CALCULUS

Trifce SANDEV 1,2 1 Radiation Safety Directorate Partizanski odredi 143, P.O. Box 22, 1020 Skopje, MACEDONIA e-mail: [email protected] 2 Research Center for Computer Science and Information Technologies Macedonian Academy of Sciences and Arts Bul. Krste Misirkov 2, 1000 Skopje, MACEDONIA The mathematical theory of continuous time random walk (CTRW), introduced by Montroll and Weiss (1965), after its application of Scher and Lax (1973) in physical problems, has became a very popular tool for descrip- tion of anomalous dynamics in complex systems. This stochastic model is based on the fact that the individual jumps are separated by independent, random waiting times. It has been shown that the CTRW process with a scale-free waiting time probability distribution function (PDF) of a power- law form, leads to the time fractional diffusion equation for anomalous subdiffusion exhibiting a monoscaling behavior. Long tailed jump length PDF leads to space fractional diffusion equations and anomalous super- diffusion. Here we present different generalized CTRW models, for which we can find the corresponding generalized(-fractional) Fokker-Planck equa- tion, that generate a broad class of anomalous nonscaling patterns. MSC 2010: 60Gxx, 26A33, 35R11 Key Words and Phrases: continuous time random walk (CTRW); fractional calculus References [1] T. Sandev, A.V. Chechkin, N. Korabel, H. Kantz, I.M. Sokolov, R. Metzler, In: Physical Review E 92 (2015), # 042117. [2] T. Sandev, A. Chechkin, H. Kantz, R. Metzler, In: Fract. Calc. Appl. Anal. 18, No 4 (2015), 1006–1038. [3] T. Sandev, I.M. Sokolov, R. Metzler, A. Chechkin, In: Chaos Solitons & Fractals 2017 (2017), doi: 10.1016/j.chaos.2017.05.001. 56

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

GAUSS-LUCAS THEOREM FOR POLYNOMIALS WITH REAL COEFFICIENTS Blagovest SENDOV 1,§, Hristo SENDOV 2 1 Institute of Information and Communication Technology Bulgarian Academy of Sciences Acad. G. Bonchev str., Sofia 1113, BULGARIA e-mail: [email protected] 2 Department of Statistical and Actuarial Sciences University of Western Ontario London, Ontario, CANADA N6A 5B7 e-mail: [email protected]

Recently we have proved, see [1], the so-called Sector theorem: Let p(z) be a polynomial of degree n with non-negative coefficients and all zeros on the sector S(φ) = {z : | arg(z)| ≥ φ} from the complex plane. Then, the critical points of p(z) are also on S(φ). The main goal of the lecture is to present an application of the Sector theorem to straighten the Gauss-Lucas theorem in case of polynomials with real coefficients. References [1] Bl. Sendov, H.S. Sendov, On the zeros and critical points of poly- nomials with nonnegative coefficients: A nonconvex analogue of the Gauss-Lucas theorem, Constr. Approx. (2017); publ. online 12 April 2017, 13 pp.; doi:10.1007/s00365-017-9374-6. MSC 2010: 30C10 Key Words and Phrases: polynomials with real coefficients; zeros; critical points; Gauss-Lucas theorem; Sector theorem Acknowledgements. A work partially supported by Project FIN I 02/20 “Efficient Parallel Algorithms for Large-Scale Computational Prob- lems”. 57

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FRACTIONAL BESSEL INTEGRAL AND DERIVATIVE ON THE SEMI-AXIS Elina SHISHKINA Applied Mathematics, Informatics and Mechanics Department – Voronezh State University Universitetskaya Pl. 1, Voronezh – 394000, RUSSIA e-mail: ilina [email protected] In this talk we consider real powers of the singular Bessel differen- d2 ν d ≥ tial operator Bν = dx2 + x dx , ν 0. Definitions and representations of the fractional power α > 0 of the corresponding Bessel integral op- α erator, called fractional Bessel integral IB,ν,−, were given in [1]–[3] on the semi-axis by means of the hypergeometric Gauss function in the ker- nel. Then, the fractional Bessel derivative can be defined by the formula α n n−α ∈ [2α]+1 ∞ (DB,ν,−f)(x) = Bν (IB,ν,−f)(x), with n=[α]+1, f(x) C (0, + ). α α For the operators IB,ν,− and DB,ν,− relations with the Mellin and Han- kel transforms, group property, generalized Taylor formula with Bessel ope- rators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag–Leffler functions are obtained. Besides, the problem for α ∈ the fractional differential equation (DB,ν,−)tu(x, t) = uxx(x, t), α (0, 1/2), t > 0, x ∈ R is solved. MSC 2010: 26A33, 44A15 Key Words and Phrases: fractional calculus and fractional powers of operator; Bessel operator; Mellin transform; Hankel transform; generalized Mittag–Leffler function References [1] I.G. Sprinkhuizen-Kuyper, A fractional integral operator correspond- ing to negative powers of a certain second-order differential operator. J. Math. Anal. Appl. 72 (1979), 674–702. [2] A.C. McBride, Fractional Calculus and Integral Transforms of Gen- eralized Functions. Pitman, London (1979). [3] I.H. Dimovski, V.S. Kiryakova, Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer’s G-function. In: “Complex Analysis and Applications ’83 ”(Proc. Intern. Conf. Varna 1983 ), Sofia (1985), 45–66. 58

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH THE GENERAL CAPUTO DERIVATIVE

Chung-Sik SIN Faculty of Mathematics, Kim Il Sung University Kumsong Street, Taesong District, Pyongyang, D.P.R. KOREA e-mail: [email protected]

In this paper initial value problems of fractional functional differential equations with bounded delay are investigated. The fractional derivatives are taken as the general Caputo-type fractional derivatives as defined by Anatoly N. Kochubei. We use the Schauder fixed point theorem to establish existence and uniqueness results for local and global solutions.

MSC 2010: 26A33, 33E12, 34A08 Key Words and Phrases: general Caputo-type derivative; fractional functional differential equation with bounded delay; existence and unique- ness of solutions; initial value problem 59

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

BUSCHMAN–ERDELYI´ TRANSMUTATIONS AND APPLICATIONS Sergei SITNIK Chair of Differential Equations – Belgorod State National Research University Pobedy str., 85, Belgorod – 308015, RUSSIA e-mail: [email protected]

The ideas, methods and applications of transmutation theory now form an important part of modern mathematics, cf. [1]–[4]. Many classes of im- portant integral operators are special cases of the Buschman–Erd´elyiones: the Riemann–Liouville fractional integrals, Sonine and Poisson transmu- tations, Mehler–Fock transforms, modified Hardy–type operators. For the special choice of parameters they are unitary operators in the standard Lebesgue space. Applications of this class of transmutations are consid- ered to differential equations with Bessel–type operators. And in fact, now the theory of Buschman–Erd´elyitransmutations may be characterized as a more detailed and specialized part of the theory of Sonine–Dimovski and Poisson–Dimovski transmutations for the hyper–Bessel functions and equa- tions, cf. [2]–[3]. MSC 2010: 26A33, 44A15 Key Words and Phrases: transmutation operators; Buschman–Erd´e- lyi transmutations; Sonine–Poisson–Delsarte transmutations; Sonine–Katra- khov and Poisson–Katrakhov transmutations; fractional integrals; Bessel and hyper-Bessel operators References [1] R. Carroll, Transmutation Theory and Applications. North Holland, Amsterdam (1986). [2] I.H. Dimovski, Convolutional Calculus. Kluwer, Dordrecht (1990). [3] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Scientific, Harlow and J. Wiley, N. York (1994). [4] S.M. Sitnik, Transmutations and Applications: A survey. arXiv: 1012.3741 (2010), 141 pp. 60

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

APPROXIMATIVE ANALYTICAL SOLUTIONS FOR SEVERAL ENGINEERING PROBLEMS OBTAINED BY THE LAPLACE TRANSFORM AND POST’S INVERSION FORMULA Dragan T. SPASIC Department of Mechanics – University of Novi Sad Trg Dositeja Obradovica 6, Novi Sad – 21000, SERBIA e-mail: [email protected] In the first part a widely held belief that the Laplace transform method is particularly suited to solving only linear problems was re-examined. In doing so the Cauchy problems describing motions of a simple pendulum, the Toda oscillators without and with damping, as well as the nonlinear two-point boundary value problem describing the optimal orbital transfer were solved by use of the Laplace transform and the method of successive approximations. The second part deals with problems of system identification and sim- ulations in time domain. The parameters involved in the system of frac- tional ordinary differential equations describing a drug distribution between three compartments and exertion from two of them were determined by the Laplace transform and Post’s inversion formula. In order to show how the same tool can be applied to partial differential equations, the tempera- ture distribution in a semi-infinite medium was given as the approximative solution of the generalized Cattaneo heat conduction equation with two fractional time derivatives and an integer order spatial derivative. In do- ing so Post’s formula was applied to Green’s function of the corresponding convolution integral. A comparison is made between this solution and the exact solution given in series form comprising the Fox H-function. Finally, several questions on preparations ensuring the efficiency of Post’s formula for both integer and fractional order systems were posed. MSC 2010: 26A33, 34C15, 44A10, 80A20, 93B30 Key Words and Phrases: Laplace transform; nonlinear problems; system identification; fractional calculus; Post’s formula Partially supported by Bulgarian National Science Fund (Grant DFNI-I 02/12). 61

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

EXACT SOLUTIONS OF BOUNDARY-VALUE PROBLEMS

Ivan DIMOVSKI 1, Margarita SPIRIDONOVA 2,§ 1,2 Institute of Mathematics and Informatics – Bulgarian Acad. of Sci. Acad. G. Bonchev Str. Block 8, Sofia – 1113, Bulgaria 1 e-mail: [email protected], 2 [email protected]

A survey of an approach for obtaining of explicit formulae for the solu- tion of local and nonlocal boundary value problems for some linear partial differential equations is presented. For solving such problems an extension of the Mikusi´nskioperational calculus is used. A two-dimensional opera- tional calculus is constructed for any of the problems considered. The main steps of construction of exact (closed) solutions using such operational cal- culi are described. A combination of the Fourier method and an extension of the Duhamel principle to the space variables is used. The obtained explicit formulae of the solutions of BVPs can be used for real applications. An example is considered. Illustrative examples of nu- merical computation and visualization of the solutions using the computer algebra system Mathematica are presented.

MSC 2010: 44A35, 44A40, 44A45, 44A99, 68W30 Key Words and Phrases: nonclassical convolutions; Mikusi´nskical- culus; Duhamel principle; nonlocal boundary value problem; Computer Algebra System 62

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

FRACTIONAL AND FRACTAL DERIVATIVE MODELS FOR TRANSIENT ANOMALOUS DIFFUSION: MODEL COMPARISON HongGuang Sun a,b,§, Zhipeng Li a, Yong Zhang a,b, Wen Chen a a Institute of Soft Matter Mechanics, Dept. of Engineering Mechanics Hohai University, Nanjing, Jiangsu – 210098, CHINA e-mail §: [email protected] b Dept. of Geological Sciences, University of Alabama Tuscaloosa – AL 35487, USA Transient anomalous diffusion characterized by transition between dif- fusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian dif- fusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional dif- fusion equation (D-FDE), the tempered fractional diffusion equation (T- FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), break- through curves (exhibiting nuance in diffusive state transition), and possi- ble hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub- diffusion to normal-diffusion, and the D-FDE model only captures tran- sient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V- FDE model, which provides a flexible transition rate, is appropriate to cap- ture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications. MSC 2010: 60J60, 60G22, 34A08, etc. Key Words and Phrases: transient diffusion; distributed-order frac- tional diffusion model; tempered fractional diffusion model; variable-order fractional diffusion model; variable-order fractal derivative diffusion model 63

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

ON THE SOLUTIONS OF FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

Arpad TAKACIˇ Department of Mathematics and Informatics, University of Novi Sad Trg. “D. Obradovica” 4, Novi Sad - 21 000, SERBIA e-mail: [email protected]

We consider a fuzzy partial fractional differential equation with fuzzy boundary and initial conditions. The solution of this problem is obtained by using Zadeh’s extension principle. First, the exact and the approximate solutions of the corresponding crisp problems are constructed, in the frame of Mikusinski operators. Then, they are extended to the exact and the approximate solutions of the given fuzzy type problem.

The work is partially supported and in the frames of the project “An- alytical and numerical methods for differential and integral equations ...” under bilateral agreement between SASA and BAS.

MSC 2010: 34A07, 34A08, 26A50, 26A33 Key Words and Phrases: fuzzy differential equations; fractional partial differential equations; Mikusinski operators; approximate solutions 64

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

MATHEMATICAL MODELING IN TEACHING – TACOMA BRIDGE

Djurdjica TAKACIˇ Department of Mathematics and Informatics, University of Novi Sad Trg. “D. Obradovica” 4, Novi Sad - 21 000, SERBIA e-mail: [email protected]

Mathematical modeling is one of the strategies in education, describing the real world and its interactions through mathematics, and it is considered as a tool for illustrating mathematical contents and for motivating students. Several interesting mathematical modeling processes will be presented including falling of Tacoma bridge.

The work is partially supported and in the frames of the project “An- alytical and numerical methods for differential and integral equations ...” under bilateral agreement between SASA and BAS.

MSC 2010: 97C70, 97DXX, 97MXX Key Words and Phrases: teaching in mathematics; mathematical models; Tacoma bridge 65

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

EXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS FOR SECOND-ORDER AND FRACTIONAL p-LAPLACIAN EQUATIONS

Stepan TERSIAN Department of Mathematics – University of Ruse Studentska 8, Ruse 7017, BULGARIA e-mail: [email protected]

p−2 Let 0 < α < 1, p > 1, q > r > 1, φp (t) = |t| t. First, we consider the existence of periodic solutions for the following one-dimensional p-Laplacian equation ′ ′ (φp(u (x))) − a(x)φq(u(x)) + b(x)φr(u(x)) = 0, x ∈ (0,T ), coupled with the periodic boundary condition u(T ) − u(0) = 0. Then, we consider the fractional p-Laplacian problem α α − ∈ tDT (φp(0Dt u(x))) a(x)φq(u(x)) + b(x)φr(u(x)) = 0, x (0,T ), α α with the boundary condition u(T ) = u(0) = 0. Here tDT and 0Dt are the left and right Riemann-Liouville fractional derivatives of order α and a = a(t), b = b(t) are positive continuous T -periodic functions on [0,T ]. Variational method is applied using minimization and extended Clark’s theorem. The second-order equations are considered in [1]. MSC 2010: Primary 34A08; Secondary 34B37, 58E05, 58E30, 26A33 Key Words and Phrases: fractional calculus; variational methods; minimization theorem; Clark’s theorem Acknowledgements: This work is in the frames of the bilateral re- search project between Bulgarian and Serbian Academies of Sciences, “An- alytical and numerical methods for differential and integral equations and mathematical models of arbitrary (fractional or high) order”. References: [1] P. Dr´abek, M. Langerov´a,S. Tersian, Existence and multiplicity of periodic solutions to one-dimensional p-Laplacian. Electronic Journal of Qualitative Theory of Differential Equations 30 (2016), 1–9. 66

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

EXPLICIT SOLUTION OF A BOUNDARY VALUE PROBLEM FOR A HEAVY STRING

Ivan DIMOVSKI 1, Yulian TSANKOV 2,§ 1 Institute of Mathematics and Informatics Bulgarian Academy of Sciences ”Acad. G. Bonchev” Str., Block 8, 1113 – Sofia, BULGARIA e-mail: [email protected] 2 Faculty of Mathematics and Informatics Sofia University “St. Kliment Ohridsky” Blvd. James Bourchier, 5, Sofia – 1164, BULGARIA e-mail: [email protected]fia.bg

In this paper we find an explicit solution of a problem for hanging chain, described by PDE ( ) ∂2 ∂ ∂ v = x v, 0 < x < 1, 0 < t, (1) ∂t2 ∂x ∂x with standard initial and boundary conditions. This problem arise when we consider heavy string, fixed on the top and free at the bottom. We solve this problem in the following way. First, by a transformation operator T of Sonine type we reduce this problem to such a BVP, but with constant coefficients and with a nonlocal boundary condition. Then we use a two-dimensional operational calculus to find an explicit Duhamel type representation of the solution. In the end, by the inverse transformation T −1 of the operator T we find explicit representation of the solution of the BVP for PDE (1).

MSC 2010: 35C05, 34B10, 44A35, 44A40 Key Words and Phrases: heavy string; nonlocal BVP; non-classical convolution; Bessel functions; Mikusi´nski-type operational calculus; Duhamel principle 67

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

SOME RESULTS ABOUT A FILTRATION OF STARLIKE FUNCTIONS Nikola TUNESKI 1,§, David SHOIKHET 2, Mark ELIN 3 1 Ss. Cyril and Methodius University in Skopje Karpos 2 b.b., Skopje, R. MACEDONIA e-mail: [email protected] 2 Holon Institute of Technology P.O. Box 305, Holon, ISRAEL e-mail: [email protected] 3 Department of Mathematics, ORT Braude College P.O. Box 78, Karmiel 21982, ISRAEL e-mail: mark [email protected] Let A be the class of functions f that are analytic in the open unit disk ∆ and are normalized such that f(0) = f ′(0) − 1 = 0. Also, let S∗ be the class of normalized starlike{ univalent[ functions] } zf ′(z) S∗ = f ∈ A : Re > 0, z ∈ ∆ . f(z) Now, using the operator [ ] f(z) f(z)f ′′(z) D[f](t, z) = t|z|2 + (1 − t) 1 − (1 − |z|2) zf ′(z) [f ′(z)]2 ∈ S∗ ∈ (t is real and z ∆) we define class t , t [0, 1], consisting of functions f ∈ A such that Re D[f](t, z) ≥ 0, z ∈ ∆. S∗ ∈ It turns out that the family of classes t , t [0, 1], is a filtration of the S∗ class of starlike functions. In addition, for a function f from t we give a f(z) result over the real part of and an approximation property of f. zf ′(z) MSC 2010: 30C45, 30C55, 30C80 Key Words and Phrases: starlike functions; filtration; embedding; real part estimate; growth estimate; approximation Work in frames of bilateral agreement between MANU and BAS. 68

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

NON-LOCAL WAVE EQUATION USING FRACTIONAL STRESS-GRADIENT ERINGEN’S CONSTITUTIVE LAW G¨unther HORMANN¨ 1, Ljubica OPARNICA 2, DuˇsanZORICA 3,4,§ 1 Faculty of Mathematics – University of Vienna Oskar-Morgenstern-Platz 1, Wien – 1090, AUSTRIA e-mail: [email protected] 2 Faculty of Education in Sombor – University of Novi Sad Podgoriˇcka 4, Sombor – 25000, SERBIA e-mail: [email protected] 3 Mathematical Institute – Serbian Academy of Arts and Sciences Kneza Mihaila 36, Belgrade – 11000, SERBIA 4 Department of Physics, Faculty of Sciences – University of Novi Sad Trg. D. Obradoviˇca4, Novi Sad – 21000, SERBIA e-mail: [email protected]

Existence of solution, supported in t > 0, to the generalized Cauchy problem for the fractional Eringen wave equation 2 α 2 ′ 2 ∂ u(x, t) − L ∂ u(x, t) = u0(x) ⊗ δ (t) + v0(x) ⊗ δ(t), (x, t) ∈ R , with t x x [( ) ] √ −1 α F −1 − | |α ∗ ∈ Lx w(x, t) = ξ→x 1 ξ cos(απ/2) x w(x, t), α (1, 3), is proved. Moreover, microlocal regularity properties of such solution are analyzed. Numerical examples are given in order to illustrate the time-evolution of spatial profiles of solutions. The fractional Eringen wave equation is obtained from the system of three equations: equation of motion, strain, and fractional Eringen law: 2 − α α F α | |α ∂xσ = ρ ∂t u, ε = ∂xu, σ ℓ D σ = E ε, with (D w) = ξ wˆ cos(απ/2). MSC 2010: 35B65, 35R11, 74J05, 74D05 Key Words and Phrases: wave front set; Eringen constitutive equa- tion; Cauchy problem; fractional derivatives; distributional solutions 69

“Transform Methods and Special Functions 2017” 8th International Conference, Sofia, August 27–31, 2017

LIST OF REGISTERED PARTICIPANTS / ABSTRACT AT PAGE:

Teodor ATANACKOVIC (Serbia) ...... 6 Emilia BAZHLEKOVA (Bulgaria) ...... 7 Ivan BAZHLEKOV (Bulgaria) ...... 7 Aurelian CERNEA (Romania) ...... 8 Bohdan DATSKO (Poland) ...... 9 Georgi DIMKOV (Bulgaria) ...... 10 Ivan DIMOVSKI (Bulgaria) ...... 11, 61, 66 Petia DINEVA (Bulgaria) ...... 50 Tzanko DONCHEV (Bulgaria) ...... 12 Maria FARCASEANU (Romania) ...... 13 Roberto GARRAPPA (Italy) ...... 14, 35, 46 Ivan GEORGIEV (Bulgaria) ...... 7 Katarzyna GORSKA (Poland) ...... 15 Luiz GUERREIRO LOPES (Portugal) ...... 16 Andrea GUISTI (Italy) ...... 17 Andrzej HORZELA (Poland) ...... 18 Snezhana HRISTOVA (Bulgaria) ...... 19 Nikolaj IKONOMOV (Bulgaria) Biljana JOLEVSKA-TUNESKA (Macedonia) ...... 20 Erkinjon KARIMOV (Uzbekistan) ...... 21, 54 Mokhtar KIRANE (France) ...... 22 Virginia KIRYAKOVA (Bulgaria) ...... 23 Malgorzata KLIMEK (Poland) ...... 25 Ralica KOVACHEVA (Bulgaria) ...... 27 Valmir KRASNIQI (R. Kosova) ...... 28 Daniel Cao LABORA (Spain) ...... 29, 51 Yingjie LIANG (China) ...... 30 Yuri LUCHKO (Germany) ...... 31 Vasile LUPULESCU (Romania) JA Tenreiro MACHADO (Portugal) ...... 33 Francesco MAINARDI (Italy) ...... 35 70

Ivan MATYCHYN (Poland) ...... 36 Penka MAYSTER (Bulgaria) Fatma Al-MUSALHI (Oman) ...... 37, 54 Oleg MUSKAROV (Bulgaria) Viktoriia ONYSHCHENKO (Ukraine) ...... 36 Jordanka PANEVA-KONOVSKA (Bulgaria) ...... 38 Donka PASHKOULEVA (Bulgaria) ...... 39 Irina PETRESKA (Macedonia) ...... 40 Stevan PILIPOVIC (Serbia) ...... 41 Igor PODLUBNY (Slovak R.) ...... 42 Tibor POGANY (Croatia) ...... 43 Nedyu POPIIVANOV (Bulgaria) ...... 44, 47 Marina POPOLIZIO (Italy) ...... 46 Todor POPOV (Bulgaria) ...... 47 Yuriy POVSTENKO (Poland) ...... 48 Predrag RAJKOVIC (Serbia) ...... 49 Tsviatko RANGELOV (Bulgaria) ...... 50 Rosana RODRUGUEZ-LOPEZ (Spain) ...... 29, 51 Peter RUSEV (Bulgaria) ...... 52 Nasser Al-SALTI (Oman) ...... 54 Trifce SANDEV (Macedonia) ...... 55 Blagovest SENDOV (Bulgaria) ...... 56 Elina SHISHKINA (Russia) ...... 57 Chung-Sik SIN (D.P.R. Korea) ...... 58 Sergei SITNIK (Russia) ...... 59 Dragan SPASIC (Serbia) ...... 60 Margarita SPIRIDONOVA (Bulgaria) ...... 61 HongGuang SUN (China) ...... 62 Arpad TAKACI (Serbia) ...... 63 Djurdjica TAKACI (Serbia) ...... 64 Stepan TERZIAN (Bulgaria) ...... 65 Yulian TSANKOV (Bulgaria) ...... 66 Nikola TUNESKI (Macedonia) ...... 67 Dusan ZORICA (Serbia) ...... 68

Total 69: Foreign 43; Bulgarian 22; Accompanying persons 4