Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


2015 | 13 | 1 |

Tytuł artykułu

Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

Treść / Zawartość

Warianty tytułu

Języki publikacji



Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters’ fission process.We make use of the theory of strongly continuous solution operators for fractional models (analogues of C0-semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993) to analyze and show existence results for clusters’ splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He’s homotopy perturbation (He, 1999) and Kato’s type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive.








Opis fizyczny




  • Department of Mathematical Sciences, University of South Africa,
    Florida 0003, South Africa
  • Department of Mathematical Sciences, University of South Africa,
    Florida 0003, South Africa


  • [1] Anderson W.J., Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer Verlag, New York, 1991
  • [2] Atangana A., On the singular perturbations for fractional differential equation, The Scientific World Journal, 2014, Article ID 752371, vol. 2014, 9 pages, preprint available at [Crossref]
  • [3] Atangana A., Kílíçman A., A possible generalization of acoustic wave equation using the concept of perturbed, Mathematical problems in Engineering, Article ID 696597 preprint available at [Crossref]
  • [4] Atangana A., Botha F.C., A generalized groundwater flow equation using the concept of variable-order derivative, Boundary Value Problems 2013, 2013:53 preprint available at
  • [5] Balakrishnan, A.V., Fractional powers of closed operators and semigroups generated by them, Pacific J. Math., 1960, 10, 419
  • [6] Banasiak J., Arlotti L., Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, 2006.
  • [7] Bartle R.G., The elements of integration and Lebesgue measure, Wiley Interscience, 1995
  • [8] Benson D.A., Schumer R., Meerschaert M.M., Wheatcraft S.W., Fractional Dispersion, Levy Motion, and the MADE Tracer Tests. Transport in Porous Media 2001, 42: 211-240.
  • [9] Benson D.A., Meerschaert M.M., Revielle J., Fractional calculus in hydrologic modeling: A numerical perspective. Advances in Water Resources 2013, 51, 479–497
  • [10] Bazhlekova E. G., Subordination principle for fractional evolution equations, Fractional Calculus & Applied Analysis, 2000, 3(3), 213 – 230
  • [11] Berberan-Santos Mário N., Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry, November 2005, 38(4)
  • [12] Brockmann D., Hufnagel L., Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Review Lett. 98, 2007
  • [13] Caputo M., Linear models of dissipation whose Q is almost frequency independent, Journal of the Royal Australian Historical Society, 1967, 13(2), 529–539
  • [14] Diethelm K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010.
  • [15] Demirci E., Unal A., Özalp N., A fractional order seir model with density dependent death rate, Hacettepe Journal of Mathematics and Statistics, 2011, 40(2), 287–295
  • [16] Doungmo Goufo E.F., Maritz R. , Munganga J., Some properties of Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence, Advances in Difference Equations 2014, 2014:278. preprint available at DOI: 10.1186/1687-1847-2014- 278, URL: [Crossref]
  • [17] Doungmo Goufo E.F., Mugisha S., Mathematical solvability of a Caputo fractional polymer degradation model using further generalized functions, Mathematical Problems in Engineering, Volume 2014, Article ID 392792, 5 pages, 2014. preprint available at [Crossref]
  • [18] EF Doungmo Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18(3), 2015
  • [19] EF Doungmo Goufo, Non-local and Non-autonomous Fragmentation-Coagulation Processes in Moving Media, PhD thesis, North- West University, South Africa, 2014.
  • [20] Doungmo Goufo E.F., Oukouomi Noutchie S.C., Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates, Comptes Rendus Mathematique, C.R Acad. Sci, Paris, Ser, I, 2013, preprint available at [Crossref]
  • [21] Engel K-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), Springer, 2000
  • [22] Érdelyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, Vol. III McGraw-Hill, New York, 1955
  • [23] Filippov I., On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl. 1961, 6, 275–293
  • [24] Garibotti C. R., Spiga G., Boltzmann equation for inelastic scattering, J. Phys. A 1994, 27, 2709–2717
  • [25] Gel’fand I., Shilov G., Generalized Functions, vol. I. Academic Pres5, New York, 1964.
  • [26] Gorenflo R., Luchko Y., Mainardi F., Analytical properties and applications of the Wright function, Fractional Calculus and Applied Analysis, 1999, 2(4), 383–414
  • [27] He J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech., 1999 vol. 178, pp. 257–262
  • [28] He J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. Non-Linear Mech., 2000, 35, 37–43
  • [29] Hilfer R., Application of Fractional Calculus in Physics, World Scientific, Singapore, 1999.
  • [30] Hilfer R., On new class of phase transitions, Random magnetism and High temprature superconductivity, page 85, Singapore, World Scientific publ. Co., 1994.
  • [31] Lachowicz M., Wrzosek D., A nonlocal coagulation-fragmentation model, Appl. Math. (Warsaw), 2000, 27 (1), 45–66
  • [32] Lions J.L., Peetre J., Sur une classe d’espace d’interpolation, Inst. Hautes étude Sci. Publ. Math, 1964, 19, 5–68 [Crossref]
  • [33] McLaughlin D. J., Lamb W., McBride A. C., A semigroup approach to fragmentation models, SIAM Journal on Mathematical Analysis, 1997, 28(5), 1158–1172 [Crossref]
  • [34] Majorana A., Milazzo C., Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. Anal. Appl. 2001, 259(2), 609–629
  • [35] Melzak Z.A., A Scalar Transport Equation, Trans. Amer. Math. Soc., 1957, 85, 547–560 [Crossref]
  • [36] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Willey and Sons, Inc., New York, 2003.
  • [37] Mittag-Leffler G.M., C. R. Acad. Sci. Paris (Ser. II) 1903, 137–554
  • [38] Norris J.R., Markov Chains, Cambridge University Press, Cambridge, 1998
  • [39] Oldham K. B., Spanier J., The fractional calculus, Academic Press, New York, 1999
  • [40] Oukouomi Noutchie S.C., Doungmo Goufo E. F., Exact solutions of fragmentation equations with general fragmentation rates and separable particles distribution kernels, Mathematical Problems in Engineering, 2014, vol. 2014, Article ID 789769, 5 pages, preprint available at [Crossref]
  • [41] Oukouomi Noutchie S.C., Doungmo Goufo E.F., Global solvability of a continuous model for nonlocal fragmentation dynamics in a moving medium, Mathematical Problem in Engineering, 2013, vol. 2013, Article ID 320750, 8 pages, 2013, preprint available at [Crossref]
  • [42] Özalp N., Demirci E., A fractional order SEIR model with vertical transmission Mathematical and Computer Modelling, 54(2011), 1–6
  • [43] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 44, 1983
  • [44] Podlubny, I., Fractional Differential Equations, Academic Press, California, USA, 1999
  • [45] Pooseh S., Rodrigues H.S., Torres D.F.M., Fractional derivatives in dengue epidemics. In: Simos, T.E., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) Numerical Analysis and Applied Mathematics, ICNAAM, American Institute of Physics, Melville, 2011, 739–742
  • [46] Prüss J., Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Boston–Berlin 1993
  • [47] Rudnicki R., Wieczorek R., Phytoplankton dynamics: From the behaviour of cells to a transport equation, Math. Model. Nat. Phenom, 2006, 1 (1), 83–100
  • [48] Rubin B., Fractional Integrals and potentials, Addison Wesley Longman Limited, Harlow 1996
  • [49] Samko S.G., Kilbas A.A., Marichev O.I., Franctional integrals and derivatives, Theory and Application, Gordon and Breach, Amsterdam, 1993
  • [50] Wagner W., Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab., 2005, 15(3), 2081–2112 [Crossref]
  • [51] Westphal U., ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren, Teil I: Halbgruppen-erzeuger, Compositio Math., 1970, 22, 67–103
  • [52] Wright E. M., The generalized Bessel function of order greater than one. Quarterly Journal of Mathematics (Oxford ser.), 1940, 11, 36–48
  • [53] Yosida K., Functional Analysis, Sixth Edition, Springer- Verlag, 1980
  • [54] Ziff R.M., McGrady E.D., The kinetics of cluster fragmentation and depolymerization, J. Phys. A, 1985, 18 3027–3037
  • [55] Ziff R.M., McGrady E.D., Shattering transition in fragmentation, Phys. Rev. Lett., 1987, 58(9)
  • [56] Ziff, R.M., McGrady, E.D., Kinetics of polymer degradation, Macromolecules 19, 1986, 2513–2519.

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.