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

Emile Franc Doungmo Goufo; Stella Mugisha

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 4191-4205
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Emile Franc Doungmo Goufo, and Stella Mugisha. "Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order." Open Mathematics 13.1 (2015): 4191-4205. <http://eudml.org/doc/271002>.

@article{EmileFrancDoungmoGoufo2015,
abstract = {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.},
author = {Emile Franc Doungmo Goufo, Stella Mugisha},
journal = {Open Mathematics},
keywords = {Fractional Cauchy problem; Contractive solutions; Fragmentation; Perturbation; Solution operators; Positivity; conventional derivative with a new parameter; ebola epidemic model; non-linear incidence; existence; stability},
language = {eng},
number = {1},
pages = {4191-4205},
title = {Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order},
url = {http://eudml.org/doc/271002},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Emile Franc Doungmo Goufo
AU - Stella Mugisha
TI - Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 4191
EP - 4205
AB - 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.
LA - eng
KW - Fractional Cauchy problem; Contractive solutions; Fragmentation; Perturbation; Solution operators; Positivity; conventional derivative with a new parameter; ebola epidemic model; non-linear incidence; existence; stability
UR - http://eudml.org/doc/271002
ER -

References

top
  1. [1] Anderson W.J., Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer Verlag, New York, 1991 Zbl0731.60067
  2. [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 http://dx.doi.org/10.1155/2013/752371 [Crossref] 
  3. [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 http://dx.doi.org/10.1155/2013/696597 [Crossref] Zbl1299.76229
  4. [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 http://www.boundaryvalueproblems.com/content/2013/1/53 Zbl1291.35206
  5. [5] Balakrishnan, A.V., Fractional powers of closed operators and semigroups generated by them, Pacific J. Math., 1960, 10, 419 Zbl0103.33502
  6. [6] Banasiak J., Arlotti L., Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, 2006. Zbl1097.47038
  7. [7] Bartle R.G., The elements of integration and Lebesgue measure, Wiley Interscience, 1995 Zbl0838.28001
  8. [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. [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. [10] Bazhlekova E. G., Subordination principle for fractional evolution equations, Fractional Calculus & Applied Analysis, 2000, 3(3), 213 – 230 Zbl1041.34046
  11. [11] Berberan-Santos Mário N., Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry, November 2005, 38(4) 
  12. [12] Brockmann D., Hufnagel L., Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Review Lett. 98, 2007 
  13. [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. [14] Diethelm K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. Zbl1215.34001
  15. [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 Zbl1262.92032
  16. [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: http://www.advancesindifferenceequations.com/content/2014/1/278 [Crossref] 
  17. [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 http://dx.doi.org/10.1155/2014/392792 [Crossref] 
  18. [18] EF Doungmo Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18(3), 2015 Zbl1316.26004
  19. [19] EF Doungmo Goufo, Non-local and Non-autonomous Fragmentation-Coagulation Processes in Moving Media, PhD thesis, North- West University, South Africa, 2014. 
  20. [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 http://dx.doi.org/10.1016/j.crma.2013.09.023 [Crossref] Zbl06238800
  21. [21] Engel K-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), Springer, 2000 Zbl0952.47036
  22. [22] Érdelyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, Vol. III McGraw-Hill, New York, 1955 Zbl0064.06302
  23. [23] Filippov I., On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl. 1961, 6, 275–293 
  24. [24] Garibotti C. R., Spiga G., Boltzmann equation for inelastic scattering, J. Phys. A 1994, 27, 2709–2717 Zbl0834.45011
  25. [25] Gel’fand I., Shilov G., Generalized Functions, vol. I. Academic Pres5, New York, 1964. 
  26. [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 Zbl1027.33006
  27. [27] He J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech., 1999 vol. 178, pp. 257–262 Zbl0956.70017
  28. [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 Zbl1068.74618
  29. [29] Hilfer R., Application of Fractional Calculus in Physics, World Scientific, Singapore, 1999. 
  30. [30] Hilfer R., On new class of phase transitions, Random magnetism and High temprature superconductivity, page 85, Singapore, World Scientific publ. Co., 1994. 
  31. [31] Lachowicz M., Wrzosek D., A nonlocal coagulation-fragmentation model, Appl. Math. (Warsaw), 2000, 27 (1), 45–66 Zbl0994.35054
  32. [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. [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] Zbl0892.47044
  34. [34] Majorana A., Milazzo C., Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. Anal. Appl. 2001, 259(2), 609–629 Zbl0986.35112
  35. [35] Melzak Z.A., A Scalar Transport Equation, Trans. Amer. Math. Soc., 1957, 85, 547–560 [Crossref] 
  36. [36] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Willey and Sons, Inc., New York, 2003. Zbl0789.26002
  37. [37] Mittag-Leffler G.M., C. R. Acad. Sci. Paris (Ser. II) 1903, 137–554 
  38. [38] Norris J.R., Markov Chains, Cambridge University Press, Cambridge, 1998 
  39. [39] Oldham K. B., Spanier J., The fractional calculus, Academic Press, New York, 1999 Zbl0428.26004
  40. [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 http://dx.doi.org/10.1155/2014/789769 [Crossref] 
  41. [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 http://dx.doi.org/10.1155/2013/320750 [Crossref] Zbl1296.35022
  42. [42] Özalp N., Demirci E., A fractional order SEIR model with vertical transmission Mathematical and Computer Modelling, 54(2011), 1–6 Zbl1225.34011
  43. [43] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 44, 1983 Zbl0516.47023
  44. [44] Podlubny, I., Fractional Differential Equations, Academic Press, California, USA, 1999 
  45. [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. [46] Prüss J., Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Boston–Berlin 1993 
  47. [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 Zbl1201.92062
  48. [48] Rubin B., Fractional Integrals and potentials, Addison Wesley Longman Limited, Harlow 1996 Zbl0864.26004
  49. [49] Samko S.G., Kilbas A.A., Marichev O.I., Franctional integrals and derivatives, Theory and Application, Gordon and Breach, Amsterdam, 1993 
  50. [50] Wagner W., Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab., 2005, 15(3), 2081–2112 [Crossref] Zbl1082.60075
  51. [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 Zbl0194.15401
  52. [52] Wright E. M., The generalized Bessel function of order greater than one. Quarterly Journal of Mathematics (Oxford ser.), 1940, 11, 36–48 Zbl0023.14101
  53. [53] Yosida K., Functional Analysis, Sixth Edition, Springer- Verlag, 1980 
  54. [54] Ziff R.M., McGrady E.D., The kinetics of cluster fragmentation and depolymerization, J. Phys. A, 1985, 18 3027–3037 
  55. [55] Ziff R.M., McGrady E.D., Shattering transition in fragmentation, Phys. Rev. Lett., 1987, 58(9) 
  56. [56] Ziff, R.M., McGrady, E.D., Kinetics of polymer degradation, Macromolecules 19, 1986, 2513–2519. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.