THERMAL SCIENCE

International Scientific Journal

External Links

EXTENSION OF FRAGMENTATION PROCESS IN A KINETIC-DIFFUSIVE-WAVE SYSTEM

ABSTRACT
Alternative methods are used to set conditions and investigate, in the space L1(R3 × R+ mdmdx) the well-posedness of a fractional fragmentation process in a kinetic-diffusive-wave medium. In the analysis, three separate models of diffusion are studied. Techniques like separation of variables and subordination principle are used to finally prove that the Cauchy problem for fractional fragmentation dynamics in a kinetic-diffusive-wave system is well-posed and admits a solution operator that is positive and contractive. This work brings a contribution that may lead to the full explanation of strange phenomena like shattering and sudden appearance of an infinite number of particles in a system that occur in the dynamics of fragmentation process and which remain partially unsolved.
KEYWORDS
PAPER SUBMITTED: 2014-10-10
PAPER REVISED: 2015-03-15
PAPER ACCEPTED: 2015-03-18
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S1S13D
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S13 - S23]
REFERENCES
  1. Bazhlekova, E. G., Subordination Principle for Fractional Evolution Equations, Fractional Calculus & Applied Analysis, 3 (2000) 3, pp. 213-230
  2. Bazhlekova, E. G., Perturbation and Approximation Properties for Abstract Evolution Equations of Fractional Order, Research Report RANA 00-05, Eindhoven University of Technology, Eindhoven, The Netherlands, 2000
  3. Brockmann, D., Hufnagel, L., Front Propagation in Reaction-Superdiffusion Dynamics: Taming Lévy Flights with Fluctuations, Phys. Review Lett., 98 (2007), 17, 178301
  4. Doungmo Goufo, E. F., A Biomathematical View on the Fractional Dynamics of Cellulose Degradation, Fractional Calculus and Applied Analysis, (2014) (in press).
  5. Gorenflo, R., et al., Analytical Properties and Applications of the Wright Function, Fractional Calculus and Applied Analysis, 2 (1999), 4, pp. 383-414
  6. Mainardi, F., Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitons and Fractals, 7 (1996), 9, pp. 1466-1477
  7. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, Cal., USA, 1999.
  8. Pruss, J., Evolutionary Integral Equations and Applications, Birkhauser, Basel-Boston-Berlin, Germany, 1993
  9. Atangana, A., Doungmo Goufo, E. F., Extension of Match Asymptotic Method to Fractional Boundary Layers Problems, Mathematical Problems in Engineering, 2014 (2014), 107535, dx.doi.org/10. ID 1155/2014/107535
  10. Atangana, A., On the Singular Perturbations for Fractional Differential Equation, The Scientific World Journal, 2014 (2014), ID 752371, dx.doi.org/10.1155/2013/752371
  11. Doungmo Goufo, E. F., et al., Some Properties of Kermack-McKendrick Epidemic Model with Fractional Derivative and Nonlinear Incidence, Advances in Difference Equations, 2014 (2014), 278, DOI: 10.1186/1687-1847-2014-278
  12. Samko, S. G., et al., Franctional Integrals and Derivatives, Theory and Application, Gordon and Breach, Amsterdam, The Netherlands,1993
  13. 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. 1351, 2013, pp. 753-759, dx.doi.org/10.1016/j.crma.2013.09.023
  14. Doungmo Goufo, E. F., Non-Local and Non-Autonomous Fragmentation-Coagulation Processes in Moving Media, Ph. D. thesis, North-West University, Mafikeng, South Africa, 2014
  15. 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 (2013), ID 320750, dx.doi.org/10.1155/2013/320750.
  16. Ziff, R. M., McGrady, E. D., The Kinetics of Cluster Fragmentation and Depolymerization, J. Phys. A, 18 (1985), pp. 3027-3037
  17. Wagner, W., Explosion Phenomena in Stochastic Coagulation-Fragmentation Models, Ann. Appl. Probab., 15 (2005), 3, pp. 2081-2112
  18. Ziff, R. M., McGrady, E. D., Shattering Transition in Fragmentation, Phys. Rev. Lett. 58 (1987), 9, pp. 982-985
  19. Benson, D. A., et al., Fractional Calculus in Hydrologic Modeling: A Numerical Perspective, Advances in Water Resources, 51 (2013), pp. 479-497
  20. Diethelm, K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010
  21. Caputo, M., Linear Models of Dissipation Whose Q is Almost Frequency Independent II, Geophys. J. R. Ast. Soc., 13 (1967), 5, pp. 529-539; Reprinted in: Fract. Calc. Appl. Anal., 11 (2008), 1, pp. 3-14
  22. Mittag-Leffler, G. M., C. R. Acad. Sci. Paris, II (1903), 137
  23. Erdelyi, A., et al., Higher Transcendental Functions, McGraw-Hill, Vol. III, New York, USA, 1955
  24. Wright, E. M., The Generalized Bessel Function of Order Greater than One, Quarterly Journal of Mathematics (Oxford ser.), 11 (1940), pp. 36-48
  25. Engel, K-J., Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer Verlag, New York, USA, 2000
  26. Meerschaert, M. M., et al., Fractional Cauchy Problems on Bounded Domains, Ann. Appl. Probab., 37 (2009), 3, pp. 979-1007
  27. Gorenflo, R., Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order, in: Fractals and Fractional Calculus in Continuum Mechanics (Eds.: A. Capinteri, F. Mainardi), Springer- Verlag, New York, USA, (2008), pp 223-276
  28. Henry, D., Geometric Theory of Semilinear Equations, Springer Verlag, New York, USA, 1981
  29. Adams, R. A., Sobolev Spaces, Academic Press, New York, USA, 1975
  30. Yosida, K., Functional Analysis, 6th ed., Springer-Verlag, New York, USA, 1980
  31. Banasiak, J., Kinetic-Type Models with Diffusion: Conservative and Nonconservative Solutions, Transport Theory and Statistical Physics, 36 (2007) 1, pp. 43-65

© 2020 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence