Download PDF - Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditionsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Global existence and blow-up of generalized self-similar solutions for a space-fractional diffusion equation with mixed conditions

Authors 1, 2

Affiliations

  1. Laboratoire de Mathématique et Physique Appliquées, École Normale Supérieure de Bousaada
  2. Laboratory of Pure and Applied Mathematics, Mohamed Boudiaf University of M'sila

Abstract

This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder's and Banach's fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

Keywords

ENG
fractional diffusion, generalized self-similar solution, blow-up, global existence, uniqueness

Bibliography

  1. Arioua, Yacine, and Bilal Basti, and Nouredine Benhamidouche. "Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative." Appl. Math. E-Notes 19 (2019): 397-412.
  2. Basti, Bilal, and Yacine Arioua, and Nouredine Benhamidouche. "Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations." J. Math. Appl. 42 (2019): 35-61.
  3. Basti, Bilal, and Yacine Arioua, and Nouredine Benhamidouche. "Existence results for nonlinear Katugampola fractional differential equations with an integral condition." Acta Math. Univ. Comenian. (N.S.) 89 (2020): 243-260.
  4. Basti, Bilal, and Nouredine Benhamidouche. "Existence results of self-similar solutions to the Caputo-type’s space-fractional heat equation." Surv. Math. Appl. 15 (2020): 153-168.
  5. Basti, Bilal, and Nouredine Benhamidouche. "Global existence and blow-up of generalized self-similar solutions to nonlinear degenerate dffusion equation not in divergence form." Appl. Math. E-Notes 20 (2020): 367-387.
  6. Buckwar, Evelyn, and Yurii F. Luchko. "Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations." J. Math. Anal. Appl. 227, no. 1 (1998): 81-97.
  7. Diethelm, Kai. The Analysis of Fractional Differential Equations, Vol. 2004 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2010.
  8. Granas, Andrzej and James Dugundji. Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003.
  9. Kilbas, Anatoly A., and Hari M. Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Diffrential Equations. vol. 204 of North-Holland Mathematics Studies. Elsevier Science, 2006.
  10. Luchko, Yurii F., and Rudolf Gorenflo. "Scale-invariant solutions of a partial differential equation of fractional order." Fract. Calc. Appl. Anal. 1, no. 1 (1998): 63-78.
  11. Luchko, Yurii F., et al. "Fractional models, non-locality, and complex systems." Comput. Math. Appl. 59, no. 3 (2010): 1048-1056.
  12. Metzler, Ralf, and Theo F. Nonnemacher. "Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation." Chem. Phys. 284, no. 1-2 (2002): 67-90.
  13. Miller, Kenneth S., and Bertram Ross. An Introduction to the Fractional Calculus and Differential Equations. A Wiley-Interscience Publication. New York– Chichester–Brisbane–Singapore: John Wiley & Sons Inc., 1993.
  14. Pierantozzi, Teresa, and Luis Vázquez Martínez. "An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like." J. Math. Phys. 46, no. 11 (2005): Art no. 113512.
  15. Podlubny, Igor. Fractional Differential Equations. Vol. 198 of Mathematics in Science and Engineering. New York: Academic Press, 1999.
  16. Samko, Stefan Grigor’evich, and Anatoli˘ı Aleksandrovich Kilbas, and Oleg Igorevich Marichev. Fractional Integral and Derivatives (Theory and Applications). Switzerland: Gordon and Breach, 1993.
  17. Vázquez Martínez, Luis, and Juan J. Trujillo, and María Pilar Velasco. "Fractional heat equation and the second law of thermodynamics." Fract. Calc. Appl. Anal. 14, no. 3 (2011): 334-342.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
2021/20
Export to PBN, Opens in new tab
Pages:
43-56
Main language of publication
English
Published
2021-05-29