Publication Cover
Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 15
165
Views
9
CrossRef citations to date
0
Altmetric
Articles

Global solution of space-fractional diffusion equations with nonlinear reaction source terms

, ORCID Icon & ORCID Icon
Pages 2709-2739 | Received 11 Nov 2018, Accepted 09 Feb 2019, Published online: 27 Feb 2019

References

  • Barkai E, Metzler R, Klafter J. From continuous time random walks to the fractional Fokker-Planck equation. Phys Rev E. 2000;61(1):132.
  • Metzler R, Klafter J. Boundary value problems for fractional diffusion equations. Phys A: Stat Mech Appl. 2000;278(1–2):107–125.
  • Yuste SB, Acedo L, Lindenberg K. Reaction front in an A + B → C reaction-subdiffusion process. Physical Review E. 2004;69(3):036126.
  • Benson DA, Schumer R, Meerschaert MM, et al. Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp Porous Media. 2001;42(1-2):211–240.
  • Benson DA, Wheatcraft SW, Meerschaert MM. Application of a fractional advection-dispersion equation. Water Resour Res. 2000;36(6):1403–1412.
  • Yuste SB, Lindenberg K. Subdiffusion-limited A + A reactions. Phys Rev Lett. 2001;87(11):118301.
  • Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339(1):1–77.
  • Metzler R, Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A: Math General. 2004;37(31):R161.
  • Compte A. Stochastic foundations of fractional dynamics. Phys Rev E. 1996;53(4):4191.
  • Klafter J, Lim SC, Metzler R. Fractional dynamics: recent advances. Singapore: World Scientific; 2012.
  • Metzler R, Klafter J, Sokolov IM. Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended. Phys Rev E. 1998;58(2):1621.
  • Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. London: World Scientific; 2010.
  • Berkowitz B, Cortis A, Dentz M, et al. Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev Geophys. 2006;44(2):1–49.
  • Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55(11):48–54.
  • Podlubny I. Fractional-order systems and PIλDμ-controllers. IEEE Trans Automat Contr. 1999;44(1):208–214.
  • Jin B, Rundell W. A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl. 2015;31(3):035003.
  • Dang DT, Nane E, Nguyen DM, et al. Continuity of solutions of a class of fractional equations. Potential Anal. 2018;0:1–56.
  • Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the space-time fractional diffusion equation. arXiv preprint cond-mat/0702419, 2007.
  • Gorenflo R, Mainardi F, Moretti D, et al. Discrete random walk models for space–time fractional diffusion. Chem Phys. 2002;284(1-2):521–541.
  • Mainardi F, Pagnini G, Saxena RK. Fox H functions in fractional diffusion. J Comput Appl Math. 2005;178(1):321–331. Proceedings of the Seventh International Symposium on Orthogonal Polynomials, Special Functions and Applications.
  • Saxena RK. Fractional helmholtz and fractional wave equations with riesz–feller and generalized riemann-liouville fractional derivatives. Eur J Pur Appl Math. 2014;7(3):312–334.
  • Kateregga M, Mataramvura S, Taylor D. Parameter estimation for stable distributions with application to commodity futures log-returns. Cogent Economics & Finance. 2017;5(1):1318813.
  • Aldoghaither A, Liu D, Laleg-Kirati T. Modulating functions based algorithm for the estimation of the coefficients and differentiation order for a space-fractional advection-dispersion equation. SIAM J Sci Comput. 2015;37(6):A2813–A2839.
  • Cheng J, Nakagawa J, Yamamoto M, et al. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 2009;25(11):115002.
  • Li G, Zhang D, Jia X, et al. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation. Inverse Probl. 2013;29(6):065014.
  • Brezis H. Functional analysis, Sobolev spaces and partial differential equations. New York: Springer Science & Business Media; 2011.
  • David Logan J. Transport modeling in hydrogeochemical systems. Vol. 15. New York: Springer Science & Business Media; 2001.
  • Triet LM, Quan PH, Trong DD. On a backward nonlinear parabolic equation with time and space dependent thermal conductivity: regularization and error estimates. J Invers Ill-pose Probl. 2013;22(3):375–401.
  • Quan PH, Trong DD. A nonlinearly backward heat problem: uniqueness, regularization and error estimate. Appl Anal. 2006;85(6-7):641–657.
  • Trong DD, Dien NM. Regularization of a backward heat transfer problem with a nonlinear source. Acta Math Viet-Namica. 2011;2:505–515.
  • Tuan NH, Duy Hai DN, Long LD, et al. On a Riesz–Feller space fractional backward diffusion problem with a nonlinear source. J Comput Appl Math. 2017;312:103–126.
  • Yang F, Li X-X, Li D-G, et al. The simplified tikhonov regularization method for solving a riesz–feller space-fractional backward diffusion problem. Mathematics in Computer Science. 2017 Mar;11(1):91–110.
  • Zheng GH, Wei T. Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem. Inverse Probl. 2010;26(11):115017.
  • Kirsch A. An introduction to the mathematical theory of inverse problems. Vol. 120. New York: Springer Science & Business Media; 2011.
  • Zhao Z, Meng Z. A modified Tikhonov regularization method for a backward heat equation. Inverse Probl Sci Eng. 2011;19(8):1175–1182.
  • Lattès R, Lions JL. Méthode de quasi-réversibilité et applications. Vol. 18. Paris: Dunod; 1967.
  • Dang DT, Nguyen HT. Regularization and error estimates for nonhomogeneous backward heat problems. Electron J Differ Eq. 2006;2006(04):1–10.
  • Huang Y, Zheng Q. Regularization for ill-posed cauchy problems associated with generators of analytic semigroups. J Differ Equ. 2004;203(1):38–54.
  • Miller K. Stabilized quasi-reversibilite and other nearly-best-possible methods for non-well-posed problems. Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Berlin: Springer; 1973. p. 161–176.
  • Payne LE. Some general remarks on improperly posed problems for partial differential equations. Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Berlin: Springer; 1973. p. 1–30.
  • Oppenheimer SF, Clark GW. Quasi-reversibility methods for non-well-posed problems. Electron J Differ Equat. 1994;8:1–9.
  • Denche M, Bessila K. A modified quasi-boundary value method for ill-posed problems. J Math Anal Appl. 2005;301(2):419–426.
  • Trong DD, Duy BT, Minh MN. Backward heat equations with locally Lipschitz source. Appl Anal. 2015;94(10):2023–2036.
  • Trong DD, Quan PH, Khanh TV, et al. A nonlinear 1-D case backward heat problem: regularization and error. Zeitschrift für Analysis und ihre Anwendungen. 2007;26(2):231–245.
  • Nam PT. An approximate solution for nonlinear backward parabolic equations. J Math Anal Appl. 2010;367(2):337–349.
  • Beals R. Semigroups and abstract Gevrey spaces. J Funct Anal. 1972;10(3):300–308.
  • Tuan NH, Au VV, Khoa VA, et al. Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction. Inverse Probl. 2017;33(5):055019.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.