Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2018: 1.08

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 18, Issue 4

Issues

Local Solvability of a Linear System with a Fractional Derivative in Time in a Boundary Condition

Nataliya Vasylyeva
  • Institute of Mathematics National Academy of Sciences of Ukraine Tereschenkivska, str. 3, Kiev - 4 – 01601, UKRAINE
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-08-04 | DOI: https://doi.org/10.1515/fca-2015-0058

Abstract

In this paper we analyze a linear system for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides depended on time. First, we prove existence and uniqueness of the classical solution to this problem, and provide the coercive estimates of the solution. Second, based on the obtained results we establish one-to-one solvability to a linear system of a general form in the H¨older spaces.

Keywords: elliptic equation; fractional dynamic boundary condition; Caputo derivative; coercive estimates

References

  • [1] B.V. Bazaliy, Stefan problem for the Laplace equation with regard for the curvature of the free boundary. Ukr. Math. J. 40 (1997), 1465-1484.Google Scholar

  • [2] B.V. Bazaliy, A. Friedman, The Hele-Shaw problem with surface tension in a half-plane: a model problem. J. Diff. Equations 216 (2005), 387-438.Google Scholar

  • [3] B.V. Bazaliy, N. Vasil’eva, On the solvability of the Hele-Shaw model problem in weighted H¨older spaces in a plane angle. Ukrainian Math. J. 52 (2000), 1647-1660.Google Scholar

  • [4] B.V. Bazaliy, N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension. J. Math. Phys. Anal. Geom. 10, No 1 (2014), 3-43.Google Scholar

  • [5] J.-P. Bouchard, A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195 (1990), 127-293.Google Scholar

  • [6] G. Drazer, D.H. Zanette, Experimental evidence of power-law trappingtime distributions in porous media. Phys. Rev. E 60 (1999), 5858-5864.Google Scholar

  • [7] A. Erd´elyi at al. (Eds.), Higher Transcendental Functions, Vol. 3. Mc Graw-Hill, New York (1955).Google Scholar

  • [8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, 204, Elsevier Science B.V., Amsterdam (2006).Google Scholar

  • [9] M. Kirane, N. Tatar, Nonexistence of local and global solutions of an elliptic systems with time-fractional dynamical boundary conditions. Siberian Math. J. 48 (2007), 477-488.Google Scholar

  • [10] J. Klafter, G. Zumofen, M.F. Shlesinger, In: F. Mallamace, H.E. Stanley (Eds.), The Physics of Complex Systems. IOS Press, Amsterdam (1997).Google Scholar

  • [11] A.N. Kochubei, Fractional-parabolic systems. Potential Analysis 37, No 1 (2012), 1-30.Google Scholar

  • [12] M. Krasnoschok, N. Vasylyeva, Existence and uniqueness of the solutions for some initial-boundary value problems with the fractional dynamic boundary condition. Inter. J. PDE 2013 (2013), Article ID 796430, 20p.Google Scholar

  • [13] M. Krasnoschok, N. Vasylyeva, On a nonclassical fractional boundaryvalue problem for the Laplace operator. J. Diff. Equations 257, No 6 (2014), 1814-1839.Google Scholar

  • [14] M. Krasnoschok, N. Vasylyeva, On local solvability of the twodimensional Hele-Shaw problem with a fractional derivative in time. Math. Trudy 17, No 2 (2014), 102-131.Google Scholar

  • [15] M. Krasnoschok, N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the H¨older spaces. J. Nonlinear Studies 20, No 4 (2013), 591-621.Google Scholar

  • [16] O.A. Ladyzhenskaia, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Parabolic Equations. Academic Press, New York (1968).Google Scholar

  • [17] O.A. Ladyzhenskaia, N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).Google Scholar

  • [18] B.-T. Liu, J.-P. Hsu, Theoretical analysis on diffusional release from ellipsoidal drug delivery devices. Chem. Eng. Sci. 61 (2006), 1748-1752.CrossrefGoogle Scholar

  • [19] J.Y. Liu, M. Xu, S.Wang, Analytical solutions to the moving boundary problems with space-time-fractional derivatives in drug release devices. J. Phys. A: Math. and Theor. 40 (2007), 12131-12141.Google Scholar

  • [20] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and Their Applications 16, Birkh¨auser Verlag, Basel (1995).Google Scholar

  • [21] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York (1997), 291-348. Google Scholar

  • [22] R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations. Physica A 278 (2000), 107-125.Google Scholar

  • [23] G.M. Mophou, G.M. N’Gu´er´ekata, On a class of fractional differential equations in a Sobolev space. Applicable Analysis 91 (2012), 15-34.Web of ScienceGoogle Scholar

  • [24] P.B. Mucha, On the Stefan problem with surface tension in the Lp framework. Adv. Diff. Equations 10, No 8 (2005), 861-900.Google Scholar

  • [25] J.A. Ochoa-Tapia, F.J. Valdes-Parada, J. Alvarez-Ramirez, A fractional-order Darcy’s law. Physica A 374 (2007), 1-14.Google Scholar

  • [26] A.V. Pskhu, Partial Differential Equations of the Fractional Order. Nauka, Moscow, 2005 (in Russian).Google Scholar

  • [27] A.V. Pskhu, The fundamental solution of a diffusion-wave equation of fractional order. Izvestia RAN 73 (2009), 141-182 (in Russian).Google Scholar

  • [28] K. Ritchie, X.-Y. Shan, J. Kondo, K. Iwasawa, T. Fujiwara, A. Kusumi, Detection of non-Brownian diffusion in the cell membrance in single molecule tracking. Biophys. J. 88 (2005), 2266-2277.CrossrefGoogle Scholar

  • [29] S. Roscani, E. Santillan Marcus, A new equivalence of Stefan’s problems for the time-fractional diffusion equation. Fract. Calc. Appl. Anal. 17, No 2 (2014), 371-381; DOI: 10.2478/s13540-014-0175-3; http://www.degruyter.com/view/j/fca.2014.17.issue-2/s13540-014-0175-3/s13540-014-0175-3.xml.CrossrefGoogle Scholar

  • [30] S. Roscani, A generalization of the Hopf’s lemma for the 1-D movingboundary problem for the fractional diffusion equation and its application to a fractional free boundary problem. arXiv:1502.01209.Google Scholar

  • [31] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382 (2011), 426-447.CrossrefGoogle Scholar

  • [32] V.A. Solonnikov, Estimates for the solution of the second initialboundary value problem for the Stokes system in spaces of functions with H¨older-continuous derivatives with respect to the space variables. J. Math. Sci. 109, No 5 (2002), 1997-2017.CrossrefGoogle Scholar

  • [33] N. Vasylyeva, On a local solvability of the multidimensional Muskat problem with a fractional derivative in time on the boundary condition. Fract. Differ. Calc. 4, No 2 (2014), 89-124.Google Scholar

  • [34] N. Vasylyeva, L. Vynnytska, On a multidimensional moving boundary problem governed by anomalous diffusion: analytical and numerical study. Nonlinear Differ. Equ. Appl. NoDEA, Dec. 2014; DOI: 10.1007/s00030-014-0295-9.CrossrefGoogle Scholar

  • [35] V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. Internat. J. Heat and Mass Transf. 53 (2010), 5622-5625. Google Scholar

About the article

Received: 2015-01-08

Published Online: 2015-08-04

Published in Print: 2015-08-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 4, Pages 982–1005, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2015-0058.

Export Citation

© Diogenes Co., Sofia - frontmatter and editorial.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Kambiz Razminia, Abolhassan Razminia, and Abdolnabi Hashemi
Environmental Earth Sciences, 2016, Volume 75, Number 22
[2]
Nataliya Vasylyeva and Vitalii Overko
Communications on Pure and Applied Analysis, 2016, Volume 15, Number 5, Page 1941

Comments (0)

Please log in or register to comment.
Log in