\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Well-posedness of abstract distributed-order fractional diffusion equations

Abstract Related Papers Cited by
  • In this paper, based on distributed-order fractional diffusion equation we propose the distributed-order fractional abstract Cauchy problem (DFACP) and study the well-posedness of DFACP. Using functional calculus technique, we prove that the general distributed-order fractional operator generates a bounded analytic $\alpha$-times resolvent operator family or a $C_0$-semigroup under some suitable conditions. In addition, we reveal the relation between two $\alpha$-times resolvent families generated by the sectorial operator $A$ and the special distributed-order fractional operator, $p_1A^{\beta_1}+ p_2A^{\beta_2}+\ldots +p_nA^{\beta_n}$, respectively.
    Mathematics Subject Classification: Primary: 26A33; Secondary: 47D06.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    D. Applebaum, "Lévy Processes and Stochastic Calculus," Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511755323.

    [2]

    R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks, Phys. Rev. E, 51 (1995), 4807-4822.doi: 10.1103/PhysRevE.51.4807.

    [3]

    E. Bazhlekova, "Fractional Evolution Equations in Banach Spaces," Ph.D thesis, Eindhoven University of Technology, 2001.

    [4]

    Alfred. S. Carasso and T. Kato, On subordinated holomorphic semigroups, Trans. Amer. Math. Soc., 327 (1991), 867-878.doi: 10.1090/S0002-9947-1991-1018572-4.

    [5]

    Aleksei. V. Chechkin, R. Gorenflo and Igor. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66 (2002), 046129.doi: 10.1103/PhysRevE.66.046129.

    [6]

    C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.doi: 10.1007/s00233-009-9184-7.

    [7]

    G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.doi: 10.1007/BF01163654.

    [8]

    N. Dungey, Asymptotic type for sectorial operators and an integral of fractional powers, J. Func. Anal., 256 (2009), 1387-1407.doi: 10.1016/j.jfa.2008.07.020.

    [9]

    A. Einstein, "Investigations on the Theory of the Brownian Movement," Dover Publications Inc., New York, 1956.

    [10]

    R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191.

    [11]

    T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966.doi: 10.1007/978-3-642-53393-8.

    [12]

    A. Klemm, Hans. P. Müller and R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E, 55 (1997), 4413-4422.doi: 10.1103/PhysRevE.55.4413.

    [13]

    M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, Birkh$\ddota$uzer, 2006.doi: 10.1007/3-7643-7698-8.

    [14]

    M. Li, C. Chen and F. Li, On fractional powers of generators of fractional resolvent families, J. Func. Anal., 259 (2010), 2702-2726.doi: 10.1016/j.jfa.2010.07.007.

    [15]

    K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 63 (2011), 1398-1404.doi: 10.1016/j.camwa.2011.02.038.

    [16]

    F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput., 187 (2007), 295-305.doi: 10.1016/j.amc.2006.08.126.

    [17]

    F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192.

    [18]

    Rosario. N. Mantegna and Harry. E. Stanley, Stochastic process with ultraslow convergence to a gaussian: the truncated Levy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.doi: 10.1103/PhysRevLett.73.2946.

    [19]

    Celso. M. Carracedo and Miguel. S. Alix, "The Theory of Fractional Powers of Operators," North-Holland Publishing Co., Amsterdam, 2001.

    [20]

    Mark. M. Meerschaert and Hans. P. Scheffler, Stochastic model for ultraslow diffusion, Stoch. Proc. Appl., 116 (2006), 1215-1235.doi: 10.1016/j.spa.2006.01.006.

    [21]

    Mark. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379 (2011), 216-228.doi: 10.1016/j.jmaa.2010.12.056.

    [22]

    R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.doi: 10.1016/S0370-1573(00)00070-3.

    [23]

    Elliott. W. Montroll and George. H. Weiss, Random walks on lattices. II, J. Math. Phys., 167 (1965), 167-181.doi: 10.1063/1.1704269.

    [24]

    M. Naber, Distributed order fractional subdiffusion, Fractals, 12 (2004), 23-32.doi: 10.1142/S0218348X04002410.

    [25]

    I. Podlubny, "Fractional Differential Equations," Academic Press, New York, 1999.

    [26]

    J. Pr$\ddotu$ss, "Evolutionary Integral Equations and Applications," Birkhäuser Verlag, Basel, 1993.doi: 10.1007/978-3-0348-8570-6.

    [27]

    Lewis. F. Richardson, Atmospheric diffusion shown on a distance-neighbout graph, Proc. Roy. Soc., 110 (1926), 709-737.doi: 10.1098/rspa.1926.0043.

    [28]

    Yakov. G. Sinai, The limiting behaviour of a one-dimensional random walk in a random medium, Theor. Prob. Appl., 27 (1982), 256-268.doi: 10.1137/1127028.

    [29]

    Igor. M. Sokolov, Aleksei. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Polon., 35 (2004), 1323-1341.

    [30]

    G. Zumofen and J. Klafter, Spectral random walk of a single molecule, Chem. Phys. Lett., 219 (1994), 303-309.doi: 10.1016/0009-2614(94)87062-4.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(113) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return