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

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

See all formats and pricing
More options …
Ahead of print


Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation

Rezvan SalehiORCID iD: http://orcid.org/0000-0001-7351-9223
Published Online: 2018-04-18 | DOI: https://doi.org/10.1515/cmam-2018-0009


In this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order O(τ) and O(τ1+12σ) are derived, respectively. Stability and L2 norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.

Keywords: Distributed-Order Time Fractional Sub-Diffusion Equation; Caputo’s Fractional Derivative, Moving Least Squares Reproducing Kernel Method; Meshless Methods; Convergence and Stability

MSC 2010: 65M70; 65M12; 65M15; 35R11; 60G22


  • [1]

    E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial movement analysis, Water Resour. Res. 28 (1992), no. 2, 3293–3307. CrossrefGoogle Scholar

  • [2]

    A. Angulo, L. Pérez Pozo and F. Perazzo, A posteriori error estimator and an adaptive technique in meshless finite points method, Eng. Anal. Bound. Elem. 33 (2009), no. 11, 1322–1338. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    T. M. Atanackovic, S. Pilipovic and D. Zorica, Existence and calculation of the solution to the time distributed order diffusion equation, Phys. Scr. 2009 (2009), Article ID 014012. Web of ScienceGoogle Scholar

  • [4]

    K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd ed., Texts Appl. Math. 39, Springer, Dordrecht, 2009. Google Scholar

  • [5]

    S. N. Atluri and T. Zhu, A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22 (1998), no. 2, 117–127. CrossrefGoogle Scholar

  • [6]

    T. Belytschko, Y. Y. Lu and L. Gu, Element-free Galerkin methods, Internat. J. Numer. Methods Engrg. 37 (1994), no. 2, 229–256. CrossrefGoogle Scholar

  • [7]

    W. Bu, A. Xiao and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput. 72 (2017), no. 1, 422–441. Web of ScienceCrossrefGoogle Scholar

  • [8]

    M. Caputo, Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. Roy. Astronom. Soc. 13 (1967), no. 5, 529–539. CrossrefGoogle Scholar

  • [9]

    M. Caputo, Elasticità e dissipazione, Zanichelli, Bologna, 1969. Google Scholar

  • [10]

    M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 421–442. Google Scholar

  • [11]

    A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E. 66 (2002), no. 4, Article ID 046129. Google Scholar

  • [12]

    A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal. 6 (2003), no. 3, 259–279. Google Scholar

  • [13]

    R. Cheng and Y. Cheng, Error estimates for the finite point method, Appl. Numer. Math. 58 (2008), no. 6, 884–898. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math. 225 (2009), no. 1, 96–104. Web of ScienceCrossrefGoogle Scholar

  • [15]

    N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems, Comput. Math. Appl. 64 (2012), no. 10, 2973–2981. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    G.-H. Gao, H.-W. Sun and Z.-Z. Sun, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys. 298 (2015), 337–359. Web of ScienceCrossrefGoogle Scholar

  • [17]

    G.-H. Gao and Z.-Z. Sun, Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput. 66 (2016), no. 3, 1281–1312. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    G.-H. Gao and Z.-Z. Sun, Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations, Numer. Methods Partial Differential Equations 32 (2016), no. 2, 591–615. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    R. A. Gingold and J. J. Monaghan, Smoothed particle hydrodynamics: Theory and application to non-spherical stars, Mon. Not. R. Astron. Soc. 181 (1977), no. 3, 375–389. CrossrefGoogle Scholar

  • [20]

    A. Hanyga, Anomalous diffusion without scale invariance, J. Phys. A 40 (2007), no. 21, 5551–5563. CrossrefGoogle Scholar

  • [21]

    Z. Jiao, Y. Chen and I. Podlubny, Distributed-Order Dynamic Systems, Springer Briefs Electr. Comput. Eng., Springer, London, 2012. Google Scholar

  • [22]

    B. Jin, R. Lazarov, D. Sheen and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 69–93. Web of ScienceGoogle Scholar

  • [23]

    J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys. 259 (2014), 11–22. Web of ScienceCrossrefGoogle Scholar

  • [24]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Google Scholar

  • [25]

    A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), no. 1, 252–281. Web of ScienceCrossrefGoogle Scholar

  • [26]

    S. Li and W. K. Liu, Moving least-square reproducing kernel method. II. Fourier analysis, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1–4, 159–193. CrossrefGoogle Scholar

  • [27]

    G. R. Liu and Y. T. Gu, A point interpolation method for two-dimensional solids, Int. J. Numer. Methods Eng. 50 (2001), no. 4, 937–951. CrossrefGoogle Scholar

  • [28]

    W. K. Liu, Y. Chen, S. Jun, J. S. Chen, T. Belytschko, C. Pan, R. A. Uras and C. T. Chang, Overview and applications of the reproducing kernel particle methods, Arch. Comput. Methods Eng. 3 (1996), no. 1, 3–80. CrossrefGoogle Scholar

  • [29]

    W. K. Liu, S. Jun and Y. F. Zhang, Reproducing kernel particle methods, Internat. J. Numer. Methods Fluids 20 (1995), no. 8–9, 1081–1106. CrossrefGoogle Scholar

  • [30]

    W.-K. Liu, S. Li and T. Belytschko, Moving least-square reproducing kernel methods. I. Methodology and convergence, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1–2, 113–154. CrossrefGoogle Scholar

  • [31]

    C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam. 29 (2002), no. 1–4, 57–98. CrossrefGoogle Scholar

  • [32]

    Y. Luchko, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal. 12 (2009), no. 4, 409–422. Google Scholar

  • [33]

    F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo, Time-fractional diffusion of distributed order, J. Vib. Control 14 (2008), no. 9–10, 1267–1290. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput. 187 (2007), no. 1, 295–305. Web of ScienceGoogle Scholar

  • [35]

    S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys. 315 (2016), 169–181. Web of ScienceCrossrefGoogle Scholar

  • [36]

    M. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl. 379 (2011), no. 1, 216–228. Web of ScienceCrossrefGoogle Scholar

  • [37]

    J. M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1–4, 289–314. CrossrefGoogle Scholar

  • [38]

    B. Nayroles, G. Touzot and P. Villon, Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mech. 10 (1992), no. 5, 307–315. CrossrefGoogle Scholar

  • [39]

    R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B. 133 (1986), no. 1, 425–430. CrossrefGoogle Scholar

  • [40]

    E. Oñate, S. Idelsohn, O. C. Zienkiewicz and R. L. Taylor, A finite point method in computational mechanics. Applications to convective transport and fluid flow, Internat. J. Numer. Methods Engrg. 39 (1996), no. 22, 3839–3866. CrossrefGoogle Scholar

  • [41]

    E. Oñate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor and C. Sacco, A stabilized finite point method for analysis of fluid mechanics problems, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1–4, 315–346. CrossrefGoogle Scholar

  • [42]

    I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky and Y. Chen, Matrix approach to discrete fractional calculus III: Non-equidistant grids, variable step length and distributed orders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), no. 1990, Article iD 20120153. Google Scholar

  • [43]

    A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994. Google Scholar

  • [44]

    Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193–209. CrossrefGoogle Scholar

  • [45]

    S. Umarov and R. Gorenflo, Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations. I, Z. Anal. Anwend. 24 (2005), no. 3, 449–466. Google Scholar

  • [46]

    X. X. Zhang and L. Mouchao, Persistence of anomalous dispersion in uniform porous media demonstrated by pore-scale simulations, Water. Resour. Res. 43 (2007), no. 7, 407–437. Web of ScienceGoogle Scholar

About the article

Received: 2017-08-15

Revised: 2017-12-23

Accepted: 2018-03-19

Published Online: 2018-04-18

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0009.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in