Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 18, 2018

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

Rezvan Salehi ORCID logo EMAIL logo


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.


The author is very grateful to the reviewer for carefully reading the paper and for constructive comments and suggestions which are improved the quality of the paper.


[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. 10.1029/92WR01757Search in Google 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. 10.1016/j.enganabound.2009.06.004Search in Google 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. 10.1088/0031-8949/2009/T136/014012Search in Google Scholar

[4] K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd ed., Texts Appl. Math. 39, Springer, Dordrecht, 2009. 10.1007/978-1-4419-0458-4Search in 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. 10.1007/s004660050346Search in Google 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. 10.1002/nme.1620370205Search in Google 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. 10.1007/s10915-017-0360-8Search in Google 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. 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[9] M. Caputo, Elasticità e dissipazione, Zanichelli, Bologna, 1969. Search in Google Scholar

[10] M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 421–442. Search in 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. 10.1103/PhysRevE.66.046129Search in Google Scholar PubMed

[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. Search in Google Scholar

[13] R. Cheng and Y. Cheng, Error estimates for the finite point method, Appl. Numer. Math. 58 (2008), no. 6, 884–898. 10.1016/j.apnum.2007.04.003Search in Google 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. 10.1016/ in Google 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. 10.1016/j.camwa.2012.01.053Search in Google 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. 10.1016/ in Google 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. 10.1007/s10915-015-0064-xSearch in Google 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. 10.1002/num.22020Search in Google 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. 10.1093/mnras/181.3.375Search in Google Scholar

[20] A. Hanyga, Anomalous diffusion without scale invariance, J. Phys. A 40 (2007), no. 21, 5551–5563. 10.1088/1751-8113/40/21/007Search in Google Scholar

[21] Z. Jiao, Y. Chen and I. Podlubny, Distributed-Order Dynamic Systems, Springer Briefs Electr. Comput. Eng., Springer, London, 2012. 10.1007/978-1-4471-2852-6Search in 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. 10.1515/fca-2016-0005Search in Google Scholar

[23] J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys. 259 (2014), 11–22. 10.1016/ in Google 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. Search in Google Scholar

[25] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), no. 1, 252–281. 10.1016/j.jmaa.2007.08.024Search in Google 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. 10.1016/S0045-7825(96)01082-1Search in Google 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. 10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-XSearch in Google 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. 10.1007/BF02736130Search in Google 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. 10.1002/fld.1650200824Search in Google 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. 10.1016/S0045-7825(96)01132-2Search in Google 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. 10.1023/A:1016586905654Search in Google 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. Search in 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. 10.1177/1077546307087452Search in Google 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. 10.1016/j.amc.2006.08.126Search in Google 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. 10.1016/ in Google 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. 10.1016/j.jmaa.2010.12.056Search in Google 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. 10.1016/S0045-7825(96)01087-0Search in Google 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. 10.1007/BF00364252Search in Google 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. 10.1515/9783112495483-049Search in Google 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. 10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-RSearch in Google 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. 10.1016/S0045-7825(96)01088-2Search in Google 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. 10.1098/rsta.2012.0153Search in Google Scholar

[43] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994. 10.1007/978-3-540-85268-1Search in 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. 10.1016/j.apnum.2005.03.003Search in Google 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. 10.4171/ZAA/1250Search in 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. 10.1029/2006WR005557Search in Google Scholar

Received: 2017-08-15
Revised: 2017-12-23
Accepted: 2018-03-19
Published Online: 2018-04-18
Published in Print: 2019-10-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.12.2022 from
Scroll Up Arrow