Accessible 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

Abstract

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.

Acknowledgements

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.

References

[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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search in 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. 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search in Google Scholar

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

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

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

[43] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994. Search 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. Search 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. Search 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. Search 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