An investigation of radial basis functions for fractional derivatives and their applications

  • Qingxia Liu
  • Pinghui ZhuangEmail author
  • Fawang Liu
  • Junjiang Lai
  • Vo Anh
  • Shanzhen Chen
Original Paper


In the present study, the radial basis functions (RBF) are combined with polynomial basis functions to approximate the fractional derivatives specifically. We explore two new types of local support fields for the first time in the literature. In addition, we apply the RBF, combined with an appropriate number of polynomial basis functions featuring a new type of local support field to solve the single-term, multi-term ordinary fractional equations, time–space and two-side space fractional partial differential equations. Finally, the accuracy of the presented method is demonstrated via numerical experiments.


Fractional derivatives Mixed RBF Local support fields Polynomial basis functions Jacobi–Gauss–Lobatto quadratures 



The project was partially supported by the Fundamental Research Funds for the Central Universities (Nos. 20720180003 and 20720160002). The authors also wish to acknowledge that this research was partially supported by the Australian Research Council via the Discovery Projects (Nos. DP180103858 and DP190101889) and the National Natural Science Foundation of China (Nos. 11772046, 11701467 and 11771364).


  1. 1.
    Cheng R, Sun F, Wang J (2018) Meshless analysis of two-dimensional two-sided space-fractional wave equation based on improved moving least-squares approximation. Int J Comput Math 95:540–560MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cheng R, Sun F, Wei Q, Wang J (2018) Numerical treatment for solving two-dimensional space-fractional advection–dispersion equation using meshless method. Mod Phys Lett B 32:1850073MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of a meshless method for the time fractional diffusion-wave equation. Numer Algorithm 73:445–476MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fan W, Liu F, Turner I (2017) A novel unstructured mesh finite element method for solving the time–space fractional wave equation on a two-dimensional irregular convex domain. Fract Calc Appl Anal 20:352–383MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feng L, Liu F, Turner I, Yang Q, Zhuang P (2018) Unstructured mesh finite difference/finite element method for the 2d time-space riesz fractional diffusion equation on irregular convex domains. Appl Math Model 59:441–463MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gu YT (2001) A local point interpolation method (LPIM) for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng 190:5515–5528CrossRefGoogle Scholar
  7. 7.
    Gu YT, Liu GR (2001) A coupled element free Galerkin/boundary element method for stress analysis of two-dimensional solids. Comput Methods Appl Mech Eng 190:4405–4419CrossRefGoogle Scholar
  8. 8.
    Gu YT, Liu GR (2001) A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput Mech 27:188–198CrossRefGoogle Scholar
  9. 9.
    Gu YT, Liu GR (2005) Meshfree methods and their comparisons. Int J Comput Methods 2:477–515CrossRefGoogle Scholar
  10. 10.
    Gu YT, Zhuang P (2012) Anomalous sub-diffusion equations by the meshless collocation method. Aust J Mech Eng 10:1–8CrossRefGoogle Scholar
  11. 11.
    Gu YT, Zhuang P, Liu F (2010) An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Comput Model Eng Sci 56:303–334MathSciNetzbMATHGoogle Scholar
  12. 12.
    Li C, Zeng F (2018) The finite difference methods for fractional ordinary differential equations. Numer Funct Anal Optim 34:149–179MathSciNetCrossRefGoogle Scholar
  13. 13.
    Li X, Chuanju X (2010) Existence and uniqueness of the solution of the space–time fractional diffusion euation and a spectral method approximation. Commun Comput Phys 8:1016–1051MathSciNetzbMATHGoogle Scholar
  14. 14.
    Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, BerlinGoogle Scholar
  15. 15.
    Liu GR, Gu YT (2000) Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches. Comput Mech 26:534–546zbMATHGoogle Scholar
  16. 16.
    Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 50:937–951 CrossRefGoogle Scholar
  17. 17.
    Liu GR, Gu YT (2007) Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation. Comput Mech 26:166–173CrossRefGoogle Scholar
  18. 18.
    Liu GR, Zhang GY, Gu YT, Wang Y (2001) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech 36:421–430MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu L, Zheng LC, liu F, Zhang XX (2016) An improved heat conduction model with riesz fractional Cattaneo–Christov flux. Int J Heat Mass Transf 103:1191–1197CrossRefGoogle Scholar
  20. 20.
    Liu Q, Gu YT, Zhuang P, Liu F, Nie YF (2011) An implicit RBF meshless approach for time fractional diffusion equations. Comput Mech 48:1–12MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu Q, Liu F, Gu YT, Zhuang P, Chen J, Turner I (2015) A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation. Appl Math Comput 256:930–938MathSciNetzbMATHGoogle Scholar
  22. 22.
    Liu Y, Zudeng Y, Li H, Liu F, Wang J (2018) Time two-mesh algorithm combined with finite element method for time fractional water wave model. Int J Heat Mass Transf 120:1132–1145CrossRefGoogle Scholar
  23. 23.
    Lian Y, Ying Y, Tang S, Lin S, Wagner GJ, Liu WK (2016) A Petrov–Galerkin finite element method for the fractional advection–diffusion equation. Comput Methods Appl Mech Eng 309:388–410MathSciNetCrossRefGoogle Scholar
  24. 24.
    Luan S, Lian Y, Ying Y, Tang S, Wagner GJ, Liu WK (2017) An enriched finite element method to fractional advection–diffusion equation. Comput Mech 60:181–201MathSciNetCrossRefGoogle Scholar
  25. 25.
    Podlubny I (1999) Fractional differential equations. Academic Press, CambridgezbMATHGoogle Scholar
  26. 26.
    Qin S, Liu F, Turner I (2018) A 2d multi-term time and space fractional Bloch–Torrey model based on bilinear rectangular finite elements. Commun Nonlinear Sci Numer Simul 56:270–286MathSciNetCrossRefGoogle Scholar
  27. 27.
    Shen J, Tang T, Wang L-L (2011) Spectral methods, algorithms, analysis and applications. Springer, BerlinzbMATHGoogle Scholar
  28. 28.
    Tang S, Ying Y, Lian Y, Lin S, Yang Y, Wagner GJ, Liu WK (2016) Differential operator multiplication method for fractional differential equations. Comput Mech 58:879–888MathSciNetCrossRefGoogle Scholar
  29. 29.
    Uddin M, Kamran K, Usman M, Ali A (2018) On the Laplace-transformed-based local meshless method for fractional-order diffusion equation. Numer Algorithm 19:221–225MathSciNetGoogle Scholar
  30. 30.
    Yuan ZB, Nie YF, Liu F, Turner I, Gu YT (2016) An advanced numerical modeling for riesz space fractional advection–dispersion equations by a meshfree approach. Appl Math Model 40:7816–7829MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ying Y, Lian Y, Tang S, Liu WK (2016) Enriched reproducing kernel particle method for fractional advection–diffusion equation. Acta Mech Sin 34:515–527MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ying Y, Lian Y, Tang S, Liu WK (2017) High-order central difference scheme for Caputo fractional derivative. Comput Methods Appl Mech Eng 317:42–54MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zeng F, Liu F, Li C, Burrage K, Turner IW, Anh VV (2014) A Crank–Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction–diffusion equation. SIAM J Numer Anal 120:2599–2622MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang H, Liu F, Chen S, Anh V, Chen J (2018) Fast numerical simulation of a new time–space fractional option pricing model governing European call option. Appl Math Comput 339:186–198MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhang H, Liu F, Jiang X, Zeng F, Turner I (2018) A Crank–Nicolson adi spectral method for a two-dimensional riesz space distributed-order advection–diffusion equation. Comput Math Appl 76:2460–2476MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zheng M, Liu F, Anh VV, Turner IW (2016) A high-order spectral method for the multi-term time-fractional diffusion equations. Appl Math Model 120:4970–4985MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zheng M, Liu F, Turner IW, Anh VV (2015) A novel high order space–time spectral method for the time fractional Fokker–Planck equation. SIAM J Sci Comput 120:A701–A724MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhuang P, Gu YT, Liu F, Turner I, Yarlagadda PKDV (2012) Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method. Int J Numer Methods Eng 88:1346–1362MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Qingxia Liu
    • 1
    • 2
  • Pinghui Zhuang
    • 1
    • 2
    Email author
  • Fawang Liu
    • 3
  • Junjiang Lai
    • 4
  • Vo Anh
    • 5
  • Shanzhen Chen
    • 6
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenPeople’s Republic of China
  2. 2.Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputationXiamen UniversityXiamenPeople’s Republic of China
  3. 3.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  4. 4.College of Mathematics and Data ScienceMinjiang UniversityFuzhouPeople’s Republic of China
  5. 5.Faculty of Science, Engineering and TechnologySwinburne University of TechnologyHawthornAustralia
  6. 6.School of Economic Mathematics SwufeSouthwestern University of Finance and EconomicsChengduPeople’s Republic of China

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