Skip to main content
SpringerLink
Log in

A class of moving Kriging interpolation-based DQ methods to simulate multi-dimensional space Galilei invariant fractional advection-diffusion equation

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The current work concerns to develop a new local meshless procedure to simulate the one-, two- and three-dimensional space Galilei invariant fractional advection-diffusion (GI-FAD) equations. The fractional derivative is discretized by a second-order finite difference formula. The unconditional stability and rate of convergence of the time-discrete scheme are analytically studied. Then, we employ the shape functions of moving Kriging interpolation for the differential quadrature (DQ) method. According to this combination, we can derive a new local meshless technique. At the end, some examples are solved to confirm the theoretical results and efficiency of the proposed meshless method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Abbaszadeh, M., Dehghan, M., Khodadadian, A., Heitzinger, C.: Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher–Kolmogorov equation. Comput. Math. Appl. 80, 247–262 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abbaszadeh, M.: Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl. Math. Lett. 88, 179–185 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms 75(1), 173–211 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abbaszadeh, M., Dehghan, M.: Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method. Appl. Numer. Math. 145, 488–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abdelkawy, M., Zaky, M., Bhrawy, A., Baleanu, D.: Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model. Rom. Rep. Phys 67(3), 773–791 (2015)

    Google Scholar 

  6. Bellman, R., Kashef, B., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10(1), 40–52 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhrawy, A.H., Baleanu, D.: A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection-diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219–233 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bhrawy, A., Zaky, M.: An improved collocation method for multi-dimensional space–time variable-order fractional schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.: Numerical solution of the two-sided space–time fractional telegraph equation via chebyshev tau approximation. J. Optim. Theory Appl. 174(1), 321–341 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York Dordrecht (2011)

    Book  MATH  Google Scholar 

  11. Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293, 264–279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buhmann, M.D., De Marchi, S., Perracchione, E.: Analysis of a new class of rational RBF expansions. IMA J. Numer. Anal. 40(3), 1972–1993 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bui, T.Q., Zhang, C.: moving Kriging interpolation-based meshfree method for dynamic analysis of structures. Proc. Appl. Math. Mech. 11, 197–198 (2011)

    Article  Google Scholar 

  14. Bui, T.Q., Nguyen, M.N., Zhang, C.: A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis. Comput. Methods Appl. Mech. Engin. 200, 1354–1366 (2011)

    Article  MATH  Google Scholar 

  15. Bui, T.Q., Nguyen, T.N., Nguyen-Dang, H.: A moving Kriging interpolation-based meshless method for numerical simulation of Kirchhoff plate problems. Int. J. Numer. Methods Eng. 77(10), 1371–1395 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bui, T.Q., Nguyen, M.N., Zhang, C. h.: Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng. Anal. Bound. Elem. 35, 1038–1053 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bui, T.Q., Nguyen, M.N., Zhang, C.: An efficient meshfree method for vibration analysis of laminated composite plates, Comput. Comput. Mech. 48, 175–193 (2011)

    Article  MATH  Google Scholar 

  18. Bui, T.Q., Nguyen, M.N., Zhang, C., Pham, D.A.: An efficient meshfree method for analysis of two-dimensional piezoelectric structures. Smart Mater. Struct. 20(6), 065016 (2011)

    Article  Google Scholar 

  19. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  20. Chen, C.S., Karageorghis, A., Local, RBF: Algorithms for elliptic boundary value problems in annular domains. Commun. Comput. Phys. 25(1), 41 (2019)

    Article  MathSciNet  Google Scholar 

  21. Chen, L., Liew, K.M.: A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems. Comput. Mech. 47, 455–467 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, C. -M., Liu, F., Anh, V., Turner, I.: Numerical simulation for the variable-order galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput. 217(12), 5729–5742 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Chen, S., Liu, F., Jiang, X., Turner, I., Burrage, K.: Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients. SIAM J. Numer. Anal. 54, p606–624 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dehghan, M., Abbaszadeh, M., Deng, W.: Fourth-order numerical method for the space–time tempered fractional diffusion-wave equation. Appl. Math. Lett. 73, 120–127 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dehestani, H., Ordokhani, Y., Razzaghi, M.: A novel direct method based on the Lucas multiwavelet functions for variable–order fractional reaction-diffusion and subdiffusion equations. Numer. Linear Algebra Appl., e2346. https://doi.org/10.1002/nla.2346 (2020)

  26. Deng, K., Chen, M., Sun, T.: A weighted numerical algorithm for two and three dimensional two-sided space fractional wave equations. Appl. Math. Comput. 257, 264–273 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Deville, M.O., Fischer, P.F., Fischer, P.F., Mund, E., et al.: High-order Methods for Incompressible Fluid Flow, vol. 9. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  28. Dai, B.D., Cheng, J., Zheng, B.J.: Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method. Appl. Math. Comput. 219, 10044–10052 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Dai, B.D., Cheng, J., Zheng, B.J.: A moving Kriging interpolation-based meshless local Petrov-Galerkin method for elastodynamic analysis. Int. J. Appl. Mech. 5(1), 1350011–1350021 (2013)

    Article  Google Scholar 

  30. Ding, H., Li, C.: High-order algorithms for Riesz derivative and their applications (iii). Fract. Calc. Appl. Anal. 19(1), 19–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (ii). J. Comput. Phys. 293, 218–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Doha, E.H., Abd–Elhameed, W.M., Elkot, N.A., Youssri, Y.H.: Integral spectral Tchebyshev approach for solving space Riemann-Liouville and Riesz fractional advection–dispersion problems. Adv. Differ. Equ. 2017(1), 284 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gu, Y.T., Wang, Q.X., Lam, K.Y.: A meshless local Kriging method for large deformation analyses. Comput. Methods Appl. Mech. Engin. 196, 1673–1684 (2007)

    Article  MATH  Google Scholar 

  34. Kumar, A., Bhardwaj, A., Kumar, B.V.R.: A meshless local collocation method for time fractional diffusion wave equation. Comput. Math. Appl. 78 (6), 1851–1861 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kutanaei, S.S., Roshan, N., Vosoughi, A., Saghafi, S., Barari, A., Soleimani, S.: Numerical solution of Stokes flow in a circular cavity using mesh-free local RBF-DQ. Eng. Anal. Bound. Elem. 36(5), 633–638 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, X.G., Dai, B.D., Wang, L.H.: A moving Kriging interpolation-based boundary node method for two-dimensional potential problems. Chin. Phys. B 19(12), 120202–120207 (2010)

    Article  Google Scholar 

  37. Li, M., Huang, C., Ming, W.: Mixed finite-element method for multi-term time-fractional diffusion and diffusion-wave equations. Comput. Appl. Math. 37, 2309–2334 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete Continuous Dyn. Syst. B 24(4), 1989–2015 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, London (2015)

    Book  MATH  Google Scholar 

  40. Li, H., Cao, J., Li, C.: High-order approximation to caputo derivatives and caputo-type advection–diffusion equations (iii). J. Comput. Appl. Math. 299, 159–175 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Li, C.P., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383–406 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Giraldo, F.X.: Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations. Comput. Math. Appl. 45, 97–121 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Maerschalck, B.D.: Space-time least-squares spectral element method for unsteady flows application and valuation linear and non-linear hyperbolic scalar equations, Master Thesis, Department of Aerospace Engineering at Delft University of Technology (2003)

  46. Oruc, Ö. : A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics. Numer. Methods. Partial Differ. Equ. 36(6), 1699–1717 (2020)

    Article  MathSciNet  Google Scholar 

  47. Ray, S.S., Sahoo, S.: Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives. Math. Methods Appl. Sci. 38 (13), 2840–2849 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Racz, D., Bui, T.Q.: Novel adaptive meshfree integration techniques in meshless methods. Int. J. Numer. Methods Eng. 90(11), 1414–1434 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sarra, S.A.: A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains. Appl. Math. Comput. 218(19), 9853–9865 (2012)

    MathSciNet  MATH  Google Scholar 

  50. Shen, J.: Efficient spectral-Galerkin method I. Direct solvers of second-and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shu, C.: Differential Quadrature and its Application in Engineering. Springer Science & Business Media, Berlin (2012)

  52. Shu, C., Ding, H., Chen, H., Wang, T.: An upwind local RBF-DQ method for simulation of inviscid compressible flows. Comput. Methods Appl. Mech. Eng. 194(18), 2001–2017 (2005)

    Article  MATH  Google Scholar 

  53. Shu, C., Ding, H., Yeo, K.: Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 192(7), 941–954 (2003)

    Article  MATH  Google Scholar 

  54. Shu, C., Ding, H., Yeo, K.: Solution of partial differential equations by a global radial basis function-based differential quadrature method. Eng. Anal. Bound. Elem. 28(10), 1217–1226 (2004)

    Article  MATH  Google Scholar 

  55. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. Tornabene, F., Fantuzzi, N., Bacciocchi, M., Neves, A.M., Ferreira, A.J.: MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells. Compos. B Eng. 99, 30–47 (2016)

    Article  Google Scholar 

  57. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, New York (1997)

    MATH  Google Scholar 

  58. Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wang, T., Guo, B., Zhang, L.: New conservative difference schemes for a coupled nonlinear Schrodinger system. Appl. Math. Comput. 217, 1604–1619 (2010)

    MathSciNet  MATH  Google Scholar 

  60. Wu, X., Deng, W., Barkai, E.: Tempered fractional feynman-kac equation, arXiv:1602.00071

  61. Zaky, M., Baleanu, D., Alzaidy, J., Hashemizadeh, E.: Operational matrix approach for solving the variable-order nonlinear galilei invariant advection–diffusion equation. Adv. Differ. Equ. 2018(1), 102 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zaky, M.A., Ameen, I.G.: A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and Volterra-Fredholm integral equations with smooth solutions. Numer. Algorithms. In press (2019)

  63. Zayernouri, M., Karniadakis, G.E.: Fractional sturm–liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhang, G., Huang, C., Li, M.: A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrodinger equations. Eur. Phys. J. Plus 133, 155 (2018). https://doi.org/10.1140/epjp/i2018-11982-3

    Article  Google Scholar 

  65. Zheng, B., Dai, B.D.: A meshless local moving Kriging method for two-dimensional solids, Appl. Math. Comput. 218, 563–573 (2011)

    MathSciNet  MATH  Google Scholar 

  66. Zeng, F., Li, C., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37, A55–A78 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high-order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J. Sci. Comput. 37, A701–A724 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. Zhu, W., Kopriva, D.A.: A spectral element method to price european options, I. Single asset with and without jump diffusion. J. Sci. Comput. 39, 222–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Zhu, W., Kopriva, D.A.: A spectral element approximation to price European options with one asset and stochastic volatility. J. Sci. Comput. 42, 426–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  70. Zhu, P., Zhang, L.W., Liew, K.M.: Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov-Galerkin approach with moving Kriging interpolation. Compos. Struct. 107, 298–314 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

We would like to appreciate reviewers for their beneficial comments and suggestions that have improved our paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mostafa Abbaszadeh & Mehdi Dehghan

Authors
  1. Mostafa Abbaszadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehdi Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mostafa Abbaszadeh.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbaszadeh, M., Dehghan, M. A class of moving Kriging interpolation-based DQ methods to simulate multi-dimensional space Galilei invariant fractional advection-diffusion equation. Numer Algor 90, 271–299 (2022). https://doi.org/10.1007/s11075-021-01188-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01188-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation