Skip to main content
SpringerLink
Log in

Error Analysis of the Reduced RBF Model Based on POD Method for Time-Fractional Partial Differential Equations

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper, we present a new reduced order model based on radial basis functions (RBFs) and proper orthogonal decomposition (POD) methods for fractional advection-diffusion equations with a Caputo fractional derivative in time. In the proposed scheme, the number of basis functions in the usual RBFs method reduces by the POD technique. Therefore, the computational cost of the RBF-POD method decreases in comparison with usual RBFs method, while the accuracy completely maintains. In the sequel, we provide a complete error analysis in the \(L_{2}\) norm between the exact solution and the RBFs solution, as well as between the exact solution and the proposed RBF-POD model by using the properties of the native space and projection operators. Also, the obtained error estimation is used to choose the number of POD bases for constructing the RBF-POD model with the required accuracy. Numerical examples are given to confirm the accuracy and efficiency of the proposed scheme.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. An, J., Luo, Z., Lie, H., Sun, P.: Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation. Front. Math. China 10(5), 1025–1040 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33, 1–19 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Atwell, J.A., King, B.B.: Reduced order controllers for spatially distributed systems via proper orthogonal decomposition. SIAM J. Sci. Comput. 26, 128–151 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Azeez, M.F.A., Vakalis, A.F.: Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations. J. Sound Vib. 240(5), 859–889 (2001)

    Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains Scientific Computation. Springer, Berlin (2007)

    MATH  Google Scholar 

  6. Dacorogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004)

    MATH  Google Scholar 

  7. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions. Comput. Math. Appl. 68, 212–237 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2004)

    Google Scholar 

  10. Fakhr Kazemi, B., Ghoreishi, F.: Error estimate in fractional differential equations using multiquadratic radial basis functions. J. Comput. Appl. Math. 245, 133–147 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific, Hackensack (2007)

    MATH  Google Scholar 

  12. Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38, 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  13. Franke, C., Schback, R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8(4), 381–399 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Fukunaga, K.: Introduction to Statistical Recognition. Academic Press, New York (1990)

    MATH  Google Scholar 

  15. Gao, G.H., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Gao, G., Sun, Z., Zhang, Y.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231, 2865–2879 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Ghaffari, R., Ghoreishi, F.: Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations. Appl. Numer. Math. 137, 62–79 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Ghaffari, R., Ghoreishi, F.: Reduced collocation method for time-dependent parametrized partial differential equations. Bull. Iran. Math. Soc. (2019). https://doi.org/10.1007/s41980-019-00210-w

    Article  MathSciNet  MATH  Google Scholar 

  19. Ghaffari, R., Hosseini, S.M.: Obtaining artificial boundary conditions for fractional sub-diffusion equation on space two-dimensional unbounded domains. Comput. Math. Appl. 68, 13–26 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in random media. Physica A 185(1–4), 87–97 (1992)

    Google Scholar 

  21. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion. Chem. Phys. 284(1–2), 521–541 (2002)

    MATH  Google Scholar 

  22. Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915 (1971)

    Google Scholar 

  23. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  24. Holmes, P., Lumley, J., Berkooz, G., Rowley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  25. Hosseini, S.M., Ghaffari, R.: Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations. Appl. Math. Model. 38, 3554–3566 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Jankeviciute, G., Leonaviciene, T., Ciegis, R., Bugajev, A.: Reduced order models based on pod method for Schrodinger equations. Math. Model. Anal. 18(5), 694–707 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Jin, B., Zhou, Z.: An analysis of the Galerkin proper orthogonal decomposition for subdiffusion. ESAIM: Math. Model. Numer. Anal. 51(1), 89–113 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Jin, B.T., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Kerschen, G., Golinval, J.C., Varkakis, A.F.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41, 147–169 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Li, X., Wong, P.J.Y.: A higher order non-polynomial spline method for fractional sub-diffusion problems. J. Comput. Phys. 328, 46–65 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Liu, J., Li, H., Fang, Zh., Liu, Y.: Application of low-dimensional finite element method to fractional diffusion equation. Int. J. Model. Simul. Sci. Comput. 5(4), 1450022 (2014)

    Google Scholar 

  37. Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38, 3871–3878 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Luo, Z.D., Chen, J., Sun, P., Yang, X.Z.: Finite element formulation based on proper orthogonal decomposition for parabolic equations. Sci. China Ser. A, Math. 52, 585–596 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Luo, Z., Zhou, Y., Yang, X.: A reduced finite element formulation based on proper orthogonal decomposition for Burgers equation. Appl. Numer. Math. 59, 1933–1946 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Luo, Zh., Ou, Q., Xie, Zh.: Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation. Appl. Math. Mech. 32(7), 847–858 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Luo, Zh., Xie, Zh., Shang, Y., Chen, J.: A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems. J. Comput. Appl. Math. 235, 2098–2111 (2011)

    MathSciNet  MATH  Google Scholar 

  42. Luo, Z., Li, H., Sun, P.: A reduced-order Crank-Nicolson finite volume element formulation based on POD method for parabolic equations. Appl. Math. Comput. 219, 5887–5900 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Luo, Zh., Li, H., Sun, P., Anc, J., Navon, I.M.: A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems. Math. Comput. Simul. 89, 50–68 (2013)

    MathSciNet  Google Scholar 

  44. Luo, Zh., Gao, J., Xie, Zh.: Reduced-order finite difference extrapolation model based on proper orthogonal decomposition for two-dimensional shallow water equations including sediment concentration. J. Math. Anal. Appl. 429, 901–923 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Luo, Zh., Jin, Sh., Chen, J.: A reduced order extrapolation central difference scheme based on POD for two-dimensional fourth-order hyperbolic equations. Appl. Math. Comput. 289, 396–408 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Madych, W.R., Nelson, S.A.: Multivariate interpolation and conditionally positive definite functions. Approx. Theory Appl. 4, 77–89 (1988)

    MathSciNet  MATH  Google Scholar 

  47. Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24, 175–186 (2006)

    MathSciNet  MATH  Google Scholar 

  48. Piret, C., Hanert, E.: A radial basis functions method for fractional diffusion equations. J. Comput. Phys. 238, 71–81 (2013)

    MathSciNet  MATH  Google Scholar 

  49. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  50. San, O., Iliescu, T.: Proper orthogonal decomposition closure models for fluid flows: Burgers equation. Int. J. Numer. Anal. Model. 1(1), 1–18 (2013)

    MATH  Google Scholar 

  51. Schaback, R.: Native Hilbert spaces for radial basis functions. I. In: New Developments in Approximation Theory, Dortmund, 1998. Inter. Ser. Numer. Math., vol. 132, pp. 255–282 (1999)

    Google Scholar 

  52. Schaback, R.: A unified theory of radial basis functions native Hilbert spaces for radial basis functions. II. Numerical analysis in the 20th century, vol. I, approximation theory. J. Comput. Appl. Math. 121, 165–177 (2000)

    MathSciNet  MATH  Google Scholar 

  53. Sun, P., Luo, Zh., Zhou, Y.: Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Appl. Numer. Math. 60, 154–164 (2010)

    MathSciNet  MATH  Google Scholar 

  54. Uddin, M., Haq, S.: RBFs approximation method for time fractional partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 4208–4214 (2011)

    MathSciNet  MATH  Google Scholar 

  55. Wendland, H.: Sobolev-type error estimates for interpolation by radial basis functions. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 337–344. Vanderbilt University Press, Nashville (1997)

    Google Scholar 

  56. Wendland, H.: Scattered Data Approximation. Cambridge Monograph on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  57. Wu, Z., Schback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)

    MathSciNet  Google Scholar 

  58. Yoon, J.: \(L_{p}\)-Error estimates for “shifted” surface spline interpolation on Sobolev space. Math. Comput. 72(243), 1349–1367 (2003)

    MATH  Google Scholar 

  59. Yoon, J.: Improved accuracy of \(L_{p}\)-approximation to derivatives by radial basis function interpolation. Appl. Math. Comput. 161, 109–119 (2005)

    MathSciNet  MATH  Google Scholar 

  60. Zahra, W.K., Elkholy, M.S.: Quadratic spline solution boundary value problem of fractional order. Numer. Algorithms 59, 373–391 (2012)

    MathSciNet  MATH  Google Scholar 

  61. Zhuang, P., Liu, F.: Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 1(1), 1–15 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    Rezvan Ghaffari & Farideh Ghoreishi

Authors
  1. Rezvan Ghaffari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Farideh Ghoreishi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farideh Ghoreishi.

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

Ghaffari, R., Ghoreishi, F. Error Analysis of the Reduced RBF Model Based on POD Method for Time-Fractional Partial Differential Equations. Acta Appl Math 168, 33–55 (2020). https://doi.org/10.1007/s10440-019-00278-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-019-00278-w

Keywords

Mathematics Subject Classification

Search

Navigation