Regularization of Highly Ill-Conditioned RBF Asymmetric Collocation Systems in Fractional Models

Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)


While attempting to approximate differential equations using Kansa’s radial basis function (RBF) collocation, we need to solve a non-symmetric, highly ill-conditioned system. There are many attempts to evaluate RBF interpolant in a more stable manner using approaches like Laurent series expansion, regularization, QR algorithms, etc. In this article, we modify the regularization method and obtain regularization parameter that reduces the ill-conditioning and provide stable solutions for fractional differential equations using Kansa’s asymmetric collocation. Numerical results are provided to illustrate the algorithm.


  1. 1.
    M. Arghand and M. Amirfakhrian: A meshless method based on the fundamental solution and radial basis function for solving an inverse heat conduction problem. Adv. Math. Phys., pages Art. ID 256726, 8, (2015).Google Scholar
  2. 2.
    A. Ben-Israel and T. N. Greville: Generalized Inverses: Theory and Applications, volume 15. Springer Science & Business Media, (2003).Google Scholar
  3. 3.
    J. P. Boyd and K. W. Gildersleeve: Numerical experiments on the condition number of the interpolation matrices for radial basis functions. Appl. Numer. Math., 61(4), 443–459, (2011).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. D. Buhmann: Radial basis functions: theory and implementations. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge, New York, (2003).CrossRefGoogle Scholar
  5. 5.
    S. K. Damarla and M. Kundu: Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices. Appl. Math. Comput., 263, 189–203, (2015).MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Di Paola, F. P. Pinnola, and M. Zingales: Fractional differential equations and related exact mechanical models. Comput. Math. Appl., 66(5), 608–620, (2013).MathSciNetCrossRefGoogle Scholar
  7. 7.
    B. Fakhr Kazemi and F. Ghoreishi: Error estimate in fractional differential equations using multiquadratic radial basis functions. J. Comput. Appl. Math., 245, 133–147, (2013).MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. E. Fasshauer and M. J. McCourt: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput., 34(2), A737–A762, (2012).MathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Fornberg, E. Larsson, and N. Flyer: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput., 33(2), 869–892, (2011).MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Fornberg and C. Piret: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput., 30(1), 60–80, (2007/08).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Fornberg and G. Wright: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl., 48(5–6), 853–867, (2004).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Golbabai, E. Mohebianfar, and H. Rabiei: On the new variable shape parameter strategies for radial basis functions. Comput. Appl. Math., 34(2), 691–704, (2015).MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. F. Gómez-Aguilar, H. Yépez-Martí nez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza, and J. Reyes-Reyes: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model., 40(21–22), 9079–9094, (2016).MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Gonzalez-Rodriguez, V. Bayona, M. Moscoso, and M. Kindelan: Laurent series based RBF-FD method to avoid ill-conditioning. Eng. Anal. Bound. Elem., 52, 24–31, (2015).MathSciNetCrossRefGoogle Scholar
  15. 15.
    P. C. Hansen: Regularization Tools version 4.0 for Matlab 7.3. Numer. Algorithms, 46(2), 189–194, (2007).MathSciNetCrossRefGoogle Scholar
  16. 16.
    J.-H. He: Application of homotopy perturbation method to nonlinear wave equations. Chaos. Soliton. Fract., 26(3), 695–700, (2005).MathSciNetCrossRefGoogle Scholar
  17. 17.
    X. Hei, W. Chen, G. Pang, R. Xiao, and C. Zhang: A new visco–elasto-plastic model via time–space fractional derivative. Mech. Time-Depend. Mater., pages 1–13, (2017).Google Scholar
  18. 18.
    E. J. Kansa: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl., 19(8), 147–161, (1990).MathSciNetCrossRefGoogle Scholar
  19. 19.
    E. Larsson, E. Lehto, A. Heryudono, and B. Fornberg: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J. Sci. Comput., 35(4), A2096–A2119, (2013).MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Lin, W. Chen, and K. Sze: A new radial basis function for Helmholtz problems. Eng. Anal. Bound. Elem., 36(12), 1923–1930, (2012).MathSciNetCrossRefGoogle Scholar
  21. 21.
    J. Lin, W. Chen, and F. Wang: A new investigation into regularization techniques for the method of fundamental solutions. Math. Comput. Simulation, 81(6), 1144–1152, (2011).MathSciNetCrossRefGoogle Scholar
  22. 22.
    S. Momani and Z. Odibat: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math., 207(1), 96–110, (2007).MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. D. Paola and M. Zingales: Exact mechanical models of fractional hereditary materials. J. Rheol., 56(5), 983–1004, (2012).CrossRefGoogle Scholar
  24. 24.
    I. Podlubny: Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198. Academic Press, (1998).Google Scholar
  25. 25.
    F. A. Rihan: Numerical modeling of fractional-order biological systems. Abstr. Appl. Anal., pages Art. ID 816803, 11, (2013).Google Scholar
  26. 26.
    S. A. Sarra and D. Sturgill: A random variable shape parameter strategy for radial basis function approximation methods. Eng. Anal. Bound. Elem., 33(11), 1239–1245, (2009).MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. Schaback: Comparison of radial basis function interpolants. In Multivariate approximation: from CAGD to wavelets (Santiago, 1992), volume 3 of Ser. Approx. Decompos., pages 293–305. World Sci. Publ., River Edge, NJ, (1993).Google Scholar
  28. 28.
    N. H. Sweilam, A. M. Nagy, and A. A. El-Sayed: Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation. Chaos Solitons Fractals, 73, 141–147, (2015).MathSciNetCrossRefGoogle Scholar
  29. 29.
    V. Turut and N. Güzel: On solving partial differential equations of fractional order by using the variational iteration method and multivariate padé approximations. Eur. J. Pure Appl. Math., 6(2), 147–171, (2013).MathSciNetzbMATHGoogle Scholar
  30. 30.
    Y.-F. Zhang and C.-J. Li: A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems. Inverse Probl. Sci. Eng., 24(9), 1606–1646, (2016).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National Institute of Technology KarnatakaSurathkalIndia

Personalised recommendations