Cluster Computing

, Volume 22, Supplement 4, pp 8131–8139 | Cite as

The simulation by using bivariate splines for solving two dimensional non-classical diffusion problem

  • Kai QuEmail author
  • Jiawei Xuan
  • Ning Wang
  • Mengdi Zhang


In this paper, a high order method using bivariate spline finite elements on domains defined by NURBS is proposed for solving two dimensional non-classical diffusion problem. Bivariate spline proper subspace of \(S_4^{2,3} (\Delta _{mn}^{(2)} )\) satisfying homogeneous boundary conditions on type-2 triangulations and quadratic B-spline interpolating boundary functions are primarily constructed. Two examples are solved to assess the accuracy of the method. The simulation obtained, indicates that spline method is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods.


Bivariate spline Finite element Non-classical diffusion problem NURBS 



The authors acknowledge the National Natural Science Foundation of China (Grants Nos. 11601056 and 51009017), the Fundamental Research Funds for the Central Universities (Grants Nos. 3132017055 and 3132016314), the National Scholarship Foundation of China.


  1. 1.
    Li, Z., Jieqing, T., Xianyu, G., Guo, Z.: Generalized B-splines’ geometric iterative fitting method with mutually different weights. J. Comput. Appl. Math. 329, 331–343 (2018)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Mingzeng, L., Baojun, L., Qingjie, G., Zhu Chungang, H., Ping, S.Y.: Progressive iterative approximation for regularized least square bivariate B-spline surface fitting. J. Comput. Appl. Math. 327, 175–187 (2018)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Christopher, P.: B-splines collocation for plate bending eigenanalysis. J. Mech. Mater. Struct. 12(4), 353–371 (2017)MathSciNetGoogle Scholar
  4. 4.
    Alaattin, E., Orkun, T.: Numerical solution of time fractional Schrödinger equation by using quadratic B-spline finite elements. Ann. Math. Sil. 31(1), 83–98 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Jalil, R., Sanaz, J.: Collocation method based on modified cubic B-spline for option pricing models. Math. Commun. 22(1), 89–102 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kai, Q., Wang, Z., Jiang, B.: A finite element method by using bivariate splines for one dimensional heat equations. J. Inf. Comput. Sci. 10(12), 3659–3666 (2013)Google Scholar
  7. 7.
    Ole, C.: Goh Say Song: From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa. Appl. Comput. Harmon. Anal. 36(2), 198–214 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Annalisa, B., Carlotta, G.: Adaptive isogeometric methods with hierarchical splines: optimality and convergence rates. Math. Models Methods Appl. Sci. 27(14), 2781–2802 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Annalisa, B., Garau, E.M.: Refinable spaces and local approximation estimates for hierarchical splines. IMA J. Numer. Anal. 37(3), 1125–1149 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Annalisa, B., Carlotta, G.: Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Math. Models Methods Appl. Sci. 26(1), 1–25 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Andrea, B., Annalisa, B., Giancarlo, S.: Characterization of analysis-suitable T-splines. Comput. Aided Geom. Design 39, 17–49 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Annalisa, B., Vázquez, R.H., Sangalli, G., Beirão da Veiga, L.: Approximation estimates for isogeometric spaces in multipatch geometries. Numer. Methods Partial Differ. Equ. 31(2), 422–438 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Deepesh, T., Hendrik, S., Hughes Thomas, J.R.: Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Eng. 327, 411–458 (2017)MathSciNetGoogle Scholar
  14. 14.
    Deepesh, T., Hendrik, S., Hiemstra René, R., Hughes Thomas, J.R.: Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 316, 1005–1061 (2017)MathSciNetGoogle Scholar
  15. 15.
    Kamensky, D., Hsu, M.-C., Yu, Y., Evans, J.A., Sacks, M.S., Hughes, T.J.R.: Immersogeometric cardiovascular fluid-structure interaction analysis with divergence-conforming B-splines. Comput. Methods Appl. Mech. Eng. 314, 408–472 (2017)MathSciNetGoogle Scholar
  16. 16.
    Kruse, R., Nguyen-Thanh, N., De Lorenzis, L., Hughes, T.J.R.: Isogeometric collocation for large deformation elasticity and frictional contact problems. Comput. Methods Appl. Mech. Eng. 2296, 73–112 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kanca, F.: The inverse problem of the heat equation with periodic boundary and integral over determination conditions. J. Inequal. Appl. 18, 1–9 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Martín, V.J., Queiruga, D.A., Encinas, A.H.: Numerical algorithms for diffusion-reaction problems with non-classical conditions. Appl. Math. Comput. 218(9), 5487–5495 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math. 52, 39–62 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Martin-Vaquero, J., Vigo-Aguiar, J.: A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. Appl. Numer. Math. 59(6), 1258–1264 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Khaliq, A.Q.M., Martín, V.J., Wade, B.A., Yousuf, M.: Smoothing schemes for reaction-diffusion systems with nonsmooth data. J. Comput. Appl. Math. 223(1), 374–386 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, X., Wu, B.: New algorithm for non-classical parabolic problems based on the reproducing kernel method. Math. Sci. 7, 4–8 (2013)zbMATHGoogle Scholar
  23. 23.
    Dehghan, M.: On the numerical solution of the diffusion equation with a nonlocal boundary condition. Math. Probl. Eng. 2, 81–92 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dehghan, M.: A computational study of the one-dimensional parabolic equation subject to non-classical boundary specifications. Numer. Methods Partial Differ. Equ. 22, 220–257 (2006)zbMATHGoogle Scholar
  25. 25.
    Tatari, M., Dehghan, M.: On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model. 33, 1729–1738 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Golbabai, A., Javidi, M.: A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method. Appl. Math. Comput. 190, 179–185 (2007)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Raunak, B., Charbel, F., Radek, T.: A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 327, 93–117 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Abbasbandy, S., Shirzadi, A.: MLPG method for two-dimensional diffusion equation with Neumann’s and non-classical boundary conditions. Appl. Numer. Math. 61(2), 170–180 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wang, R.H., Li, C.J.: Bivariate quartic spline spaces and quasi-interpolation operators. J. Comput. Appl. Math. 190, 325–338 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceDalian Maritime UniversityDalianChina
  2. 2.School of Marine Electrical EngineeringDalian Maritime UniversityDalianChina

Personalised recommendations