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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Christopher, P.: B-splines collocation for plate bending eigenanalysis. J. Mech. Mater. Struct. 12(4), 353–371 (2017)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  5. Jalil, R., Sanaz, J.: Collocation method based on modified cubic B-spline for option pricing models. Math. Commun. 22(1), 89–102 (2017)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Annalisa, B., Garau, E.M.: Refinable spaces and local approximation estimates for hierarchical splines. IMA J. Numer. Anal. 37(3), 1125–1149 (2017)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Andrea, B., Annalisa, B., Giancarlo, S.: Characterization of analysis-suitable T-splines. Comput. Aided Geom. Design 39, 17–49 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  19. Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math. 52, 39–62 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, X., Wu, B.: New algorithm for non-classical parabolic problems based on the reproducing kernel method. Math. Sci. 7, 4–8 (2013)

    Article  MATH  Google Scholar 

  23. Dehghan, M.: On the numerical solution of the diffusion equation with a nonlocal boundary condition. Math. Probl. Eng. 2, 81–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, R.H., Li, C.J.: Bivariate quartic spline spaces and quasi-interpolation operators. J. Comput. Appl. Math. 190, 325–338 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. College of Science, Dalian Maritime University, Dalian, China

    Kai Qu, Jiawei Xuan & Mengdi Zhang

  2. School of Marine Electrical Engineering, Dalian Maritime University, Dalian, China

    Ning Wang

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  1. Kai Qu
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  2. Jiawei Xuan
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  3. Ning Wang
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  4. Mengdi Zhang
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Correspondence to Kai Qu.

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Qu, K., Xuan, J., Wang, N. et al. The simulation by using bivariate splines for solving two dimensional non-classical diffusion problem. Cluster Comput 22 (Suppl 4), 8131–8139 (2019). https://doi.org/10.1007/s10586-017-1636-3

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  • DOI: https://doi.org/10.1007/s10586-017-1636-3

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