A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs

Abstract

In this work, we present two approaches for simulation of fourth-order parabolic partial differential equations. In the first method, cubic B-spline quasi-interpolation is used to approximate the spatial derivative of the dependent variable and forward difference to approximate the time derivative. In the second method, we have used modified cubic B-spline functions-based differential quadrature method (DQM) for space discretization to get a system of ODEs and then this system is solved by SSP-RK43 method to get the results at knots. The numerical results demonstrate the accuracy of the proposed method. The stability analysis of the methods has also been discussed. It is observed that quasi-interpolation-based method is unconditionally stable, whereas for DQM, the stability has to be checked for a large number of space points. Moreover, for the small number of grid points, DQM gives better results, while for a large number of grid points, quasi-interpolation-based method is better.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Collatz L (1973) Hermitian methods for initial value problems in partial differential equations topics in numerical analysis. Academic Press, London, pp 41–61

    Google Scholar 

  2. 2.

    Conte SD (1957) A stable implicit finite difference approximation to a fourth order parabolic equation. J Assoc Comput Mech 4:18–23

    MathSciNet  Article  Google Scholar 

  3. 3.

    Crandall SH (1954) Numerical treatment of a fourth order partial differential equations. J Assoc Comput Mech 1:111–118

    MathSciNet  Article  Google Scholar 

  4. 4.

    Evans DJ (1965) A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation. Comput J 8:280–287

    MathSciNet  Article  Google Scholar 

  5. 5.

    Todd J (1956) A direct approach to the problem of stability in the numerical solution of partial differential equations. Commun Pure Appl Math 9:597–612

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jain MK, Iyengar SRK, Lone AG (1976) Higher order difference formulas for a fourth order parabolic partial differential equation. Int J Numer Methods Eng 10:1357–1367

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fairweather G, Gourlay AR (1967) Some stable difference approximations to a fourth order parabolic partial differential equation. Math Comput 21:1–11

    MathSciNet  Article  Google Scholar 

  8. 8.

    Lees M (1961) Alternate direction and semi explicit difference methods for solving parabolic partial differential equation. Numer Math 3:398–412

    MathSciNet  Article  Google Scholar 

  9. 9.

    Mohanty RK, McKee S, Kaur D (2017) A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation. Appl Math Comput 309:272–280

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Mittal RC, Jain RK (2011) B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations. Appl Math Comput 217:9741–9755

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Dehghan M, Manafian J (2009) The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Zeitschrift fr Naturforschung A 64:420–430

    ADS  Article  Google Scholar 

  12. 12.

    Sablonniere P (2005) Univariate spline quasi-interpolants and applications to numerical analysis. Rend Semin Mat Univ Politec Torino 63:211–222

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Sablonniere P (2007) A quadrature formula associated with a univariate spline quasi interpolant. BIT 47:825–837

    MathSciNet  Article  Google Scholar 

  14. 14.

    Zhu CG, Kang WS (2010) Applying cubic B-spline quasi-interpolation to solve hyperbolic conservation laws. UPB Sci Bull Ser 72:49–58

    Google Scholar 

  15. 15.

    Zhu CG, Kang WS (2010) Numerical solution of Burgers–Fisher equation by cubic B-spline quasi-interpolation. Appl Math Comput 216:2679–2686

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kumar R, Baskar S (2016) B-spline quasi-interpolation based numerical methods for some sobolev type equations. J Comput Appl Math 292:41–66

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Quan JR, Chang CT (1989) New insights in solving distributed system equations by quadrature methods-I. Comput Chem Eng 13:779–788

    Article  Google Scholar 

  19. 19.

    Mittal RC, Jiwari R (2009) Numerical study of Fisher’s equation by using differential quadrature method. Int J Inf Syst Sci 5:143–160

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mittal RC, Jiwari R (2012) A differential quadrature method for numerical solutions of Burgers’-type equations. Int J Numer Methods Heat Fluid Flow 22:880–895

    MathSciNet  Article  Google Scholar 

  21. 21.

    Falco MD, Gaeta M, Loia V, Rarita L (2016) Differential quadrature based numerical solutions of a fluid dynamic model for supply chains. Commun Math Sci 14:1467–1476

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ghasemi M (2018) On the numerical solution of high order multi-dimensional elliptic PDEs. Comput Math Appl 76:1228–1245

    MathSciNet  Article  Google Scholar 

  23. 23.

    Jiwari R, Tomasiello S, Tornabene F (2018) A numerical algorithm for computational modelling of coupled advection–diffusion–reaction systems. Eng Comput 35:1383–1401

    Article  Google Scholar 

  24. 24.

    Macias-Diaz JE, Tomasiello S (2016) A differential quadrature-based approach a la Picard for systems of partial differential equations associated to fuzzy differential equations. J Comput Appl Math 299:15–23

    MathSciNet  Article  Google Scholar 

  25. 25.

    Dehghan M, Abbaszadeh M (2016) Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion. Comput Methods Appl Mech Eng 300:770–797

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Dehghan M, Abbaszadeh M (2018) Solution of multi-dimensional Klein–Gordon–Zakharov and Schrodinger/Gross–Pitaevskii equations via local radial basis functions–differential quadrature (RBF-DQ) technique on non-rectangular computational domains. Eng Anal Bound Elem 92:156–170

    MathSciNet  Article  Google Scholar 

  27. 27.

    Dehghan M, Mohammadi V (2015) The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods. Eng Anal Bound Elem 51:74–100

    MathSciNet  Article  Google Scholar 

  28. 28.

    Dehghan M, Nilpour A (2013) Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl Math Model 37:8578–8599

    MathSciNet  Article  Google Scholar 

  29. 29.

    Lakestani M, Dehghan M (2009) Numerical solution of Fokker–Planck equation using the cubic B-spline scaling functions. Numer Methods Partial Differ Equ 25:418–429

    MathSciNet  Article  Google Scholar 

  30. 30.

    Schumaker L (2007) Spline functions: basic theory. Cambridge University Press, Cambridge

    Google Scholar 

  31. 31.

    Sablonniere P (2003) Quadratic spline quasi-interpolants on bounded domains of Rd, d = 1; 2; 3. Rend Semin Mat Univ Politec Torino 61:229–246

    MATH  Google Scholar 

  32. 32.

    Shu C (2000) Differential quadrature and its application in engineering. Springer, London

    Google Scholar 

Download references

Acknowledgements

SK thanks Council of Scientific and Industrial Research (CSIR), Government of India [File No: 09/143(0889)/2017-EMR-I], for the financial support given during this work.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sudhir Kumar.

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

Verify currency and authenticity via CrossMark

Cite this article

Mittal, R.C., Kumar, S. & Jiwari, R. A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (2020). https://doi.org/10.1007/s40010-020-00684-y

Download citation

Keywords

  • Cubic B-spline functions
  • Quasi-interpolation
  • Differential quadrature method
  • Stability