Skip to main content

A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems

Abstract

This study deals with the numerical solution of a general class of nonlinear singular boundary value problems (SBVPs). Firstly, we modify the original model problem at the singular point and then we construct a numerical technique based on quartic trigonometric B-spline functions to solve the resulting problem. Numerical experiments are performed to demonstrate the applicability and efficiency of the method. More specifically, we consider three real-life problems: (1) thermal explosion in a cylindrical vessel; (2) isothermal gas sphere; (3) oxygen diffusion in a spherical cell. The computed results are compared with the results obtained by the compact finite difference method (CFDM) (Roul et al. in Appl Math Comput 350:283–304, 2019) and the B-spline collocation method (Thula and Roul in Mediterr J Math 15(4):176, 2018) in order to justify the advantage of present method. The proposed method is a promising one to handle the general class of SBVPs.

This is a preview of subscription content, access via your institution.

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

References

  1. 1.

    R.K. Pandey, On a class of regular singular two point boundary value problems. J. Math. Anal. Appl. 208, 388–403 (1997)

    Article  Google Scholar 

  2. 2.

    R.K. Pandey, On a class of weakly regular singular two-point boundary value problems, II. J. Differ. Equ. 127, 110–123 (1996)

    Article  Google Scholar 

  3. 3.

    H.S. Fogler, Elements of Chemical Reaction Engineering, 2nd edn. (Prentice-Hall Inc, Englewood Cliffs, 1992)

    Google Scholar 

  4. 4.

    D.L.S. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. J. Theor. Biol. 71, 255–263 (1978)

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  5. 5.

    H.S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J. Theor. Biol. 60, 449–457 (1976)

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  6. 6.

    P.L. Chambre, On the solution of the Poisson–Boltzmann equation with the application to the theory of thermal explosions. J. Chem. Phys. 20, 1795–1797 (1952)

    Article  CAS  Google Scholar 

  7. 7.

    U. Flesch, The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 54, 285–287 (1975)

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  8. 8.

    B.F. Gray, The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 82, 473–476 (1980)

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  9. 9.

    B. Manimegalai, M.E.G. Lyons, L. Rajendran, A kinetic model for amperometric immobilized enzymes at planar, cylindrical and spherical electrodes: the Akbari–Ganji method. J. Electroanal. Chem. 880(2), 114921 (2021)

    Article  CAS  Google Scholar 

  10. 10.

    P. Roul, H. Madduri, R. Agarwal, A fast-converging recursive approach for Lane-Emden type initial value problems arising in astrophysics. J. Comput. Appl. Math. 359, 182–195 (2019)

    Article  Google Scholar 

  11. 11.

    S. Mckee, R. Watson, J.A. Cuminato, J. Caldwell, M.S. Chen, Calculation of electro-hydrodynamic flow in a circular cylindrical conduit. Z. Angew. Math. Mech. 77, 457–465 (1997)

    Article  Google Scholar 

  12. 12.

    H. Madduri, P. Roul, A fast-converging iterative scheme for solving a system of Lane–Emden equations arising in catalytic diffusion reactions. J. Math. Chem. 57(2), 570–582 (2019)

    Article  CAS  Google Scholar 

  13. 13.

    A. Mastroberardino, Homotopy analysis method applied to electro-hydrodynamic flow. Commun. Nonlinear. Sci. Numer. Simul. 16, 2730–2736 (2011)

    Article  Google Scholar 

  14. 14.

    P. Roul, H. Madduri, A new approximate method and its convergence for a strongly nonlinear problem governing electrohydrodynamic flow of a fluid in a circular cylindrical conduit. Appl. Math. Comput. 341, 335–347 (2019)

    Google Scholar 

  15. 15.

    P. Roul, A fourth-order non-uniform mesh optimal B-spline collocation method for solving a strongly nonlinear singular boundary value problem describing electrohydrodynamic flow of a fluid. Appl. Numer. Math. 153, 558–574 (2020)

    Article  Google Scholar 

  16. 16.

    J.V. Baxley, S.B. Robinson, Nonlinear boundary value problems for shallow membrane caps, II. J. Comput. Appl. Math. 88(1), 203–224 (1998)

    Article  Google Scholar 

  17. 17.

    I. Rachunkov, G. Pulverer, E. Weinmller, A unified approach to singular problems arising in the membrane theory. Appl. Math. 55(1), 47–75 (2010)

    Article  Google Scholar 

  18. 18.

    J.A. Adam, A simplified mathematical model of tumor growth. Math. Biosci. 81, 224–229 (1986)

    Article  Google Scholar 

  19. 19.

    J.A. Adam, A mathematical model of tumor growth II: effect of geometry and spatial nonuniformity on stability. Math. Biosci. 86, 183–211 (1987)

    Article  Google Scholar 

  20. 20.

    J.A. Adam, S.A. Maggelakis, Mathematical model of tumor growth IV: effect of necrotic core. Math. Biosci. 97, 121–136 (1989)

    PubMed  Article  CAS  PubMed Central  Google Scholar 

  21. 21.

    A.C. Burton, Rate of growth of solid tumor as a problem of diffusion. Growth 30, 157–176 (1966)

    PubMed  CAS  PubMed Central  Google Scholar 

  22. 22.

    A.S.V.R. Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189, 2017–2022 (2007)

    Google Scholar 

  23. 23.

    R.K. Pandey, A.K. Singh, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology. J. Comput. Appl. Math. 166, 553–564 (2004)

    Article  Google Scholar 

  24. 24.

    M.M. Chawla, S. Mckee, G. Shaw, Order \(h^2\) method for singular two-point boundary value problem. BIT 26, 318–326 (1986)

    Article  Google Scholar 

  25. 25.

    M. Abukhaled, S.A. Khuri, A. Sayfy, A numerical approach for solving a class of singular boundary value problems arising in physiology. Int. J. Numer. Anal. Model. 8(2), 353–363 (2010)

    Google Scholar 

  26. 26.

    J. Goh, A.A. Majid, A.IMd. Ismail, A quartic B-spline for second-order singular boundary value problems. Comput. Math. Appl. 64, 115–120 (2012)

    Article  Google Scholar 

  27. 27.

    A.S.V.R. Kanth, Y.N. Reddy, Higher order finite difference method for a class of singular boundary value problems. Appl. Math. Comput. 155, 249–258 (2004)

    Google Scholar 

  28. 28.

    M.K. Kadalbajoo, V. Kumar, B-spline method for a class of singular two-point boundary value problems using optimal grid. Appl. Math. Comput. 188, 1856–1869 (2007)

    Google Scholar 

  29. 29.

    A.S.V. Ravikanth, V. Bhattacharya, Cubic spline for a class of nonlinear singular boundary-value problems arising in physiology. Appl. Math. Comput. 174(1), 768–774 (2006)

    Google Scholar 

  30. 30.

    S.A. Khuri, A. Sayfy, Numerical solutions for the nonlinear Emden–Fowler type equations by a fourth-order adaptive method. Int. J. Comput. Methods 11(1), 1350052–1350072 (2014)

    Article  Google Scholar 

  31. 31.

    H. Caglar, N. Caglar, M. Ozer, B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fractals 39, 1232–1237 (2009)

    Article  Google Scholar 

  32. 32.

    P. Roul, V.M.K.P. Goura, R. Agarwal, A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions. Appl. Math. Comput. 350, 283–304 (2019)

    Google Scholar 

  33. 33.

    K. Thula, P. Roul, A high-order B-spline collocation method for solving nonlinear singular boundary value problems arising in engineering and applied science. Mediterr. J. Math. 15(4), 176 (2018)

    Article  Google Scholar 

  34. 34.

    M. Chawla, R. Subramanian, H.L. Sathi, A fourth order method for a singular two point boundary value problem. BIT Numer. Math. 28, 88–97 (1988)

    Article  Google Scholar 

  35. 35.

    P. Roul, U. Warbhe, New approach for solving a class of singular boundary value problem arising in various physical models. J. Math. Chem. 54, 1255–1285 (2016)

    Article  CAS  Google Scholar 

  36. 36.

    P. Roul, U. Warbhe, A new homotopy perturbation scheme for solving singular boundary value problems arising in various physical models. Z. Naturforschung A 72, 733–743 (2017)

    Article  CAS  Google Scholar 

  37. 37.

    M. Inc, D.J. Evans, The decomposition method for solving of a class of singular two-point boundary value problems. Int. J. Comput. Math. 80, 869–882 (2003)

    Article  Google Scholar 

  38. 38.

    M. Kumar, N. Singh, Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng. 34, 1750–1760 (2010)

    Article  CAS  Google Scholar 

  39. 39.

    P. Roul, On the numerical solution of singular boundary value problem: a domain decomposition homotopy perturbation approach. Math. Methods Appl. Sci. 40, 7396–7409 (2017)

    Article  Google Scholar 

  40. 40.

    P. Roul, A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems. Int. J. Comput. Math. 96, 51–72 (2019)

    Article  Google Scholar 

  41. 41.

    P. Roul, D. Biswal, A new numerical approach for solving a class of singular two point boundary value problems. Numer. Algorithms 75, 531–552 (2017)

    Article  Google Scholar 

  42. 42.

    P. Roul, A new mixed MADM-collocation approach for solving a class of Lane-Emden singular boundary value problems. J. Math. Chem. 57, 945–969 (2019)

    Article  CAS  Google Scholar 

  43. 43.

    N.S. Asaithambi, J.B. Garner, Pointwise solution bounds for a class of singular diffusion problems in physiology. Appl. Math. Comput. 30, 215–222 (1989)

    Google Scholar 

  44. 44.

    P. Roul, K. Thula, V.M.K.P. Goura, An optimal sixth-order quartic B-spline collocation method for solving Bratu-type and Lane–Emden-type problems. Math. Methods Appl. Sci. 42(8), 2613–2630 (2019)

    Article  Google Scholar 

  45. 45.

    P. Roul, K. Thula, R. Agarwal, Non-optimal fourth-order and optimal sixth-order B-spline collocation methods for Lane–Emden boundary value problems. Appl. Numer. Math. 145, 342–360 (2019)

    Article  Google Scholar 

  46. 46.

    U. Yucel, M. Sari, Differential quadrature method (DQM) for a class of singular two-point boundary value problem. Int. J. Comput. Math. 86(3), 465–475 (2009)

    Article  Google Scholar 

  47. 47.

    P. Koch, T. Lyche, M. Neamtu, L. Schumaker, Control curves and knot insertion for trigonometric splines. Adv. Comput. Math. 3(4), 405–424 (1995)

    Article  Google Scholar 

  48. 48.

    G. Walz, Identities for trigonometric B-splines with an application to curve design. BIT Numer. Math. 37(1), 189–201 (1997)

    Article  Google Scholar 

  49. 49.

    R.U. Rani, L. Rajendran, Taylor’s series method for solving the nonlinear reaction-diffusion equation in the electroactive polymer film. Chem. Phys. Lett. 754, 137573 (2020)

Download references

Acknowledgements

The authors are very grateful for financial support from CSIR, India in the form of Project No. \(25(0286)/18/EMR-II\).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pradip Roul.

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

Roul, P., Kumari, T. A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems. J Math Chem (2021). https://doi.org/10.1007/s10910-021-01293-9

Download citation

Keywords

  • Singular boundary value problems
  • Trigonometric B-spline
  • Fourth order accuracy
  • Thermal explosion in cylindrical vessel
  • Oxygen diffusion problem