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An efficient numerical approach for solving a general class of nonlinear singular boundary value problems

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Abstract

This paper is concerned with the development of a collocation method based on the Bessel polynomials for numerical solution of a general class of nonlinear singular boundary value problems (SBVPs). Due to the existence of singularity at the point \(x=0,\) we first modify the problem at the singular point. The proposed method is then developed for solving the resulting regular boundary value problem. To demonstrate the effectiveness and accuracy of the method, we apply it on several numerical examples. The numerical results obtained confirm that the present method has an advantage in terms of numerical accuracy over the uniform mesh cubic B-spline collocation (UCS) method (Roul and Goura in Appl Math Comput 341:428–450, 2019), non-standard finite difference (NSFD) method (Verma and Kayenat in J Math Chem 56:1667–1706, 2018), three-point finite difference methods (FDMs) (Pandey and Singh in Int J Comput Math 80:1323–1331, 2003; Pandey and Singh in J Comput Appl Math 205:469–478, 2007) and the cubic B-spline collocation (CBSC) method (Caglar et al. in Chaos Solitons Fractals 39:1232–1237, 2009)

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References

  1. 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, 353–363 (2010)

    Google Scholar 

  2. 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 

  3. 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 

  4. 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 

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

    Google Scholar 

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

    Article  CAS  Google Scholar 

  7. 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, 570–582 (2019)

    Article  CAS  Google Scholar 

  8. S.R.K. Iyengar, P. Jain, Spline finite difference methods for singular two point boundary value problems. Numer. Math. 50, 363–376 (1987)

    Article  Google Scholar 

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

    Article  Google Scholar 

  10. J.H. He, F.Y. Ji, Taylor series solution for Lane-Emden equation. J. Math. Chem. 57(8), 1932–1934 (2019)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  12. R.K. Pandey, A finite difference methods for a class of singular two point boundary value problems arising in physiology. Int. J. Comput. Math. 65, 131–140 (1997)

    Article  Google Scholar 

  13. R.K. Pandey, On the convergence of a spline method for singular two point boundary value problems arising in physiology. Int. J. Comput. Math. 79, 357–366 (2002)

    Article  Google Scholar 

  14. 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 

  15. 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 

  16. R.K. Pandey, A.K. Singh, On the convergence of finite difference methods for general singular boundary value problems. Int. J. Comput. Math. 80, 1323–1331 (2003)

    Article  Google Scholar 

  17. R.K. Pandey, A.K. Singh, On the convergence of finite difference methods for weakly regular singular boundary value problems. J. Comput. Appl. Math. 205, 469–478 (2007)

    Article  Google Scholar 

  18. 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 

  19. 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 

  20. P. Roul, Doubly singular boundary value problems with derivative dependent source function: A fast-converging iterative approach. Math. Method Appl. Sci. 42(1), 354–374 (2019)

    Article  Google Scholar 

  21. P. Roul, V.M.K.P. Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems. Appl. Math. Comput. 341, 428–450 (2019)

    Google Scholar 

  22. P. Roul, H. Madduri, A new highly accurate domain decomposition optimal homotopy analysis method and its convergence for singular boundary value problems. Math. Method Appl. Sci. 41(16), 6625–6644 (2018)

    Article  Google Scholar 

  23. J. Rashidina, Z. Mahmoodi, M. Ghasemi, Parametric spline method for a class of singular two-point boundary value problems. Appl. Maths. Comput. 188, 58–63 (2007)

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  26. A.K. Verma, S. Kayenat, On the convergence of Mickens type nonstandard finite difference schemes on Lane-Emden type equations. J. Math. Chem. 56, 1667–1706 (2018)

    Article  CAS  Google Scholar 

  27. P. Roul, U. Warbhe, New approach for solving a class of singular boundary value problem arising in various physical Models. J. Math. Chem. 54(6), 1255–1285 (2016)

    Article  CAS  Google Scholar 

  28. A.M. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Commun. Nonlinear. Sci. Numer. Simulat. 16, 3881–3886 (2011)

    Article  Google Scholar 

  29. J.H. He, Y.O. El-Dib, Homotopy perturbation method for Fangzhu oscillator. J. Math. Chem. 58(10), 2245–2253 (2020)

    Article  CAS  Google Scholar 

  30. J.H. He, Y.O. El-Dib, The reducing rank method to solve third-order Duffing equation with the homotopy perturbation. Numer. Methods Part. Differ. Equ. 37(2), 1800–1808 (2020)

    Article  Google Scholar 

  31. J.H. He, Approximate analytical solution for seepage flow with fractional derivative with porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    Article  Google Scholar 

  32. P. Roul, U. Warbhe, A novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems. J. Comput. Appl. Math. 296, 661–676 (2016)

    Article  Google Scholar 

  33. P. Roul, On the numerical solution of singular boundary value problem: a domain decomposition homotopy perturbation approach. Math. Method Appl. Sci. 40(18), 7396–7409 (2017)

    Article  Google Scholar 

  34. P. Roul, U. Warbhe, A new homotopy perturbation scheme for solving singular boundary value problems arising in various physical models. Z. Naturforsch. A. 72(8), 733–743 (2017)

    Article  CAS  Google Scholar 

  35. P. Roul, An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems. Eur. Phys. J. Plus 131, 209 (2016)

    Article  Google Scholar 

  36. P. Roul, A new efficient recursive technique for solving singular boundary problem arising in various physical models. Eur. Phys. J. Plus 131, 105 (2016)

    Article  Google Scholar 

  37. J.H. He, X.H. Wu, Variational iteration method: new development and applications. Comput. Math. Appl. 54, 881–894 (2007)

    Article  Google Scholar 

  38. P. Roul, Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie modified Reimann-Liouville derivative. Math. Method Appl. Sci. 34, 1025–1035 (2011)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the CSIR, India in the form of project no.\(25(0286)/18/EMR-11\).

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  1. Department of Mathematics, VNIT, Nagpur, 440010, India

    Pradip Roul, Trishna Kumari & V. M. K. Prasad Goura

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  1. Pradip Roul
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  2. Trishna Kumari
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  3. V. M. K. Prasad Goura
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Correspondence to Pradip Roul.

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Roul, P., Kumari, T. & Goura, V.M.K.P. An efficient numerical approach for solving a general class of nonlinear singular boundary value problems. J Math Chem 59, 1977–1993 (2021). https://doi.org/10.1007/s10910-021-01279-7

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