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A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems

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Abstract

A high-order numerical scheme based on collocation of a quintic B-spline over finite element is proposed for the numerical solution of a class of nonlinear singular boundary value problems (SBVPs) arising in various physical models in engineering and applied sciences. Five illustrative examples are presented to illustrate the applicability and accuracy of the method. In order to justify the advantage of the proposed numerical scheme, the computed results are compared with the results obtained by two other fourth-order numerical methods, namely the finite difference method (Chawla et al. in BIT 28(1):88–97, 1988) and B-spline collocation method (Goh et al. in Comput Math Appl 64:115–120, 2012).

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References

  1. E. Celik, H. Tunc, M. Sari, An efficient multi-derivative numerical method for chemical boundary value problems. J. Math. Chem. (2023). https://doi.org/10.1007/s10910-023-01556-7

    Article  Google Scholar 

  2. H.S. Fogler, HS Elements of Chemical Reaction Engineering, 2nd edn. (Prentice hall-Hall Inc, New Jersey, 1997)

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  7. M. Chawla, R. Subramanian, H. Sathi, A fourth order method for a singular two-point boundary value problem. BIT 28(1), 88–97 (1988)

    Article  Google Scholar 

  8. 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 

  9. P. Roul, V.M.K.P. Goura, K. Kassner, A high accuracy numerical approach for electro-hydrodynamic flow of a fluid in an ion-drag configuration in a circular cylindrical conduit. Appl. Numer. Math. 165, 303–321 (2021)

    Article  Google Scholar 

  10. J.E. Paullet, On the solutions of electrohydrodynamic flow in a circular cylindrical conduit. Z. Angew. Math. Mech.. 79, 357–360 (1999)

  11. 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 

  12. P. Roul, U. Warbhe, A Novel Numerical approach and its convergence for numerical solutionsof nonlinear doubley singular boundary value problems. J. Comput. Appl. Math. 226, 661–676 (2016)

    Article  Google Scholar 

  13. P. Roul, T. Kumari, A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems. J. Math. Chem. 60(1), 128–144 (2021)

    Article  Google Scholar 

  14. P. Roul, T. Kumari, V.M.K.P. Goura, An efficient numerical approach for solving a general class of nonlinear singular boundary value problems. J. Math. Chem. 59(9), 1977–1993 (2021)

    Article  CAS  Google Scholar 

  15. 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)

  16. R.K. Pandey, A note on a finite difference method for a class of singular boundary-value problems in physiology. Int. J. Comput. Math. 74(1), 127–132 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. M.M. Chawla, R. Subramanian, A new spline method for singular two-point boundary value problems. Int. J. Comput. Math. 24(3), 291–310 (1988)

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  23. 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 

  24. P. Roul, A new efficient recursive technique forsolving singular boundary value problems arising in various physical models. Eur. Phys. J. Plus 131(4), 1–15 (2016)

    Article  Google Scholar 

  25. P. Roul, An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems. Eur. Phys. J. Plus 131(6), 1–15 (2016)

    Article  Google Scholar 

  26. 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 

  27. 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(11), 1750–1760 (2010)

    Article  CAS  Google Scholar 

  28. P. Roul, T. Kumari, V.M.K.P. Goura, An efficient numerical method based on exponential B-spline basis functions for solving a class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions. Math. Method Appl. Sci. 44(5), 3376–3395 (2021)

    Article  Google Scholar 

  29. S.A. Khuri, A. Safy, A novel approach for the solution of a class of singular boundary value pröblems arising in physiology. Math. Comput. Model. 52, 626–636 (2010)

    Article  Google Scholar 

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

    Article  Google Scholar 

  31. P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options. Appl. Numer. Math. 151, 472–493 (2020)

    Article  Google Scholar 

  32. R.C. Mittal, S. Kumar, R. Jiwari, A cubic B-spline quasi-interpolation method for solving two-dimensional unsteady advection diffusion equations. Int. J. Numer. Methods Heat Fluid Flow 30(9), 4281–4306 (2020)

    Article  Google Scholar 

  33. R. Jiwari, S. Pandit, M.E. Koksal, A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems. Comput. Appl. Math. 38, 140 (2019). https://doi.org/10.1007/s40314-019-0918-1

    Article  Google Scholar 

  34. R. Jiwari, Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput. Phys. Commun. 193, 55–65 (2015)

    Article  CAS  Google Scholar 

  35. A.S. Alshomrani, S. Pandit, A.K. Alzahrani, M.S. Alghamdi, R. Jiwari, A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic type wave equations. Eng. Comput. 34(4), 1257–1276 (2017)

    Article  Google Scholar 

  36. P. Roul, V. Rohil, A high-accuracy computational technique based on \(L2-1_{\sigma }\) and B-spline schemes for solving the nonlinear time-fractional Burgers’ equation. Soft Comput. (2023). https://doi.org/10.1007/s00500-023-09413-0

    Article  Google Scholar 

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  1. Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010, India

    Pradip Roul

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Pradip Roul: Conceptualization, Formal analysis, Resources, Writing- original draft, Investigation, Software, Writing-original draft.

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Roul, P. A high-order B-spline collocation method for solving a class of nonlinear singular boundary value problems. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01590-z

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