Abstract
In this paper, we describe an optimal B-spline collocation method to solve one-dimensional non-linear Bratu problem. Convergence result of the method is established through Green’s function technique and it is proved that the method has sixth order convergence. The applicability and accuracy of the method are demonstrated through two numerical examples. It is shown that the computational result is consistent with theoretical prediction. Moreover, the numerical results obtained by the present method are compared with those obtained by two other B-spline collocation approaches reported in the literature. The computed results reveal that our method is accurate and easy to implement.
Similar content being viewed by others
References
H. Caglar, N. Caglar, M. Ozer, A. Valarstos, A.N. Anagnostopoulos, B-spline method for solving Bratu’s problem. Int. J. Comput. Math. 87, 1885–1891 (2010)
P. Roul, K. Thula, A fourth order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Comput. Math. 96, 85–104 (2019)
D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics (Princeton University Press, Princeton, NJ, 1955)
Y.Q. Wan, Q. Guo, N. Pan, Thermo-electro-hydro dynamic model for electro spinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004)
N. Das, R. Singh, A.M. Wazwaz, J. Kumar, An algorithm based on the variational iteration technique for the Bratu-type and the Lane–Emden problems. J. Math. Chem. 54, 527–551 (2016)
J.H. He, H.Y. Kong, R.X. Chen, M.S. Hu, Q.L. Chen, Variational iteration method for Bratu-like equation arising in electrospinning. Carbohydr. Polym. 105, 229–230 (2014)
S. Chandrasekhar, Introduction to the Study of Stellar Structure (Dover, New York, 1967)
S. Li, S.J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)
A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)
J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)
J.H. He, Variational iteration method-A kind of non-linear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)
J.H. He, Some asymptotic methods for strongly non linear equations. Int. J. Mod. Phys. B 20, 1141–1199 (2006)
J.H. He, Variational iteration method-some recent results and new interpretations. J. Comput. Appl. Math. 207, 3–17 (2007)
J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 22, 3487–3578 (2008)
J.H. He, Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. Article ID 916793 (2012)
J.H. He, X.H. Wu, Variational iteration method: new development and applications. Comput. Math. Appl. 54, 881–894 (2007)
P. Roul, Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie-modified Riemann–Liouville derivative. Math. Methods Appl. Sci. 34, 1025–1035 (2011)
P. Roul, H. Madduri, An optimal iterative algorithm for solving Bratu-type problems. J. Math. Chem. 57, 583–598 (2018)
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, 2613–2630 (2019)
P. Roul, V.M.K. Prasad 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)
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, 176 (2018)
P. Roul, K. Thula, A new high-order numerical method for solving singular two-point boundary value problems. J. Comput. Appl. Math. 343, 1–18 (2018)
M.K. Kadalbajoo, P. Arora, B-spline collocation method for the singular-perturbation problem using artificial viscosity. Comput. Math. Appl. 57, 650–663 (2009)
R.C. Mittal, R. Rohila, A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations. Wave Motion 80, 47–68 (2018)
P. Roul, A high accuracy numerical method and its convergence for time-fractional Black–Scholes equation governing European options. Appl. Numer. Math. (2019). https://doi.org/10.1016/j.apnum.2019.11.004
P. Roul, V.M.K. Prasad Goura, R. Agarwal, A new high order numerical approach for a class of nonlinear derivative dependent singular boundary value problems. Appl. Numer. Math. 145, 315–341 (2019)
P. Roul, V.M.K. Prasad Goura, A high order numerical method and its convergence for time-fractional fourth order partial differential equations. Appl. Math. Comput. 366, 124727 (2020)
C. De Boor, A Practical Guide to Splines (Springer, New York, 1978)
D.J. Fyfe, Linear dependence relations connecting equal interval Nth degree splines and their derivatives. J. Inst. Math. Appl. 7, 398–406 (1971)
T.R. Lucas, Error bound for interpolating cubic spline under various end conditions. SIAM. J. Numer. Anal. 3, 569–584 (1974)
W.D. Hoskins, D.S. Meek, Linear dependence relations for polynomial splines at midknots. BIT 15, 272–276 (1975)
R.D. Russell, L.F. Shampine, A collocation method for boundary value problems. Numer. Math. 19, 1–28 (1971)
Acknowledgements
The authors are very grateful to anonymous referees for their valuable suggestions and comments and thankfully acknowledge the financial support provided by the CSIR, India in the form of Project No. \(25(0286)/18/EMR-II\).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
The elements of the coefficient matrix P are given as follows:
Rights and permissions
About this article
Cite this article
Roul, P., Goura, V.M.K.P. A sixth order optimal B-spline collocation method for solving Bratu’s problem. J Math Chem 58, 967–988 (2020). https://doi.org/10.1007/s10910-020-01105-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-020-01105-6