A new approach for solving nonlinear singular boundary value problems
DOI: https://doi.org/10.3846/mma.2018.003Abstract
In this paper, an e_cient method based on Quasi-Newton's method and the simpli_ed reproducing kernel method is proposed for solving nonlinear singular boundary value problems. For the Quasi-Newton's method the convergence order is studied. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the e_ciency and stability of the proposed method. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.
Keywords:
nonlinear singular boundary value problem, numerical solution, Quasi-Newton's method, reproducing kernel methodHow to Cite
Zhu, H., Niu, J., Zhang, R., & Lin, Y. (2018). A new approach for solving nonlinear singular boundary value problems. Mathematical Modelling and Analysis, 23(1), 33-43. https://doi.org/10.3846/mma.2018.003
Share
License
Copyright (c) 2018 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
H. Çağlar, N. Çağlar and M. Özer. B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons and Fractals, 39(3):1232{1237, 2009. <a target="_blank" href="https://doi.org/10.1016/j.chaos.2007.06.007"> https://doi.org/10.1016/j.chaos.2007.06.007</a>
S.H. Chang. Taylor series method for solving a class of nonlinear singular boundary value problems arising in applied science. Applied Mathematics and Computation, 235:110{117, 2014. <a target="_blank" href="https://doi.org/10.1016/j.amc.2014.02.094"> https://doi.org/10.1016/j.amc.2014.02.094</a>
M. Chawla, R. Subramanian and H. Sathi. A fourth order method for a singular two-point boundary value problem. BIT, 28(1):88{97, 1988. <a target="_blank" href="https://doi.org/10.1007/BF01934697"> https://doi.org/10.1007/BF01934697</a>
M.G. Cui. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, NewYork, 2009.
A. Daniel. Reproducing Kernel Spaces and Applications. Springer, NewYork, 2013.
A.S.V. Ravi Kanth and K. Aruna. He's variational iteration method for treating nonlinear singular boundary value problems. Computers and Mathematics with Applications, 60(3):821{829, 2010. <a target="_blank" href="https://doi.org/10.1016/j.camwa.2010.05.029"> https://doi.org/10.1016/j.camwa.2010.05.029</a>
A.S.V. Ravi Kanth and V. Bhattacharya. Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Applied Mathematics and Computation, 174(1):768{774, 2006. <a target="_blank" href="https://doi.org/10.1016/j.amc.2005.05.022"> https://doi.org/10.1016/j.amc.2005.05.022</a>
S.A. Khuri and A. Sayfy. A novel approach for the solution of a class of singular boundary value problems arising in physiology. Mathematical and Computer Modelling, 52(3{4):626{636, 2010. <a target="_blank" href="https://doi.org/10.1016/j.mcm.2010.04.009"> https://doi.org/10.1016/j.mcm.2010.04.009</a>
Y.Z. Lin, J. Niu and M.G. Cui. A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 218(14):7362{7368, 2012. <a target="_blank" href="https://doi.org/10.1016/j.amc.2011.11.009"> https://doi.org/10.1016/j.amc.2011.11.009</a>
L.J.Xie, C.L.Zhou and S.Xu. An e_ective numerical method to solve a class of nonlinear singular boundary value problems using improved di_erential transform method. SpringerPlus, 5:1066{1084, 2016. <a target="_blank" href="https://doi.org/10.1186/s40064-016-2753-9"> https://doi.org/10.1186/s40064-016-2753-9</a>
L.J.Xie, C.L.Zhou and S.Xu. A new algorithm based on di_erential transform method for solving multi-point boundary value problems. International Journal of Computer Mathematics, 93(6):981{994, 2016. <a target="_blank" href="https://doi.org/10.1080/00207160.2015.1012070"> https://doi.org/10.1080/00207160.2015.1012070</a>
P.R. Mcgilliuray and D.W. Olenburg. Methods for calculating Frechet derivatives and senstivities for the non-linear inversion problem. Geophysical Prospecting, 38(5):499{524, 1990. <a target="_blank" href="https://doi.org/10.1111/j.1365-2478.1990.tb01859.x"> https://doi.org/10.1111/j.1365-2478.1990.tb01859.x</a>
M. Mohsenyzadeh, K. Maleknejad and R. Ezzati. A numerical approach for the solution of a class of singular boundary value problems arising in physiology. Advances in Di_erence Equations, 231(1):231, 2015. <a target="_blank" href="https://doi.org/10.1186/s13662-015-0572-x"> https://doi.org/10.1186/s13662-015-0572-x</a>
J. Niu, Y.Z. Lin and C.P. Zhang. Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 17(2):190{202, 2012. <a target="_blank" href="https://doi.org/10.3846/13926292.2012.660889"> https://doi.org/10.3846/13926292.2012.660889</a>
J. Niu, Y.Z. Lin and C.P. Zhang. Numerical solution of nonlinear three-point boundary value problem on the positive half-line. Mathematical Methods in the Applied Sciences, 35:1601{1610, 2012. <a target="_blank" href="https://doi.org/10.1002/mma.2549"> https://doi.org/10.1002/mma.2549</a>
R.K. Pandey and Arvind K.Singh. On the convergence of a _nite di_erence method for a class of singular boundary value problems arising in physiology. Journal of Computational and Applied Mathematics, 166(2):553{564, 2004. <a target="_blank" href="https://doi.org/10.1016/j.cam.2003.09.053"> https://doi.org/10.1016/j.cam.2003.09.053</a>
J. Rashidinia, R. Mohammadi and R. Jalilian. The numerical solution of non-linear singular boundary value problems arising in physiology. Applied Mathematics and Computation, 185(1):360{367, 2007. <a target="_blank" href="https://doi.org/10.1016/j.amc.2006.06.104"> https://doi.org/10.1016/j.amc.2006.06.104</a>
R. Singh and J. Kumar. An e_cient numerical technique for the solution of nonlinear singular boundary value problems. Computer Physics Communications, 185(4):1282{1289, 2014. <a target="_blank" href="https://doi.org/10.1016/j.cpc.2014.01.002"> https://doi.org/10.1016/j.cpc.2014.01.002</a>
Z.H. Zhao, Y.Z. Lin and J. Niu. Convergence order of the reproducing kernel method for solving boundary value problems. Mathematical Modelling and Analysis, 21(4):466{477, 2016. <a target="_blank" href="https://doi.org/10.3846/13926292.2016.1183240"> https://doi.org/10.3846/13926292.2016.1183240</a>
S.H. Chang. Taylor series method for solving a class of nonlinear singular boundary value problems arising in applied science. Applied Mathematics and Computation, 235:110{117, 2014. <a target="_blank" href="https://doi.org/10.1016/j.amc.2014.02.094"> https://doi.org/10.1016/j.amc.2014.02.094</a>
M. Chawla, R. Subramanian and H. Sathi. A fourth order method for a singular two-point boundary value problem. BIT, 28(1):88{97, 1988. <a target="_blank" href="https://doi.org/10.1007/BF01934697"> https://doi.org/10.1007/BF01934697</a>
M.G. Cui. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, NewYork, 2009.
A. Daniel. Reproducing Kernel Spaces and Applications. Springer, NewYork, 2013.
A.S.V. Ravi Kanth and K. Aruna. He's variational iteration method for treating nonlinear singular boundary value problems. Computers and Mathematics with Applications, 60(3):821{829, 2010. <a target="_blank" href="https://doi.org/10.1016/j.camwa.2010.05.029"> https://doi.org/10.1016/j.camwa.2010.05.029</a>
A.S.V. Ravi Kanth and V. Bhattacharya. Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Applied Mathematics and Computation, 174(1):768{774, 2006. <a target="_blank" href="https://doi.org/10.1016/j.amc.2005.05.022"> https://doi.org/10.1016/j.amc.2005.05.022</a>
S.A. Khuri and A. Sayfy. A novel approach for the solution of a class of singular boundary value problems arising in physiology. Mathematical and Computer Modelling, 52(3{4):626{636, 2010. <a target="_blank" href="https://doi.org/10.1016/j.mcm.2010.04.009"> https://doi.org/10.1016/j.mcm.2010.04.009</a>
Y.Z. Lin, J. Niu and M.G. Cui. A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 218(14):7362{7368, 2012. <a target="_blank" href="https://doi.org/10.1016/j.amc.2011.11.009"> https://doi.org/10.1016/j.amc.2011.11.009</a>
L.J.Xie, C.L.Zhou and S.Xu. An e_ective numerical method to solve a class of nonlinear singular boundary value problems using improved di_erential transform method. SpringerPlus, 5:1066{1084, 2016. <a target="_blank" href="https://doi.org/10.1186/s40064-016-2753-9"> https://doi.org/10.1186/s40064-016-2753-9</a>
L.J.Xie, C.L.Zhou and S.Xu. A new algorithm based on di_erential transform method for solving multi-point boundary value problems. International Journal of Computer Mathematics, 93(6):981{994, 2016. <a target="_blank" href="https://doi.org/10.1080/00207160.2015.1012070"> https://doi.org/10.1080/00207160.2015.1012070</a>
P.R. Mcgilliuray and D.W. Olenburg. Methods for calculating Frechet derivatives and senstivities for the non-linear inversion problem. Geophysical Prospecting, 38(5):499{524, 1990. <a target="_blank" href="https://doi.org/10.1111/j.1365-2478.1990.tb01859.x"> https://doi.org/10.1111/j.1365-2478.1990.tb01859.x</a>
M. Mohsenyzadeh, K. Maleknejad and R. Ezzati. A numerical approach for the solution of a class of singular boundary value problems arising in physiology. Advances in Di_erence Equations, 231(1):231, 2015. <a target="_blank" href="https://doi.org/10.1186/s13662-015-0572-x"> https://doi.org/10.1186/s13662-015-0572-x</a>
J. Niu, Y.Z. Lin and C.P. Zhang. Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 17(2):190{202, 2012. <a target="_blank" href="https://doi.org/10.3846/13926292.2012.660889"> https://doi.org/10.3846/13926292.2012.660889</a>
J. Niu, Y.Z. Lin and C.P. Zhang. Numerical solution of nonlinear three-point boundary value problem on the positive half-line. Mathematical Methods in the Applied Sciences, 35:1601{1610, 2012. <a target="_blank" href="https://doi.org/10.1002/mma.2549"> https://doi.org/10.1002/mma.2549</a>
R.K. Pandey and Arvind K.Singh. On the convergence of a _nite di_erence method for a class of singular boundary value problems arising in physiology. Journal of Computational and Applied Mathematics, 166(2):553{564, 2004. <a target="_blank" href="https://doi.org/10.1016/j.cam.2003.09.053"> https://doi.org/10.1016/j.cam.2003.09.053</a>
J. Rashidinia, R. Mohammadi and R. Jalilian. The numerical solution of non-linear singular boundary value problems arising in physiology. Applied Mathematics and Computation, 185(1):360{367, 2007. <a target="_blank" href="https://doi.org/10.1016/j.amc.2006.06.104"> https://doi.org/10.1016/j.amc.2006.06.104</a>
R. Singh and J. Kumar. An e_cient numerical technique for the solution of nonlinear singular boundary value problems. Computer Physics Communications, 185(4):1282{1289, 2014. <a target="_blank" href="https://doi.org/10.1016/j.cpc.2014.01.002"> https://doi.org/10.1016/j.cpc.2014.01.002</a>
Z.H. Zhao, Y.Z. Lin and J. Niu. Convergence order of the reproducing kernel method for solving boundary value problems. Mathematical Modelling and Analysis, 21(4):466{477, 2016. <a target="_blank" href="https://doi.org/10.3846/13926292.2016.1183240"> https://doi.org/10.3846/13926292.2016.1183240</a>
View article in other formats
CrossMark check
Published
2018-02-20
Issue
Section
Articles
Copyright
Copyright (c) 2018 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
Zhu, H., Niu, J., Zhang, R., & Lin, Y. (2018). A new approach for solving nonlinear singular boundary value problems. Mathematical Modelling and Analysis, 23(1), 33-43. https://doi.org/10.3846/mma.2018.003