Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOLVING SINGULAR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS IN THE REPRODUCING KERNEL SPACE

  • Geng, Fazhan (Department of Mathematics Harbin Institute of Technology Weihai) ;
  • Cui, Minggen (Department of Mathematics Harbin Institute of Technology Weihai)
  • Published : 2008.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we present a new method for solving a nonlinear two-point boundary value problem with finitely many singularities. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation $u_n(x)$ to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Keywords

References

  1. R. P. Agarwal and D. O'Regan, Second-order boundary value problems of singular type, J. Math. Anal. Appl. 226 (1998), no. 2, 414-430 https://doi.org/10.1006/jmaa.1998.6088
  2. M. K. Kadalbajoo and V. K. Aggarwal, Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Appl. Math. Comput. 160 (2005), no. 3, 851-863 https://doi.org/10.1016/j.amc.2003.12.004
  3. P. Kelevedjiev, Existence of positive solutions to a singular second order boundary value problem, Nonlinear Anal. 50 (2002), no. 8, Ser. A: Theory Methods, 1107-1118 https://doi.org/10.1016/S0362-546X(01)00803-3
  4. C. Li and M. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl. Math. Comput. 143 (2003), no. 2-3, 393-399 https://doi.org/10.1016/S0096-3003(02)00370-3
  5. Y. Liu and A. Qi, Positive solutions of nonlinear singular boundary value problem in abstract space, Comput. Math. Appl. 47 (2004), no. 4-5, 683-688 https://doi.org/10.1016/S0898-1221(04)90055-7
  6. Y. Liu and H. Yu, Existence and uniqueness of positive solution for singular boundary value problem, Comput. Math. Appl. 50 (2005), no. 1-2, 133-143 https://doi.org/10.1016/j.camwa.2005.01.022
  7. R. K. Mohanty, P. L. Sachdev, and N. Jha, An O($h^4$) accurate cubic spline TAGE method for nonlinear singular two point boundary value problems, Appl. Math. Comput. 158 (2004), no. 3, 853-868 https://doi.org/10.1016/j.amc.2003.08.145
  8. A. S. V. Ravi Kanth and Y. N. Reddy, Higher order finite difference method for a class of singular boundary value problems, Appl. Math. Comput. 155 (2004), no. 1, 249-258 https://doi.org/10.1016/S0096-3003(03)00774-4
  9. A. S. V. Ravi Kanth and Y. N. Reddy, Cubic spline for a class of singular two-point boundary value problems, Appl. Math. Comput. 170 (2005), no. 2, 733-740 https://doi.org/10.1016/j.amc.2004.12.049
  10. J. Wang, W. Gao, Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl. 233 (1999), no. 1, 246-256 https://doi.org/10.1006/jmaa.1999.6290
  11. X. Xu and J. Ma, A note on singular nonlinear boundary value problems, J. Math. Anal. Appl. 293 (2004), no. 1, 108-124 https://doi.org/10.1016/j.jmaa.2003.12.017
  12. X. Zhang and L. Liu, Positive solutions of superlinear semipositone singular Dirichlet boundary value problems, J. Math. Anal. Appl. 316 (2006), no. 2, 525-537 https://doi.org/10.1016/j.jmaa.2005.04.081

Cited by

  1. Solving a system of linear Volterra integral equations using the new reproducing kernel method vol.219, pp.20, 2013, https://doi.org/10.1016/j.amc.2013.03.123
  2. Solving the Dym initial value problem in reproducing kernel space 2017, https://doi.org/10.1007/s11075-017-0381-2
  3. Homotopy perturbation–reproducing kernel method for nonlinear systems of second order boundary value problems vol.235, pp.8, 2011, https://doi.org/10.1016/j.cam.2010.10.040
  4. Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations vol.2012, 2012, https://doi.org/10.1155/2012/839836
  5. Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods vol.11, pp.2, 2010, https://doi.org/10.1016/j.nonrwa.2008.10.033
  6. The exact solution of a class of Volterra integral equation with weakly singular kernel vol.217, pp.18, 2011, https://doi.org/10.1016/j.amc.2011.02.059
  7. Construction and calculation of reproducing kernel determined by various linear differential operators vol.215, pp.2, 2009, https://doi.org/10.1016/j.amc.2009.05.063
  8. A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space vol.34, pp.1, 2011, https://doi.org/10.1002/mma.1327
  9. An Approximation Method for Solving Volterra Integrodifferential Equations with a Weighted Integral Condition vol.2015, 2015, https://doi.org/10.1155/2015/758410
  10. An iterative reproducing kernel method in Hilbert space for the multi-point boundary value problems vol.328, 2018, https://doi.org/10.1016/j.cam.2017.07.015
  11. RKM for solving Bratu-type differential equations of fractional order vol.39, pp.6, 2016, https://doi.org/10.1002/mma.3588
  12. Approximate Solutions to Three-Point Boundary Value Problems with Two-Space Integral Condition for Parabolic Equations vol.2012, 2012, https://doi.org/10.1155/2012/414612
  13. A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM vol.217, pp.9, 2011, https://doi.org/10.1016/j.amc.2010.11.020
  14. Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method vol.215, pp.6, 2009, https://doi.org/10.1016/j.amc.2009.08.002
  15. Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations vol.20, pp.6, 2015, https://doi.org/10.3846/13926292.2015.1111953
  16. Solving Fredholm integro–differential equations using reproducing kernel Hilbert space method vol.219, pp.17, 2013, https://doi.org/10.1016/j.amc.2013.03.006
  17. A novel method for nonlinear singular fourth order four-point boundary value problems vol.62, pp.1, 2011, https://doi.org/10.1016/j.camwa.2011.04.029
  18. A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems vol.279, 2015, https://doi.org/10.1016/j.cam.2014.11.014
  19. A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems vol.213, pp.1, 2009, https://doi.org/10.1016/j.amc.2009.02.053
  20. New Implementation of Reproducing Kernel Method for Solving Functional-Differential Equations vol.07, pp.10, 2016, https://doi.org/10.4236/am.2016.710095
  21. Application of reproducing kernel method to third order three-point boundary value problems vol.217, pp.7, 2010, https://doi.org/10.1016/j.amc.2010.09.009
  22. Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method vol.2013, 2013, https://doi.org/10.1155/2013/196308
  23. A numerical approach for solving the high-order nonlinear singular Emden–Fowler type equations vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1529-7
  24. Multiplicity results by shooting reproducing kernel Hilbert space method for the catalytic reaction in a flat particle vol.17, pp.02, 2018, https://doi.org/10.1142/S0219633618500207