Opuscula Math. 43, no. 4 (2023), 575-601

Opuscula Mathematica

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Nikhil Sriwastav
Amit K. Barnwal
Abdul-Majid Wazwaz
Mehakpreet Singh

Abstract. Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

Keywords: Bernstein polynomials, collocation method, three-point singular BVPs, convergence analysis, error estimate.

Mathematics Subject Classification: 34B05, 34B15, 34B16, 34B18, 34B27, 34B60.

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  1. R.P. Agarwal, D. O'Regan, Singular Differential and Integral Equations with Applications, Springer Science & Business Media, 2010.
  2. R. Agarwal, H. Thompson, C. Tisdell, Three-point boundary value problems for second-order discrete equations, Comput. Math. Appl. 45 (2003), 1429-1435. https://doi.org/10.1016/S0898-1221(03)00098-1
  3. I. Ahmad, M.A.Z. Raja, H. Ramos, M. Bilal, M. Shoaib, Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically, Neural Computing and Applications 33 (2021), 5753-5769. https://doi.org/10.1007/s00521-020-05355-y
  4. M. Alves, Singular functional differential equation of the second order: Diss. kand. phys.-matem. nauk, Perm University, Perm, 1999.
  5. N.V. Azbelev, V.P. Maksimov, L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, Hindawi Publishing Corporation, 2007.
  6. A.K. Barnwal, P. Pathak, Successive iteration technique for singular nonlinear system with four-point boundary conditions, J. Appl. Math. Comput. 62 (2020), no. 1-2, 301-324. https://doi.org/10.1007/s12190-019-01285-8
  7. A.K. Barnwal, N. Sriwastav, A technique for solving system of generalized Emden-Fowler equation using legendre wavelet, TWMS J. App. and Eng. Math. 13 (2023), 341-361.
  8. A.S. Bataineh, A.A. Al-Omari, O. Rasit Isik, I. Hashim, Multistage Bernstein collocation method for solving strongly nonlinear damped systems, J. Vib. Control 25 (2019), no. 1, 122-131.
  9. M. Cecchi, Z. Došlá, I. Kiguradze, M. Marini, On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems, Differential Integral Equations 20 (2007), 1081-1106.
  10. M. Chawla, R. Subramanian, A new spline method for singular two-point boundary value problems, Int. J. Comput. Math. 24 (1988), 291-310. https://doi.org/10.1080/00207168808803650
  11. F. Geng, A numerical algorithm for nonlinear multi-point boundary value problems, J. Comput. Appl. Math. 236 (2012), 1789-1794. https://doi.org/10.1016/j.cam.2011.10.010
  12. M. Greguš, F. Neuman, F. Arscott, Three-point boundary value problems in differential equations, J. London Math. Soc. 2 (1971), 429-436. https://doi.org/10.1112/jlms/s2-3.3.429
  13. J.-H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (2006), 87-88. https://doi.org/10.1016/j.physleta.2005.10.005
  14. F. Jin, B. Yan, Existence of positive solutions for singular Dirichlet boundary value problems with impulse and derivative dependence, Bound. Value Probl. 2020 (2020), Article no. 157. https://doi.org/10.1186/s13661-020-01454-w
  15. A.R. Kanth, K. Aruna, He's variational iteration method for treating nonlinear singular boundary value problems, Comput. Math. Appl. 60 (2010), 821-829. https://doi.org/10.1016/j.camwa.2010.05.029
  16. A.R. Kanth, V. Bhattacharya, Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput. 174 (2006), 768-774. https://doi.org/10.1016/j.amc.2005.05.022
  17. S.A. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology, Math. Comput. Modelling 52 (2010), 626-636. https://doi.org/10.1016/j.mcm.2010.04.009
  18. I. Kiguradze, On a singular multi-point boundary value problem, Ann. Mat. Pura Appl. 86 (1970), 367-399. https://doi.org/10.1007/BF02415727
  19. I.T. Kiguradze, B.L. Shekhter, Singular boundary value problems for ordinary second-order differential equations, J. Soviet Math. 43 (1988), 2340-2417.
  20. K.M. Levasseur, A probabilistic proof of the Weierstrass approximation theorem, Amer. Math. Monthly 91 (1984), 249-250. https://doi.org/10.2307/2322960
  21. K. Maleknejad, B. Basirat, E. Hashemizadeh, A Bernstein operational matrix approach for solving a system of high order linear Volterra--Fredholm integro-differential equations, Math. Comput. Modelling 55 (2012), 1363-1372. https://doi.org/10.1016/j.mcm.2011.10.015
  22. V.E. Mkrtchian, C. Henkel, Green function solution of generalised boundary value problems, Phys. Lett. A 364 (2020), 126573. https://doi.org/10.1016/j.physleta.2020.126573
  23. Y. Öztürk, M. Gülsu, An approximation algorithm for the solution of the Lane-Emden type equations arising in astrophysics and engineering using Hermite polynomials, Comput. Appl. Math. 33 (2014), 131-145. https://doi.org/10.1007/s40314-013-0051-5
  24. R.K. Pandey, On a class of weakly regular singular two-point boundary value problems, II, J. Differential Equations 127 (1996), 110-123. https://doi.org/10.1006/jdeq.1996.0064
  25. S.V. Parter, Solutions of a differential equation arising in chemical reactor processes, SIAM J. Appl. Math. 26 (1974), 687-716. https://doi.org/10.1137/0126063
  26. S.V. Parter, M.L. Stein, P.R. Stein, On the multiplicity of solutions of a differential equation arising in chemical reactor theory, Studies in Applied Mathematics 54 (1975), 293-314.
  27. R. Rach, J. Duan, A.M. Wazwaz, Solving the two-dimensional Lane-Emden type equations by the Adomian decomposition method, Journal of Applied Mathematics and Statistics 3 (2016), 15-26.
  28. H. Ramos, G. Singh, V. Kanwar, S. Bhatia, An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems, Numer. Algorithms 57 (2017), 509-529. https://doi.org/10.1007/s11075-016-0209-5
  29. H. Ramos, J. Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comput. Appl. Math. 204 (2007), 124-136. https://doi:10.1016/j.cam.2006.04.033
  30. J. Rashidinia, R. Mohammadi, R. Jalilian, The numerical solution of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput. 185 (2007), 360-367. https://doi.org/10.1016/j.amc.2006.06.104
  31. P. Roul, D. Biswal, A new numerical approach for solving a class of singular two-point boundary value problems, Numer. Algorithms 75 (2017), 531-552. https://doi.org/10.1007/s11075-016-0210-z
  32. R. Russell, L. Shampine, Numerical methods for singular boundary value problems, SIAM J. Numer. Anal. 12 (1975), 13-36. https://doi.org/10.1137/0712002
  33. J. Shahni, R. Singh, Numerical solution of system of Emden-Fowler type equations by Bernstein collocation method, Journal of Mathematical Chemistry 59 (2021), 1117-1138.
  34. K. Singh, A.K. Verma, M. Singh, Higher order Emden-Fowler type equations via uniform Haar wavelet resolution technique, J. Comput. Appl. Math. 376 (2020), 112836. https://doi.org/10.1016/j.cam.2020.112836
  35. M. Singh, A.K. Verma, An effective computational technique for a class of Lane-Emden equations, J. Math. Chem. 54 (2016), 231-251. https://doi.org/10.1007/s10910-015-0557-8
  36. M. Singh, A.K. Verma, R.P. Agarwal, Maximum and anti-maximum principles for three point SBVPs and nonlinear three point SBVPs, J. Appl. Math. Comput. 47 (2015), 249-263. https://doi.org/10.1007/s12190-014-0773-6
  37. M. Singh, A.K. Verma, R.P. Agarwal, On an iterative method for a class of 2 point & 3 point nonlinear sbvps, J. Appl. Anal. Comput. 9 (2019), 1242-1260. https://doi.org/10.11948/2156-907X.20180213
  38. O.P. Singh, R.K. Pandey, V.K. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified homotopy analysis method, J. Appl. Anal. Comput. 180 (2009), 1116-1124. https://doi.org/10.1016/j.cpc.2009.01.012
  39. R. Singh, Analytic solution of singular Emden-Fowler-type equations by Green’s function and homotopy analysis method, The European Physical Journal Plus 134 (2019), Article no. 583. https://doi.org/10.1140/epjp/i2019-13084-2
  40. R. Singh, M. Singh, An optimal decomposition method for analytical and numerical solution of third-order Emden-Fowler type equation, J. Comput. Sci. 63 (2022), 101790. https://doi.org/10.1016/j.jocs.2022.101790
  41. N. Sriwastav, A.K. Barnwal, Numerical solution of Lane-Emden pantograph delay differential equation: stability and convergence analysis, International Journal of Mathematical Modelling and Numerical Optimisation 13 (2023), 64-83. https://doi.org/10.1504/IJMMNO.2023.127839
  42. N. Sriwastav, A.K. Barnwal, A.M. Wazwaz, M. Singh, A novel numerical approach and stability analysis for a class of pantograph delay differential equation, J. Comput. Sci. 67 (2023), 101976. https://doi.org/10.1016/j.jocs.2023.101976
  43. Y. Sun, L. Liu, J. Zhang, R.P. Agarwal, Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math. 230 (2009), 738-750. https://doi.org/10.1016/j.cam.2009.01.003
  44. S. Tomar, M. Singh, K. Vajravelu, H. Ramos, Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier, Math. Comput. Simulation 204 (2022), 640-644. https://doi.org/10.1016/j.matcom.2022.09.003
  45. Umesh, M. Kumar, Numerical solution of singular boundary value problems using advanced Adomian decomposition method, Engineering with Computers 37 (2020), 2853-2863. https://doi.org/10.1007/s00366-020-00972-6
  46. A.K. Verma, N. Kumar, M. Singh, R.P. Agarwal, A note on variation iteration method with an application on Lane-Emden equations, Engineering with Computers 38 (2020), 3932-3943. https://doi.org/10.1108/EC-10-2020-0604
  47. J. Vigo-Aguiar, H. Ramos, Variable stepsize implementation of multistep methods for \(y''= f (x, y, y')\), J. Comput. Appl. Math. 192 (2006), 114-131. https://doi.org/10.1016/j.cam.2005.04.043
  48. D.S. Watkins, Fundamentals of Matrix Computations. Third Edition, John Wiley & Sons, 2004.
  49. A.M. Wazwaz, R. Rach, J.S. Duan, Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Appl. Math. Comput. 219 (2013), 5004-5019. https://doi.org/10.1016/j.amc.2012.11.012
  50. L.R. Williams, R.W. Leggett, Multiple fixed point theorems for problems in chemical reactor theory, J. Math. Anal. Appl. 69 (1979), 180-193. https://doi.org/10.1016/0022-247X(79)90187-2
  51. A. Yıldırım, T. Öziş, Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Anal. 70 (2009), 2480-2484. https://doi.org/10.1016/j.na.2008.03.012
  52. Ş. Yüzbaşı, A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations, Appl. Math. Comput. 273 (2016), 142-154. https://doi.org/10.1016/j.amc.2015.09.091
  53. Y. Zou, Q. Hu, R. Zhang, On numerical studies of multi-point boundary value problem and its fold bifurcation, Appl. Math. Comput. 185 (2007), 527-537. https://doi.org/10.1016/j.amc.2006.07.064
  • Communicated by Alexander Domoshnitsky.
  • Received: 2022-07-25.
  • Revised: 2023-03-31.
  • Accepted: 2023-05-24.
  • Published online: 2023-06-13.
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Cite this article as:
Nikhil Sriwastav, Amit K. Barnwal, Abdul-Majid Wazwaz, Mehakpreet Singh, Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis, Opuscula Math. 43, no. 4 (2023), 575-601, https://doi.org/10.7494/OpMath.2023.43.4.575

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