SpringerLink
Log in

Bernstein and Gegenbauer-wavelet collocation methods for Bratu-like equations arising in electrospinning process

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

We propose two numerical techniques for solving Bratu-like equations arising in the electrospinning and vibration-electrospinning process. Due to the presence of parameter \(\delta \) as well as strong nonlinearity, these problems pose difficulties in obtaining their solutions. The first numerical method is based on the Bernstein polynomials, and the second method is based on the Gegenbauer-wavelets method. To establish numerical algorithms, we consider the equivalent integral form of the Bratu-like equation with suitable boundary conditions. Using the approximation theory based on the Bernstein polynomials and the Gegenbauer wavelets combined with the collocation technique, we convert the integral equation into a nonlinear system of equations. The Newton–Raphson method is implemented to analyze the resulting nonlinear system of equations. Three examples of Bratu-like equations are provided to demonstrate the accuracy, applicability, and efficiency of the respective techniques. The obtained results of numerical solutions and residual errors are compared. We observe that the collocation technique based on the Bernstein polynomial provides a better result than the Gegenbauer wavelets and other known methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. W. Gilbert, De Magnete (Courier Corporation, Chelmsford, 1958)

    Google Scholar 

  2. H. Fong, I. Chun, D.H. Reneker, Beaded nanofibers formed during electrospinning. Polymer 40(16), 4585–4592 (1999)

    Article  CAS  Google Scholar 

  3. A.S. Mounim, B. De Dormale, From the fitting techniques to accurate schemes for the Liouville–Bratu–Gelfand problem. Numer. Methods Partial Differ. Equ. Int. J. 22(4), 761–775 (2006)

    Article  Google Scholar 

  4. W.Y. Liu, Y.P. Yu, J.H. He, S.Y. Wang, Effect of tension compensator on Sirofil yarn properties. Text. Res. J. 77(4), 195–199 (2007)

    Article  CAS  Google Scholar 

  5. Y.Q. Wan, J. Qiang, L.N. Yang, Q.Q. Cao, M.Z. Wang, Vibration and heat effect on electrospinning modeling, in Advanced Materials Research, vol. 843, (Trans Tech Publ, Zurich, 2014), pp. 9–13

    Google Scholar 

  6. A. Tokarev, O. Trotsenko, I.M. Griffiths, H.A. Stone, S. Minko, Magnetospinning of nano-and microfibers. Adv. Mater. 27(23), 3560–3565 (2015)

    Article  CAS  Google Scholar 

  7. Y.Q. Wan, Q. Guo, N. Pan et al., Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5(1), 5–8 (2004)

    Article  Google Scholar 

  8. G. Bratu, On nonlinear integral equations. Bull. Soc. Math. France 42, 113–142 (1914)

    Article  Google Scholar 

  9. J. Jacobsen, K. Schmitt, The Liouville–Bratu–Gelfand problem for radial operators. J. Differ. Equ. 184(1), 283–298 (2002)

    Article  Google Scholar 

  10. H. Liu, P. Wang, A short remark on Bratu-like equation arising in electrospinning and vibration-electrospinning process. Carbohydr. Polym. 105, 229–230 (2014)

    Article  Google Scholar 

  11. H.Y. Liu, P. Wang, A short remark on WAN model for electrospinning and bubble electrospinning and its development. Int. J. Nonlinear Sci. Numer. Simul. 16(1), 1–2 (2015)

    Article  Google Scholar 

  12. J.S. McGough, Numerical continuation and the Gelfand problem. Appl. Math. Comput. 89(1–3), 225–239 (1998)

    Google Scholar 

  13. S.A. Khuri, A new approach to Bratu’s equation. Appl. Math. Comput. 147, 131–136 (2004)

  14. R. Buckmire, Application of a Mickens finite-difference scheme to the cylindrical Bratu–Gelfand problem. Numer. Methods Partial Differ. Equ. Int. J. 20(3), 327–337 (2004)

    Article  Google Scholar 

  15. A.-M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166(3), 652–663 (2005)

    Google Scholar 

  16. M.I. Syam, A. Hamdan, An efficient method for solving Bratu equations. Appl. Math. Comput. 176(2), 704–713 (2006)

    Google Scholar 

  17. R. Jalilian, Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181(11), 1868–1872 (2010)

  18. Y. Aksoy, M. Pakdemirli, New perturbation-iteration solutions for Bratu-type equations. Comput. Math. Appl. 59(8), 2802–2808 (2010)

    Article  Google Scholar 

  19. S. Abbasbandy, M. Hashemi, C.-S. Liu, The Lie-group shooting method for solving the Bratu equation. Commun. Nonlinear Sci. Numer. Simul. 16(11), 4238–4249 (2011)

    Article  Google Scholar 

  20. A.-M. Wazwaz, A reliable study for extensions of the Bratu problem with boundary conditions. Math. Methods Appl. Sci. 35(7), 845–856 (2012)

    Article  Google Scholar 

  21. C. Yang, J. Hou, Chebyshev wavelets method for solving Bratu’s problem. Boundary Value Probl. 2013(1), 142 (2013)

  22. A. Kazemi Nasab, Z. Pashazadeh Atabakan, A. Kılıçman, An efficient approach for solving nonlinear Troesch’s and Bratu’s problems by wavelet analysis method. Math. Probl. Eng. (2013)

  23. H.N. Hassan, M.S. Semary, Analytic approximate solution for the Bratu’s problem by optimal homotopy analysis method. Commun. Numer. Anal. 2013, 1–14 (2013)

  24. J.-H. He, H.Y. Kong, R.X. Chen, M.S. Hu, Q.I. Chen, , Variational iteration method for Bratu-like equation arising in electrospinning. Carbohydr. Polym. 105, 229–230 (2014)

  25. Z. Abo-Hammour, O. Abu Arqub, S. Momani, N. Shawagfeh, Optimization solution of Troesch’s and Bratu’s problems of ordinary type using novel continuous genetic algorithm. Discrete Dyn. Nat. Soc. (2014)

  26. A. Mohsen, A simple solution of the Bratu problem. Comput. Math. Appl. 67(1), 26–33 (2014)

    Article  Google Scholar 

  27. W. Abd-Elhameed, Y. Youssri, E. Doha, A novel operational matrix method based on shifted legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations. Math. Sci. 9(2), 93–102 (2015)

    Article  Google Scholar 

  28. R. Singh, G. Nelakanti, J. Kumar, Approximate solution of two-point boundary value problems using Adomian decomposition method with Green’s function. Proc. Natl. Acad. Sci., India Sect. A 85(1), 51–61 (2015)

    Article  Google Scholar 

  29. H. Kafri, S.A. Khuri, Bratu’s problem: A novel approach using fixed-point iterations and Green’s functions. Comput. Phys. Commun. 198, 97–104 (2016)

  30. N. Das, R. Singh, A.-M. Wazwaz, J. Kumar, An algorithm based on the variational iteration technique for the Bratu-type and the Lne-Emden problems. J. Math. Chem. 54(2), 527–551 (2016)

    Article  CAS  Google Scholar 

  31. H. Temimi, M. Ben-Romdhane, An iterative finite difference method for solving Bratu’s problem. J. Comput. Appl. Math. 292, 76–82 (2016)

  32. M. Al-Mazmumy, A. Al-Mutairi, K. Al-Zahrani, An efficient decomposition method for solving Bratu’s boundary value problem. Am. J. Comput. Math. 7(1), 84–93 (2017)

  33. B.S. Kashkari, S.S. Abbas, Solution of initial value problem of Bratu-type equation using modifications of homotopy perturbation method. Int. J. Comput. Appl. 162(5), 0975–8887 (2017)

  34. H. Muzara, S. Shateyi, G.T. Marewo, On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem. Open Phys. 16(1), 554–562 (2018)

  35. L. Koudahoun, J. Akande, D. Adjaï, Y. Kpomahou, M. Monsia, On the general solution to the Bratu and generalized Bratu equations. J.Math. Stat. 14(1), 193–200 (2018)

    Article  Google Scholar 

  36. R. Singh, An iterative technique for solving a class of local and nonlocal elliptic boundary value problems. J. Math. Chem. 58(9), 1874–1894 (2020)

    Article  CAS  Google Scholar 

  37. H. Singh, F. Akhavan Ghassabzadeh, E. Tohidi, C. Cattani, Legendre spectral method for the fractional Bratu problem. Math. Methods Appl. Sci. 43(9), 5941–5952 (2020)

    Article  Google Scholar 

  38. M.R. Ali, W.X. Ma, New exact solutions of Bratu Gelfand model in two dimensions using Lie symmetry analysis. Chin. J. Phys. 65, 198–206 (2020)

    Article  Google Scholar 

  39. M. Abdelhakem, Y. Youssri, Two spectral Legendre’s derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems. Appl. Numer. Math. 169, 243–255 (2021)

  40. G. Swaminathan, G. Hariharan, V. Selvaganesan, S. Bharatwaja, A new spectral collocation method for solving Bratu-type equations using Genocchi polynomials. J. Math. Chem. (2021)

  41. H. Kafri, S.A. Khuri, A. Sayfy, Bratu-like equation arising in electrospinning process: a green’s function fixed-point iteration approach. Int. J. Comput. Sci. Math. 8(4), 364–373 (2017)

  42. S. Bernšteın, Démonstration du théoreme de weierstrass fondée sur le calcul des probabilities. Comm. Soc. Math. Kharkov 13, 1–2 (1912)

  43. D.D. Bhatta, M.I. Bhatti, Numerical solution of KdV equation using modified Bernstein polynomials. Appl. Math. Comput. 174(2), 1255–1268 (2006)

    Google Scholar 

  44. M.I. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205(1), 272–280 (2007)

    Article  Google Scholar 

  45. B.N. Mandal, S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput. 190(2), 1707–1716 (2007)

    Google Scholar 

  46. K. Maleknejad, E. Hashemizadeh, R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Commun. Nonlinear Sci. Numer. Simul. 16(2), 647–655 (2011)

  47. S.S. Ray, S. Singh, Numerical solution of nonlinear stochastic Itô-Volterra integral equation driven by fractional Brownian motion. Eng. Comput. (2020)

  48. Ş Yüzbaşı, A numerical approach for solving a class of the nonlinear Lane–Emden type equations arising in astrophysics. Math. Methods Appl. Sci. 34(18), 2218–2230 (2011)

    Article  Google Scholar 

  49. O.R. Isik, M. Sezer, Bernstein series solution of a class of Lane-Emden type equations. Math. Probl. Eng. (2013)

  50. J. Shahni, R. Singh, An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method. Eur. Phys. J. Plus 135(06), 1–21 (2020)

    Article  Google Scholar 

  51. J. Shahni, R. Singh, Numerical results of Emden-Fowler boundary value problems with derivative dependence using the Bernstein collocation method. Eng. Comput. (2020)

  52. J. Shahni, R. Singh, Numerical solution of system of Emden–Fowler type equations by Bernstein collocation method. J. Math. Chem. 2020, 1–22 (2020)

    Google Scholar 

  53. W. Abd-Elhameed, Y. Youssri, New ultraspherical wavelets spectral solutions for fractional riccati differential equations, in Abstract and Applied Analysis (2014)

  54. M. Rehman, U. Saeed, Gegenbauer wavelets operational matrix method for fractional differential equations. J. Korean Math. Soc. 52(5), 1069–1096 (2015)

    Article  Google Scholar 

  55. M. Usman, M. Hamid, R.U. Haq, W. Wang, An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations. Eur. Phys. J. Plus 133(8), 327 (2018)

    Article  Google Scholar 

  56. M. Usman, M. Hamid, M.M. Rashidi, Gegenbauer wavelets collocation-based scheme to explore the solution of free bio-convection of nanofluid in 3d nearby stagnation point. Neural Comput. Appl. 31(11), 8003–8019 (2019)

    Article  Google Scholar 

  57. H. Srivastava, F. Shah, R. Abass, An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley–Torvik equation. Russ. J. Math. Phys. 26(1), 77–93 (2019)

    Article  Google Scholar 

  58. N. Ozdemir, A. Secer, M. Bayram, The Gegenbauer wavelets-based computational methods for the coupled system of burgers’ equations with time-fractional derivative. Mathematics 7(6), 486 (2019)

  59. İ Çelik, Squeezing flow of nanofluids of cu-water and kerosene between two parallel plates by Gegenbauer wavelet collocation method. Eng. Comput. (2019)

  60. M. ur Rehman, U. Saeed, , Gegenbauer wavelets operational matrix method for fractional differential equations. J. Korean Math. Soc. 52(5), 1069–1096 (2015)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology Mesra, Ranchi, 835215, India

    Julee Shahni & Randhir Singh

Authors
  1. Julee Shahni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Randhir Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Randhir Singh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shahni, J., Singh, R. Bernstein and Gegenbauer-wavelet collocation methods for Bratu-like equations arising in electrospinning process. J Math Chem 59, 2327–2343 (2021). https://doi.org/10.1007/s10910-021-01290-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-021-01290-y

Keywords

Search

Navigation