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On the accurate discretization of a highly nonlinear boundary value problem

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Abstract

The aim of this manuscript is to investigate an accurate discretization method to solve the one-, two-, and three-dimensional highly nonlinear Bratu-type problems. By discretization of the nonlinear equation via a fourth-order nonstandard compact finite difference formula, the considered problem is reduced to the solution of a highly nonlinear algebraic system. To solve the derived nonlinear system, a modified nonlinear solver is used. The new scheme is accurate, fast, straightforward and very effective to find the lower and upper branches of the Bratu’s problem. Numerical simulations and comparative results for the one-, two-, and three-dimensional cases verify that the new technique is easy to implement and more accurate than the other existing methods in the literature.

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References

  1. Bratu, G.: Sur les équations intégrales non linéaires. Bull. Math. Soc. France 42, 113–142 (1914)

    Article  MathSciNet  Google Scholar 

  2. Chandrasekhar, S.: Introduction to the Study of Stellar Structure. Dover, New York (1967)

    Google Scholar 

  3. Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfand problem for radial operators. J. Differential Equations 184, 283–298 (2002)

    Article  MathSciNet  Google Scholar 

  4. Mohsen, A.: A simple solution of the Bratu problem. Comp. Math. App. 67, 26–33 (2014)

    Article  MathSciNet  Google Scholar 

  5. Buckmire, R.: Applications of Mickens finite differences to several related boundary value problems. Advances in the Applications of Nonstandard Finite Difference Schemes 47–87 (2005)

  6. Khuri, S.: A new approach to Bratu’s problem. Appl. Math. Comput. 147(1), 131–136 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Caglar, H., Caglar, N., Ozer, M., Valaristos, A., Anagnostopoulos, A.N.: B-spline method for solving Bratu’s problem. Int. J. Comput. Math. 87(8), 1885–1891 (2010)

    Article  MathSciNet  Google Scholar 

  8. Boyd, J.: Chebychev polynomial expansions for simultaneous approximation of two branches of a function with application of the one-dimensional Bratu equation. Appl. Math. Comput. 142, 189–200 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Boyd, J.P.: An analytical and numerical study of the two-dimensional Bratu equation. J. Sci. Comput. 1, 183–206 (1986)

    Article  Google Scholar 

  10. Deeba, E., Khuri, S., Xie, S.: An algorithm for solving boundary value problems. J. Comput. Phys. 159, 125–138 (2000)

    Article  MathSciNet  Google Scholar 

  11. Aksoy, Y., Pakdemirli, M.: New perturbation iteration solutions for Bratutype equations. Comput. Math. Appl. 59(8), 2802–2808 (2010)

    Article  MathSciNet  Google Scholar 

  12. Syam, M., Hamdan, A.: An efficient method for solving Bratu equation. Appl. Math. Comput. 176, 704–713 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Syam, M.I., Attili, B.: Weighted residual method for obtaining positive solutions of two-point nonlinear boundary value problems. Appl. Math. Comput. 176, 775–784 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Abbasbandy, M.H.S., Liu, C.: The lie-group shooting method for solving the bratu equation. Commun. Nonlinear Sci. Numer. Simul. 16, 4238–4249 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Motsa, S.S., Sibanda, P.: Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs. Numer. Algor. 63(3), 399–417 (2013)

    Article  MathSciNet  Google Scholar 

  17. Mohsen, A.: On the integral solution of the one-dimensional Bratu problem. J. Comput. Appl. Math. 251, 61–66 (2013)

    Article  MathSciNet  Google Scholar 

  18. Liu, X., Zhou, Y., Wang, X., Wang, J.: A wavelet method for solving a class of nonlinear boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 18, 1939–1948 (2013)

    Article  MathSciNet  Google Scholar 

  19. Yang, Z., Shijun, L.: A HAM-based wavelet approach for nonlinear partial differential equations: two dimensional Bratu problem as an application. Commun. Nonlinear Sci. Numer. Simul. 53, 249–262 (2017)

    Article  MathSciNet  Google Scholar 

  20. Saeed, U., Rehman, M.: Wavelet-galerkin quasilinearization method for nonlinear boundary value problems. Abstr. Appl. Anal. (2014):868934. https://doi.org/10.1155/2014/868934 (2014)

    Article  MathSciNet  Google Scholar 

  21. Ullah, M.Z., Serra-Capizzano, S., Ahmad, F.: An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes. Appl. Math. Comput. 250, 249–259 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Doha, E., Bhrawy, A., Baleanu, D., Hafez, R.: Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type. Comput. Math. Math. Phys. 53(9), 1292–1302 (2013)

    Article  MathSciNet  Google Scholar 

  23. Chan, T., Keller, H.B.: Arc-length continuation and multi-grid techniques for nonlinear elliptic eigenvalue problems. SIAM J. Sci. Stat. Comput. 3, 173–194 (1984)

    Article  Google Scholar 

  24. Hackbusch, W.: Comparison of different multi-grid variants for nonlinear equations. ZAMM Z. Angew. Math. Mech. 72, 148–151 (1992)

    Article  MathSciNet  Google Scholar 

  25. Rashidinia, J., Maleknejad, K., Taheri, N.: Sinc-galerkin method for numerical solution of the Bratu’s problems. Numer. Algor. 62, 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  26. Zarebnia, M., Sarvari, Z.: Parametric spline method for solving Bratu’s problem. Int. J. Nonlin. Sci. 14, 3–10 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181, 1868–187 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Kumar, M., Yadav, N.: Numerical solution of Bratu’s problem using multilayer perceptron Neural Network method. Natl. Acad. Sci. Lett. 38(5), 425–428 (2015)

    Article  MathSciNet  Google Scholar 

  30. Li, S., Liao, S.J.: An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Xu, Y., Li, X., Zhang, L.: The particle swarm shooting method for solving the Bratu’s problem. Journal of Algorithms & Computational Technology 9(3), 291–302 (2014)

    Article  MathSciNet  Google Scholar 

  32. Odejide, S., Aregbesola, Y.: A note on two-dimensional Bratu problem. Kragujevac J. Math. 29, 49–56 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Lele, S.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)

    Article  MathSciNet  Google Scholar 

  34. Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  35. Mickens, R.E.: Applications of Nonstandard Finite-Difference Schemes. World Scientific, River Edge (2000)

    Book  Google Scholar 

  36. Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (1977)

    MATH  Google Scholar 

  37. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1993)

    Book  Google Scholar 

  38. Jarausch, H., Mackens, W.: Computing solution branches by use of a condensed Newton-supported Picard iteration scheme. ZAMM Z. Angew. Math. Mech. 64, 282–284 (1981)

    MathSciNet  MATH  Google Scholar 

  39. Simpson, R.: A method for the numerical determination of bifurcation states of nonlinear systems of equations. SIAM J. Numer. Anal. 12, 439–451 (1975)

    Article  MathSciNet  Google Scholar 

  40. Karkowski, J.: Numerical experiments with the Bratu equation in one, two and three dimensions. Comp. Appl. Math. 32, 231–244 (2013)

    Article  MathSciNet  Google Scholar 

  41. Shampine, L.F., Reichelt, M.W., Kierzenka, J.: Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. https://www.mathworks.com/help/matlab/ref/bvp4c.html

  42. Mohsen, A., Sedeek, L.F., Mohamed, S.A.: New smoother to enhance multigrid-based methods for Bratu problem. Appl. Math. Comput. 204, 325–339 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Moore, G., Spence, A.: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17(4), 567–576 (1980)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to express their deep gratitude to the editor and anonymous referees for their careful reading and valuable suggestions to improve the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran

    Mojtaba Hajipour

  2. Department of Electrical Engineering, University of Bojnord, P.O. Box: 94531-1339, Bojnord, Iran

    Amin Jajarmi

  3. Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530, Ankara, Turkey

    Dumitru Baleanu

  4. Institute of Space Sciences, P.O.Box, MG-23, 76900, Magurele-Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Mojtaba Hajipour
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  2. Amin Jajarmi
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  3. Dumitru Baleanu
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Correspondence to Mojtaba Hajipour.

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Hajipour, M., Jajarmi, A. & Baleanu, D. On the accurate discretization of a highly nonlinear boundary value problem. Numer Algor 79, 679–695 (2018). https://doi.org/10.1007/s11075-017-0455-1

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  • DOI: https://doi.org/10.1007/s11075-017-0455-1

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