Journal of Mathematical Chemistry

, Volume 57, Issue 2, pp 583–598 | Cite as

An optimal iterative algorithm for solving Bratu-type problems

  • Pradip RoulEmail author
  • Harshita Madduri
Original Paper


Very Recently, Das et al. (J Math Chem 54:527–551, 2016) proposed a method based on variational iteration method for solving Bratu-type problems. In this work, we design a fast iterative scheme based on the optimal homotopy analysis method to obtain series solution for such problem. The proposed method contains a convergence control parameter which adjusts the interval of convergence of the series solution. This parameter is computed by minimizing the sum of the squared residual of the governing equation and with the help of which we obtain an optimal series solution to the problem. Further, our method avoids the usual time-consuming procedure to find the root of a nonlinear algebraic or transcendental equation for the undetermined coefficients. Besides the design of the method, convergence analysis and error estimate of the proposed method are supplemented. Two Bratu type problems are solved to demonstrate the efficiency and accuracy of the proposed method. Numerical results reveal the superiority of the proposed iterative scheme over a newly developed iterative method based on the variational iteration method (Das et al. 2016).


Bratu-type problems Nonlinear integral equation Optimal homotopy analysis method Undetermined coefficients Variational iteration method 



The authors thankfully acknowledge the financial support provided by the SERB, Department of science and Technology, New Delhi, India in the form of Project Number SB/S4/MS/877/2014.


  1. 1.
    N. Das, R. Singh, A.M. Wazwaz, J. Kumar, An algorithm based on the variational iteration technique for the Bratu-type and Lane-Emden problems. J. Math. Chem. 54, 527–551 (2016)CrossRefGoogle Scholar
  2. 2.
    S. Chandrasekhar, Introduction to the Study of Stellar Structure (Dover, New York, 1967)Google Scholar
  3. 3.
    D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics (Princeton University Press, Princeton, 1955)CrossRefGoogle Scholar
  4. 4.
    Y.Q. Wan, Q. Guo, N. Pan, Thermo-electro-hydro dynamic model for electro spinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004)Google Scholar
  5. 5.
    U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations (SIAM, Philadelphia, 1995)CrossRefGoogle Scholar
  6. 6.
    R. Buckmire, Application of a Mickens finite-difference scheme to the cylindrical BratuGelfand problem. Numer. Methods Part. Differ. Equ. 20(3), 327–337 (2004)CrossRefGoogle Scholar
  7. 7.
    H. Caglar, N. Caglar, M. Ozer, A. Valarstos, A.N. Anagnostopoulos, B-spline method for solving Bratus problem. Int. J. Comput. Math. 87(8), 1885–1891 (2010)CrossRefGoogle Scholar
  8. 8.
    S.A. Khuri, A new approach to Bratus problem. Appl. Math. Comput. 147, 131–136 (2004)Google Scholar
  9. 9.
    A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)Google Scholar
  10. 10.
    J.H. He, H.Y. Kong, R.X. Chen, M.S. Hu, Q.L. Chen, Variational iteration method for Bratu-like equation arising in electrospinning. Carbohydr. Polym. 105, 229–230 (2014)CrossRefGoogle Scholar
  11. 11.
    P. Roul, K. Thula, A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Comput. Math. (2017).
  12. 12.
    S. J. Liao, The proposed homotopy analysis technique for the solution of Nonlinear problems, Ph.D. Thesis, Sanghai Jiao Tong University, Shangai (1992)Google Scholar
  13. 13.
    S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method (Chapman and Hall/CRC Press, Boca Raton, 2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsVisvesvaraya National Institute of TechnologyNagpurIndia

Personalised recommendations