Abstract
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).
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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)
S. Chandrasekhar, Introduction to the Study of Stellar Structure (Dover, New York, 1967)
D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics (Princeton University Press, Princeton, 1955)
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)
U.M. Ascher, R. Matheij, R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations (SIAM, Philadelphia, 1995)
R. Buckmire, Application of a Mickens finite-difference scheme to the cylindrical BratuGelfand problem. Numer. Methods Part. Differ. Equ. 20(3), 327–337 (2004)
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)
S.A. Khuri, A new approach to Bratus problem. Appl. Math. Comput. 147, 131–136 (2004)
A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)
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)
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). https://doi.org/10.1080/00207160.2017.2631417592264
S. J. Liao, The proposed homotopy analysis technique for the solution of Nonlinear problems, Ph.D. Thesis, Sanghai Jiao Tong University, Shangai (1992)
S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method (Chapman and Hall/CRC Press, Boca Raton, 2003)
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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.
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Roul, P., Madduri, H. An optimal iterative algorithm for solving Bratu-type problems. J Math Chem 57, 583–598 (2019). https://doi.org/10.1007/s10910-018-0965-7
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DOI: https://doi.org/10.1007/s10910-018-0965-7