Approximate Solution of Bratu Differential Equations Using Trigonometric Basic Functions
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Authors: B. AGHELI
DOI: 10.46793/KgJMat2102.203A
Abstract:
In this paper, I have proposed a method for finding an approximate function for Bratu differential equations (BDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, I define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the approximate function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get an approximate solution function with discrete derivatives of the solution function, we have determined the approximate solution function which satisfies in the Bratu differential equations (BDEs). In the end, the algorithm of the method is elaborated with several examples. In one example, I have presented an absolute error comparison of some approximate methods.
Keywords:
Trigonometric transform, Bratu differential equations, basic functions.
References:
[1] S. Abbasbandy, M. Hashemi and C.-S. Liu, The lie-group shooting method for solving the Bratu equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4238–4249.
[2] A. Akgül, F. Geng et al., Reproducing kernel Hilbert space method for solving Bratu’s problem, Bull. Malays. Math. Sci. Soc. 38 (2015), 271–287.
[3] J. P. Boyd, One-point pseudospectral collocation for the one-dimensional Bratu equation, Appl. Math. Comput. 217 (2011), 5553–5565.
[4] H. Caglar, N. Caglar, M. Özer, A. Valarıstos and A. N. Anagnostopoulos, B-spline method for solving Bratu’s problem, Int. J. Comput. Math. 87 (2010), 1885–1891.
[5] A. Colantoni and K. Boubaker, Electro-spun organic nanofibers elaboration process investigations using comparative analytical solutions, Carbohydrate Polymers 101 (2014), 307–312.
[6] M. A. Darwish and B. S. Kashkari, Numerical solutions of second order initial value problems of Bratu-type via optimal homotopy asymptotic method, American Journal of Computational Mathematics 4 (2014), 47–54.
[7] N. Das, R. Singh, A.-M. Wazwaz and J. Kumar, An algorithm based on the variational iteration technique for the Bratu-type and the lane-emden problems, J. Math. Chem. 54 (2016), 527–551.
[8] X. Feng, Y. He and J. Meng, Application of homotopy perturbation method to the Bratu-type equations, Topol. Methods Nonlinear Anal. 31 (2008), 243–252.
[9] D. A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, New Jersey, 2015.
[10] J. Jacobsen and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations 184 (2002), 283–298.
[11] R. Jalilian, Non-polynomial spline method for solving Bratu’s problem, Comput. Phys. Commun. 181 (2010), 1868–1872.
[12] S. Jator and V. Manathunga, Block Nyström type integrator for Bratu’s equation, J. Comput. Appl. Math. 327 (2018), 341–349.
[13] J. Karkowski, Numerical experiments with the Bratu equation in one, two and three dimensions, Comput. Appl. Math. 32 (2013), 231–244.
[14] E. Keshavarz, Y. Ordokhani and M. Razzaghi, The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations, Appl. Numer. Math. 128 (2018), 205–216.
[15] S. A. Khuri, A new approach to Bratu’s problem, Appl. Math. Comput. 147 (2004), 131–136.
[16] S. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math. 119 (2007), 297–354.
[17] X. Liu, Y. Zhou, X. Wang and J. Wang, A wavelet method for solving a class of nonlinear boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1939–1948.
[18] J. S. McGough, Numerical continuation and the Gelfand problem, Appl. Math. Comput. 89 (1998), 225–239.
[19] A. Mohsen, A simple solution of the Bratu problem, Comput. Math. Appl. 67 (2014), 26–33.
[20] I. Perfilieva, Fuzzy transforms: theory and applications, Fuzzy Sets and Systems 157 (2006), 993–1023.
[21] O. Ragb, L. Seddek and M. Matbuly, Iterative differential quadrature solutions for Bratu problem, Comput. Math. Appl. 74 (2017), 249–257.
[22] H. Temimi and M. Ben-Romdhane, An iterative finite difference method for solving Bratu’s problem, J. Comput. Appl. Math. 292 (2016), 76–82.
[23] A.-M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166 (2005), 652–663.
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