International Journal of Nonlinear Analysis and Applications
On solving Bratu’s type equation by perturbation method
Document Type : Research Paper
Authors
College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
Abstract
In this paper, the perturbation method is employed to obtain an approximate solution of some examples of the Bratu equation by choosing the different values of $ \varepsilon $ and comparison with the exact solutions. It can be seen that the perturbation method is an alternative technique to be considered in solving many practical problems involving differential equations.
Keywords
[1] Y. Aregbesola, Numerical solution of Bratu problem using the method of weighted residual, Elect. J. South African
Math. Soc. 3(1) (2003) 1–7.
[2] R. Aris, The mathematical Theory of Diffusion and Reaction in Permeable Catalyst, Oxford University Press,
Clarendon, 1, 1975.
[3] U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary
Differential Equations, 1st Edn., SIAM, Philadelphia, 1994.
[4] J. Bebernes and D. Eberly, Mathematical Problems From Combustion Theory, Springer Science & Business Media,
Springer-Verlag, 1989.
[5] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1967.
[6] Y. Changqing and H. Jianhua, Chebyshev wavelets method for solving Bratu’s problem, Bound. Value Prob.
2013(1) (2013) 1–9.
[7] T.L. Chow, Classical Mechanics, John Wiley and Sons Inc., USA, 1995.
[8] N. Cohen and J. Benavides, Exact solutions of bratu and liouville equations, CNMAC 2010 (2010) 750–756.
[9] A. Ezekiel, New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems, Amer. J. Comput. Math. 3(2) (2013) 110–113.
[10] H.B. Fenta and G.A. Derese, Numerical solution of second order initial value problems of Bratu-type equations
using sixth order Runge-Kutta seven stages method, Int. J. Comput. Sci. Appl. Math. 5(1) (2019).
[11] H.N. Hassan and M. S. Semary, Analytic approximate solution for the Bratu’s problem by optimal homotopy
analysis method, Commun. Numerical Anal. 2013 (2013), 1-14.
[12] M.H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, New York, 1995.
[13] A. Mohsen, A simple solution of the Bratu problem, J. Comput. Math. Appl. 67 (2014) 26–33.
[14] J. Rashidinia and N. Taher, Application of the Sinc approximation to the solution of Bratu’s problem, Int. J.
Math. Model. Comput. 2(3) (2012) 239–246.
[15] M. Saravi, M. Hermann and D. Kaiser, Solution of Bratu’s Equation by He’s variational Iteration Method, Amer.
J. Comput. Appl. Math. 3(1) (2013) 46–48.
[16] A.M. Wazwaz, Adomians decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math.
Comput. 166 (2005) 652–663.
[17] A.M. Wazwaz, The successive differentiation method for solving Bratu equation and Bratu-type equations, Rom.
J. Phys. 61 (2016) 774–783.
[18] M. Zarebnia and M. Hoshyar, Solution of Bratu-type equation via spline method, Acta Universities Apulensis 37
(2014) 61–72.
Math. Soc. 3(1) (2003) 1–7.
[2] R. Aris, The mathematical Theory of Diffusion and Reaction in Permeable Catalyst, Oxford University Press,
Clarendon, 1, 1975.
[3] U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary
Differential Equations, 1st Edn., SIAM, Philadelphia, 1994.
[4] J. Bebernes and D. Eberly, Mathematical Problems From Combustion Theory, Springer Science & Business Media,
Springer-Verlag, 1989.
[5] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1967.
[6] Y. Changqing and H. Jianhua, Chebyshev wavelets method for solving Bratu’s problem, Bound. Value Prob.
2013(1) (2013) 1–9.
[7] T.L. Chow, Classical Mechanics, John Wiley and Sons Inc., USA, 1995.
[8] N. Cohen and J. Benavides, Exact solutions of bratu and liouville equations, CNMAC 2010 (2010) 750–756.
[9] A. Ezekiel, New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems, Amer. J. Comput. Math. 3(2) (2013) 110–113.
[10] H.B. Fenta and G.A. Derese, Numerical solution of second order initial value problems of Bratu-type equations
using sixth order Runge-Kutta seven stages method, Int. J. Comput. Sci. Appl. Math. 5(1) (2019).
[11] H.N. Hassan and M. S. Semary, Analytic approximate solution for the Bratu’s problem by optimal homotopy
analysis method, Commun. Numerical Anal. 2013 (2013), 1-14.
[12] M.H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, New York, 1995.
[13] A. Mohsen, A simple solution of the Bratu problem, J. Comput. Math. Appl. 67 (2014) 26–33.
[14] J. Rashidinia and N. Taher, Application of the Sinc approximation to the solution of Bratu’s problem, Int. J.
Math. Model. Comput. 2(3) (2012) 239–246.
[15] M. Saravi, M. Hermann and D. Kaiser, Solution of Bratu’s Equation by He’s variational Iteration Method, Amer.
J. Comput. Appl. Math. 3(1) (2013) 46–48.
[16] A.M. Wazwaz, Adomians decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math.
Comput. 166 (2005) 652–663.
[17] A.M. Wazwaz, The successive differentiation method for solving Bratu equation and Bratu-type equations, Rom.
J. Phys. 61 (2016) 774–783.
[18] M. Zarebnia and M. Hoshyar, Solution of Bratu-type equation via spline method, Acta Universities Apulensis 37
(2014) 61–72.
)
Volume 13, Issue 1
March 2022Pages 2755-2763
- Receive Date: 12 September 2021
- Revise Date: 18 October 2021
- Accept Date: 24 November 2021