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# Solution of Initial Value Prpblem of Bratu – Type Equation using Modifications of Homotopy Perturbation Method

10.5120/ijca2017913311 |

Bothayna S H Kashkari and Sharifa S Z Abbas. Solution of Initial Value Prpblem of Bratu – Type Equation using Modifications of Homotopy Perturbation Method. *International Journal of Computer Applications* 162(5):44-49, March 2017. BibTeX

@article{10.5120/ijca2017913311, author = {Bothayna S. H. Kashkari and Sharifa S. Z. Abbas}, title = {Solution of Initial Value Prpblem of Bratu – Type Equation using Modifications of Homotopy Perturbation Method}, journal = {International Journal of Computer Applications}, issue_date = {March 2017}, volume = {162}, number = {5}, month = {Mar}, year = {2017}, issn = {0975-8887}, pages = {44-49}, numpages = {6}, url = {http://www.ijcaonline.org/archives/volume162/number5/27243-2017913311}, doi = {10.5120/ijca2017913311}, publisher = {Foundation of Computer Science (FCS), NY, USA}, address = {New York, USA} }

### Abstract

Homotopy perturbation method (HPM) is an effective method for solving nonlinear differential equations. In this paper, some modifications of this method has been proposed to initial value problem of Bratu - Type model. The combination of Laplace transform and homotopy perturbation (LHPM), the new homotopy perturbation method (NHPM) and Laplace new homotopy perturbation method (LNHPM) are applied, and the solutions are considered as an infinite series that converge rapidly to the exact solutions.

### References

- S. Hichar, et al., "Application of nonlinear Bratu's equation in two and three dimension to electrostatics," Reports on mathematical physics, Vol. 76, No. 3, pp. 283 – 290, 2015.
- C. Hikmet, et al., "B-spline method for solving Bratu's problem," International Journal of Computer Mathematics, Volume 87, No. 8, pp. 1885 – 1891, 2010.
- R. Jalilian, "Non-polynomial spline method for solving Bratu's problem," Computer Physics Communications, Volume 181, No. 11, pp. 1868 – 1872, 2010.
- C. Yang and J. Hou, "Chebyshev wavelets method for solving Bratu's problem", Doc 391, Boundary Value Problems, Volume 2013, No. 1, pp. 142, 2013.
- A. Kazemi, et al., "An efficient approach for solving nonlinear Troesch's and Bratu's problems by wavelet analysis method," Mathematical Problems in Engineering, Volume 2013, 2013.
- G. Venkatesh, et al., "The Legendre wavelet method for solving initial value problems of Bratu-type," Computers & Mathematics with Applications, Volume 63, No. 8, pp. 1287 – 1295, 2012.
- E. Doha, et al., "Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type," Computational Mathematics and Mathematical Physics, Volume 53 No. 9, pp. 1292 – 1302, 2013.
- M. Abd-Elhameed, et al., "New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations," Abstract and Applied Analysis, Volume 2013, pp. 1 – 9, 2013.
- M. Abd-Elhameed, et al., "A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations," Mathematical Sciences, Vol. 9, No. 2, pp. 93 – 102, 2015.
- P. John, "One-point pseudospectral collocation for the one-dimensional Bratuequation," Applied Mathematics and Computation, Vol. 217, No. 12, pp. 5553 – 5565, 2011.
- M. Syam and H. Abdelrahem, "An efficient method for solving Bratu equations," Applied Mathematics and Computation, Vol. 176, No. 2, pp. 704 – 713, 2006.
- A. Wazwaz, "Adomian decomposition method for a reliable treatment of the Bratu-type equations," Applied Mathematics and Computation, Vol. 166, No. 3, pp. 652–663, 2005.
- A. Vahidi and M. Hasanzade, "Restarted Adomian’s Decomposition Method for the Bratu-Type Problem," Applied Mathematical Sciences, Vol. 6, No. 10, pp. 479 – 486, 2012.
- O. Samuel, et al., "A New Result On Adomian Decomposition Method for Solving Bratu’s Problem," Mathematical Theory and Modeling Vol.3, No.2, 2013.
- S. Liao, "Homotopy Analysis Method in Nonlinear Differential Equations," Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2012.
- N. Hany and S. Mourad , "Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method," Communications in Numerical Analysis, Volume 2013, pp. 1 – 14, 2013.
- M. Darwish and B. Kashkari, "Numerical solutions of second order initial value problems of Bratu-type via optimal homotopy asymptotic method," American Journal of Computational Mathematics, Vol. 4, No. 2, pp. 47 – 54, 2014.
- J. He, "Homotopy perturbation technique," Comput. Methods Appl. Mech. Engrg. Vol. 178, pp. 257 – 262, 1999.
- X. Feng, et al., "Application of homotopy perturbation method to the Bratu – Type equations," Topological Methods in Nonlinear Analysis Journal of Juliusz Schauder Center, Vol. 31, pp. 243 – 252, 2008.
- A. Ezekiel, "New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems," American Journal of Computational Mathematics, Vol. 3, No. 2, pp. 110 – 113, 2013.
- H. Aminikhah H. and M. Hemmatnezhad, "An efficient method for quadratic Riccati differential equation," Commun Nonlinear Sci Numer Simulat 15, pp. 835-839, 2010.
- H. Aminikhah, "The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs," Applied Mathematical Modelling, Vol. 36, pp.3638-3644, 2012

### Keywords

Bratu – Type equation, Homotopy perturbation method, Laplace homotopy perturbation method, new homotopy perturbation method, Laplace new homotopy perturbation method