A New Homotopy Perturbation Method for Solving Systems of Nonlinear Equations of Emden-Fowler Type

Musa Rahama Gadallah; Tarig M Elzaki

Musa Rahama Gadallah Mathematics Department, Faculty of Sciences, King Saud University, P.O. 2455, Riyadh 11451, Kingdom of Saudi Arabia
Tarig M Elzaki Mathematics Department, Faculty of Sciences and Arts, Alkamil, Jeddah, University of Jeddah, Kingdom of Saudi Arabia.

Keywords: New Homotopy Perturbation Method, Adomian's polynomials, Variational Iteration Method, Emden's Fowler type equations, systems of nonlinear partial differential equations.

Abstract

In this work, we apply the new homotopy perturbation method (NHPM) to get accurate results for solving systems of nonlinear equations of Emden–Fowler type, we indicate that our method (NHPM) is equivalent to the variational iteration method (VIM) with a specific convex. Four examples are given to illustrate our proposed methods. The method is easy to carry out and gives very accurate solutions for solving linear and nonlinear differential equations.

Downloads

Download data is not yet available.

References

A. M. Wazwaz, the variational iteration method for solving systems of equations of Emden–Fowler type, International Journal of Computer Mathematics, 88(16), pp. 3406–3415 (2011).

J. Biazar, M. Eslami, "A new technique for non-linaer two-dimensional wave equations", Scientia Iranica B, 20(2), pp. 359-363 (2013).

J. Biazar and M. Eslami, “A new homotopy perturbation method for solving systems of partial differential equations,” Computers and Mathematics with Applications, pp 225–234, 62 (2011).

Mohamed Elbadri and Tarig. M. Elzaki, New modification of Homotopy Perturbation Method and Fourth- Order Parabolic Equation With Variable Coefficient, pure and Applied Mathematics Journal. Vol.4, No. 6, pp.242-247(2015).

Mostafa Eslami1 and Jafar Biazar, analytical solution of the klein-gordon equation by a new Homotopy Perturbation Method Computational Mathematics and Modeling, Vol. 25, No. 1, March, (2014).

Mohammed ELbadri, A New Homotopy Perturbation Method for Solving Laplace Equation Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 8, Number, pp.237-242, 6(2013).

J.H. He, Acoupling method of homotopy technique an perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35(1),pp.37-43(2000).

Tarig. M. Elzaki, Eman M. A. Hilal. Homotopy Perturbation and Elzaki Transform for Solving Nonlinear Partial Differential Equations, Mathematical Theory and Modeling, ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online), Vol.2, No.3, pp, 33-42, (2012).

Tarig M. Elzaki, and J. Biazar, Homotopy Perturbation Method and Elzaki Transform for Solving System of Nonlinear Partial Differential Equations, World Applied Sciences Journal 24 (7): 944-948, (2013). DOI: 10.5829/idosi.wasj.2013.24.07.1041.

Tarig. M. Elzaki, Benedict I. Ita, Solutions of Radial Diffusivity and Shock Wave Equationsby Combined Homotopy Perturbation and Elzaki Transform Methods, Asia-Pacific Science and Culture Journal, 2013(2), 18–23, (2013).

Amjad Ezoo Hamza & Tarig. M. Elzaki, Application of Homotopy Perturbation and Sumudu Transform Method for Solving Burgers Equations. American Journal of Theoretical and Applied Statistics. Vol. 4, No. 6, (2015), pp. 480-483. doi: 10.11648/j.ajtas.20150406.18.

Mohannad H. Eljaily, Tarig M. Elzaki. Solution of Linear and Nonlinear Schrodinger Equations by Combine Elzaki Transform and Homotopy Perturbation Method. American Journal of Theoretical and Applied Statistics. Vol. 4, No. 6, (2015), pp. 534-538.doi: 10.11648/j.ajtas. 20150406.24.

Abbasbandy, S. ‘‘Numerical solutions of the integral equations: homotopy perturbation method and Adomian’s decomposition method’’, Applied Mathematics and Computation, 173, pp. 493–500 (2006).

A. Janalizadeh, A. Barari, D. D. Ganji, "Application of the homotopy perturbation method for solving second order non-linear wave equation”, International Symposium on Nonlinear Dynamics (2007 ISND) IOP Publishing, Journal of Physics: Conference Series (96), pp. 1-6 (2008).

He, J.H. ‘‘A coupling method of homotopy technique and perturbation technique for nonlinear problems’’, International Journal of Non-Linear Mechanics, 35(1), pp. 37–43 (2000).

A.M. Wazwaz, a new method for solving differential equations of the Lane–Emden type, Appl. Math. Comput. 118(2/3) pp. 287–310, (2001).

A.M. Wazwaz, a new method for solving singular initial value problems in the second order ordinary differential equations, Appl. Math. Comput. 128, pp. 47–57, (2002).

A.M.Wazwaz, Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Appl. Math. Comput. 161, pp. 543–560, (2005).

A.M. Wazwaz, Analytical solution for the time-dependent Emden–Fowler type of equations by Adomian decomposition method, Appl. Math. Comput. 166, pp. 638–651, (2005).

H. Eltayeb, Coupled singular and nonsingular thermoelastic system and Double Laplace Decomposition Method, BISKA, intentional open access journal, pp. 212-222, 4(3), (2016).

A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

A.Yildirim andT. Ozis, Solutions of singular IVPs of Lane–Emden type by the variational iteration method, Nonlinear Anal. pp. 2480–2484,70 (2009).

R. Muatjetjeja and C.M. Khalique, First integrals for a generalized coupled Lane–Emden system, Nonlinear Anal. RealWorld Appl. pp. 1202–1212, 12 (2011).

M. Dehghan and F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron. 13, pp. 53–59, (2008).

El-Sayed, A.M.A., Elsaid, A., El-Kalla, I.L. and Hammad, D. ‘‘A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains’’, Applied Mathematics and Computation, 218(17), pp. 8329–8340 (2012).

Rana, M.A., Siddiqui, A.M. and Ghori, Q.K. ‘‘Application of He’s homotopy perturbation method to Sumudu transform’’, International Journal of Nonlinear Science and Numerical Simulation, 8(2), pp. 185–190 (2007).

Ozis, T. and Yildirim, A. ‘‘Traveling wave solution of Korteweg–de Vries equation using He’s homotopy perturbation method’’, International Journal of Nonlinear Science and Numerical Simulation, 8(2), pp. 239–242 (2007).

Biazar, J., Eslami, M. and Ghazvini, H. ‘‘Homotopy perturbation method for systems of partial differential equations’’, International Journal of Nonlinear Science and Numerical Simulation, 8(3), pp. 411–416 (2007).

Yildirim, A. ‘‘Application of He’s homotopy perturbation method for solving the Cauchy reaction–diffusion problem’’, Computers and Mathematics with Applications, 57(4), pp. 612–618 (2009).