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A modified homotopy perturbation method for singular time dependent Emden-Fowler equations with boundary conditions. (English) Zbl 1356.35122

Summary: In this paper, we propose a new modification of the homotopy perturbation method (HPM) for solving nonlinear and singular time-dependent Emden-Fowler-types equations with the Neumann and Dirichlet boundary conditions. We first transform the singular problem into an equivalent integral equation, and we then apply the HPM to obtain approximate series solution. This new modified HPM will be used without unknown constants while computing the successive solution components, and we also avoid solving a sequence of transcendental equations for the determination of the unknown constants. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at \(x=0\). Four illustrative examples are examined to demonstrate the accuracy and applicability of the proposed method.

MSC:

35K67 Singular parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] C. Harley, E. Momoniat, First integrals and bifurcations of a Lane-Emden equation of the second kind. J. Math. Anal. Appl. 344(2), 757-764 (2008) · Zbl 1143.80005 · doi:10.1016/j.jmaa.2008.03.014
[2] R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52(1), 255-267 (2014) · Zbl 1311.92222 · doi:10.1007/s10910-013-0260-6
[3] P.M. Lima, L. Morgado, Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics. J. Math. Chem. 48(1), 145-158 (2010) · Zbl 1196.92009 · doi:10.1007/s10910-009-9646-x
[4] R. Singh, A.M. Wazwaz, An efficient approach for solving second-order nonlinear differential equation with Neumann boundary conditions. J. Math. Chem. 53(1), 767-790 (2015) · Zbl 1331.65115 · doi:10.1007/s10910-014-0455-5
[5] H. Goenner, P. Havas, Exact solutions of the generalized Lane-Emden equation. J. Math. Phys. 41, 7029-7042 (2000) · Zbl 1009.34002 · doi:10.1063/1.1308076
[6] A.M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type. Appl. Math. Comput. 118(2), 287-310 (2001) · Zbl 1023.65067 · doi:10.1016/S0096-3003(99)00223-4
[7] J.S. Wong, On the generalized Emden-Fowler equation. SIAM Rev. 17(2), 339-360 (1975) · Zbl 0295.34026 · doi:10.1137/1017036
[8] R. Singh, A.-M. Wazwaz, J. Kumar, An efficient semi-numerical technique for solving nonlinear singular boundary value problems arising in various physical models. Int. J. Compt. Math. (2015). doi:10.1080/00207160.2015.1045888 · Zbl 1331.65115
[9] A.M. Wazwaz, Analytical solution for the time-dependent Emden-Fowler type of equations by Adomian decomposition method. Appl. Math. Comput. 166(3), 638-651 (2005) · Zbl 1073.65105 · doi:10.1016/j.amc.2004.06.058
[10] A.S. Bataineh, M. Noorani, I. Hashim, Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method. Phys. Lett. A 371(1), 72-82 (2007) · Zbl 1209.65104 · doi:10.1016/j.physleta.2007.05.094
[11] A.M. Wazwaz, A reliable iterative method for solving the time-dependent singular Emden-Fowler equations. Open Eng. 3(1), 99-105 (2013) · doi:10.2478/s13531-012-0028-y
[12] J.H. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3), 257-262 (1999) · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[13] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non-Linear Mech. 35(1), 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[14] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135(1), 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[15] J.H. He, Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 350(1), 87-88 (2006) · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[16] A. Nazari-Golshan, S. Nourazar, H. Ghafoori-Fard, A. Yildirim, A. Campo, A modified homotopy perturbation method coupled with the fourier transform for nonlinear and singular Lane-Emden equations. Appl. Math. Lett. 26(10), 1018-1025 (2013) · Zbl 1308.65134 · doi:10.1016/j.aml.2013.05.010
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