Nonlinearities distribution homotopy perturbation method applied to solve nonlinear problems: Thomas-Fermi equation as a case study. (English) Zbl 1435.65115

Summary: We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results.


65L99 Numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


[1] Landau, L.; Lifshitz, E. M., Mecánica Cuántica No Relativista (1967), Editorial Reverte, S. A.
[2] Assas, L. M. B., Approximate solutions for the generalized KdV-Burgers’ equation by He’s variational iteration method, Physica Scripta, 76, 2, 161-164 (2007) · Zbl 1128.35091
[3] Kazemnia, M.; Zahedi, S. A.; Vaezi, M.; Tolou, N., Arsenic contamination in New Orleans soil: temporal changes associated with flooding, Journal of Applied Sciences, 8, 22, 4192-4197 (2008)
[4] Noorzad, R.; Poor, A. T.; Omidvar, M., Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics, Journal of Applied Sciences, 8, 2, 369-373 (2008)
[5] Evans, D. J.; Raslan, K. R., The tanh function method for solving some important non-linear partial differential equations, International Journal of Computer Mathematics, 82, 7, 897-905 (2005) · Zbl 1075.65125
[6] Mahmoudi, J.; Tolou, N.; Khatami, I.; Barari, A.; Ganji, D. D., Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method, Journal of Applied Sciences, 8, 2, 358-363 (2008)
[7] Adomian, G., A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, 2, 501-544 (1988) · Zbl 0671.34053
[8] Babolian, E.; Biazar, J., On the order of convergence of Adomian method, Applied Mathematics and Computation, 130, 2-3, 383-387 (2002) · Zbl 1044.65043
[9] Kooch, A.; Abadyan, M., Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method, Trends in Applied Sciences Research, 7, 1, 57-67 (2012)
[10] Khan, Y.; Vázquez-Leal, H.; Faraz, N., An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations, Applied Mathematical Modelling, 37, 5, 2702-2708 (2013) · Zbl 1352.65172
[11] Vanani, S. K.; Heidari, S.; Avaji, M., A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations, Journal of Applied Sciences, 11, 20, 3504-3509 (2011)
[12] Chowdhury, S. H., A comparison between the modified homotopy perturbation method and adomian decomposition method for solving nonlinear heat transfer equations, Journal of Applied Sciences, 11, 7, 1416-1420 (2011)
[13] Zhang, L.-N.; Xu, L., Determination of the limit cycle by He’s parameter expansion for oscillators in a \(u^3/1 + u^2\) potential, Zeitschrift für Naturforschung—Section A Journal of Physical Sciences, 62, 7-8, 396-398 (2007) · Zbl 1203.34053
[14] Vazquez-Leal, H.; Castaneda-Sheissa, R.; Filobello-Nino, U.; Sarmiento-Reyes, A.; Sanchez Orea, J., High accurate simple approximation of normal distribution integral, Mathematical Problems in Engineering, 2012 (2012)
[15] He, J.-H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 10, 1141-1199 (2006) · Zbl 1102.34039
[16] Fereidoon, A.; Rostamiyan, Y.; Akbarzade, M.; Ganji, D. D., Application of He’s homotopy perturbation method to nonlinear shock damper dynamics, Archive of Applied Mechanics, 80, 6, 641-649 (2010) · Zbl 1271.70052
[17] Vázquez-Leal, H.; Filobello-Niño, U.; Castañeda-Sheissa, R.; Hernández-Martínez, L.; Sarmiento-Reyes, A., Modified HPMs inspired by homotopy continuation methods, Mathematical Problems in Engineering, 2012 (2012) · Zbl 1264.65131
[18] Khan, Y.; Vázquez-Leal, H.; Wu, Q., An efficient iterated method for mathematical biology model, Neural Computing and Applications, 23, 3-4, 677-682 (2013)
[19] Khan, Y.; Vázquez-Leal, H.; Faraz, N., An efficient new iterative method for oscillator differential equation, Scientia Iranica, 19, 6, 1473-1477 (2012)
[20] Biazar, J.; Aminikhah, H., Study of convergence of homotopy perturbation method for systems of partial differential equations, Computers & Mathematics with Applications, 58, 11-12, 2221-2230 (2009) · Zbl 1189.65246
[21] Biazar, J.; Ghazvini, H., Convergence of the homotopy perturbation method for partial differential equations, Nonlinear Analysis: Real World Applications, 10, 5, 2633-2640 (2009) · Zbl 1173.35395
[22] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers and Mathematics with Applications, 61, 8, 1963-1967 (2011) · Zbl 1219.65119
[23] Noor, M. A.; Mohyud-Din, S. T., Homotopy perturbation method for solving Thomas-Fermi equation using pade approximants, International Journal of Nonlinear Science, 8, 1, 27-31 (2009) · Zbl 1183.65086
[24] Ganji, D. D.; Sahouli, A. R.; Famouri, M., A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators, Journal of Applied Mathematics and Computing, 30, 1-2, 181-192 (2009) · Zbl 1180.34011
[25] Turkyilmazoglu, M., Some issues on HPM and HAM methods: a convergence scheme, Mathematical and Computer Modelling, 53, 9-10, 1929-1936 (2011) · Zbl 1219.65083
[26] Turkyilmazoglu, M., Convergence of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 12, 1-8, 9-14 (2011) · Zbl 1401.35024
[27] Filobello-Niño, U.; Vazquez-Leal, H.; Castañeda-Sheissa, R.; Yildirim, A.; Hernandez-Martinez, L.; Pereyra Díaz, D.; Pérez Sesma, A.; Hoyos Reyes, C., An approximate solution of Blasius equation by using HPM method, Asian Journal of Mathematics and Statistics, 5, 2, 50-59 (2012)
[28] Marinca, V.; Herisanu, N., Nonlinear Dynamical Systems in Engineering (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1244.65104
[29] Patel, T.; Mehta, M. N.; Pradhan, V. H., The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method, Asian Journal of Applied Sciences, 5, 1, 60-66 (2012)
[30] Liao, S., Advances in the Homotopy Analysis Method (2014), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA · Zbl 1283.35003
[31] Turkyilmazoglu, M., Analytic approximate solutions of rotating disk boundary layer flow subject to a uniform suction or injection, International Journal of Mechanical Sciences, 52, 12, 1735-1744 (2010)
[32] Turkyilmazoglu, M., Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer, Physics of Fluids, 21, 10 (2009) · Zbl 1183.76529
[33] Filobello-Nino, U.; Vazquez-Leal, H.; Benhammouda, B.; Hernandez-Martinez, L.; Khan, Y.; Jimenez-Fernandez, V. M.; Herrera-May, A. L.; Castaneda-Sheissa, R.; Pereyra-Diaz, D.; Cervantes-Perez, J.; Perez-Sesma, J. A. A.; Hernandez-Machuca, S. F.; Cuellar-Hernandez, L., A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Menten equation, SpringerPlus, 3, article 162 (2014)
[34] Filobello-Nino, U.; Vazquez-Leal, H.; Khan, Y.; Yildirim, A.; Jimenez-Fernandez, V. M.; Herrera-May, A. L.; Castaneda-Sheissa, R.; Cervantes-Perez, J., Using perturbation methods and Laplace-Padé approximation to solve nonlinear problems, Miskolc Mathematical Notes, 14, 1, 89-101 (2013) · Zbl 1299.37063
[35] Filobello-Niño, U.; Vazquez-Leal, H.; Khan, Y.; Perez-Sesma, A.; Diaz-Sanchez, A.; Herrera-May, A.; Pereyra-Diaz, D.; Castaneda-Sheissa, R.; Jimenez-Fernandez, V. M.; Cervantes-Perez, J., A handy exact solution for flow due to a stretching boundary with partial slip, Revista Mexicana de Física E, 59, 1, 51-55 (2013)
[36] Fernandez, F. M., Rational approximation to the Thomas-Fermi equations, Applied Mathematics and Computation, 217, 13, 6433-6436 (2011) · Zbl 1210.81133
[37] Yao, B., A series solution to the Thomas-Fermi equation, Applied Mathematics and Computation, 203, 1, 396-401 (2008) · Zbl 1161.34303
[38] Khan, H.; Xu, H., Series solution to the Thomas-Fermi equation, Physics Letters A, 365, 1-2, 111-115 (2007) · Zbl 1203.81060
[39] Parand, K.; Shahini, M., Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation, Physics Letters A, 373, 2, 210-213 (2009) · Zbl 1227.49050
[40] Hille, E., On the Thomas-Fermi equation, Proceedings of the National Academy of Sciences of the United States of America, 62, 7-10 (1969) · Zbl 0179.12903
[41] Turkyilmazoglu, M., Solution of the Thomas-Fermi equation with a convergent approach, Communications in Nonlinear Science and Numerical Simulation, 17, 11, 4097-4103 (2012) · Zbl 1316.34020
[42] Zill, D. G., A First Course in Differential Equations with Modeling Applications (2012), Brooks Cole
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