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Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method. (English) Zbl 1242.65136
Summary: A homotopy perturbation transformation method which is based on homotopy perturbation method and Laplace transform is first applied to solve the approximate solution of fractional nonlinear equations. The nonlinear terms can be easily handled by the use of He’s polynomials. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations
44A10 Laplace transform
34A34 Nonlinear ordinary differential equations and systems
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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[1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. · Zbl 1056.93542
[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, Singapore, 2000. · Zbl 1111.93302
[3] R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, p. 77, 2000. · Zbl 0984.82032
[4] S. Wang and M. Xu, “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 1087-1096, 2009. · Zbl 1167.76311
[5] X. J. Xiaoyun and X. M. Yu, “Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media,” International Journal of Non-Linear Mechanics, vol. 41, no. 1, pp. 156-165, 2006.
[6] J. H. Ma and M. Y. Liu, “Exact solutions for a generalized nonlinear fractional Fokker-Planck equation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 515-521, 2010. · Zbl 1181.35293
[7] Y.-Q. Liu and J.-H. Ma, “Exact solutions of a generalized multi-fractional nonlinear diffusion equation in radical symmetry,” Communications in Theoretical Physics, vol. 52, no. 5, pp. 857-861, 2009. · Zbl 1182.35067
[8] T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616-1621, 2009. · Zbl 1171.65449
[9] A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3381-3390, 2011. · Zbl 1222.65066
[10] Z. A. Anastassi and T. E. Simos, “New trigonometrically fitted six-step symmetric methods for the efficient solution of the Schrödinger equation,” Communications in Mathematical and in Computer Chemistry, vol. 60, no. 3, pp. 733-752, 2008. · Zbl 1199.65228
[11] Z. A. Anastassi and T. E. Simos, “Numerical multistep methods for the efficient solution of quantum mechanics and related problems,” Physics Reports, vol. 482/483, pp. 1-240, 2009. · Zbl 1182.65110
[12] G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501-544, 1988. · Zbl 0671.34053
[13] A.-M. Wazwaz and S. M. El-Sayed, “A new modification of the Adomian decomposition method for linear and nonlinear operators,” Applied Mathematics and Computation, vol. 122, no. 3, pp. 393-405, 2001. · Zbl 1027.35008
[14] J. H. He, “Variational iteration method- a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005
[15] A.-M. Wazwaz, “The variational iteration method for analytic treatment of linear and nonlinear ODEs,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 120-134, 2009. · Zbl 1166.65353
[16] V. S. Ertürk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642-1654, 2008. · Zbl 1221.34022
[17] A. Al-rabtah, V. S. Ertürk, and S. Momani, “Solutions of a fractional oscillator by using differential transform method,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1356-1362, 2010. · Zbl 1189.34068
[18] E. Yusufoglu, “Numerical solution of Duffing equation by the Laplace decomposition algorithm,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 572-580, 2006. · Zbl 1096.65067
[19] Y. Khan, “An effective modification of the laplace decomposition method for nonlinear equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1373-1376, 2009.
[20] J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502
[21] X. C. Li, M. Y. Xu, and X. Y. Jiang, “Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 434-439, 2009. · Zbl 1159.65106
[22] S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345-350, 2007. · Zbl 1203.65212
[23] J.-H. He, “Recent development of the homotopy perturbation method,” Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205-209, 2008. · Zbl 1159.34333
[24] M. A. Noor, “Iterative methods for nonlinear equations using homotopy perturbation technique,” Applied Mathematics & Information Sciences, vol. 4, no. 2, pp. 227-235, 2010. · Zbl 1192.65053
[25] M. A. Noor, “Some iterative methods for solving nonlinear equations using homotopy perturbation method,” International Journal of Computer Mathematics, vol. 87, no. 1-3, pp. 141-149, 2010. · Zbl 1182.65079
[26] M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1937-1945, 2011. · Zbl 1219.65121
[27] Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He’s polynomials,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 1963-1967, 2011. · Zbl 1219.65119
[28] A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1486-1492, 2009. · Zbl 1197.65061
[29] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalized KdV equations using he’s polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 227-233, 2009. · Zbl 1168.35427
[30] A. Yildirim, “Application of the homotopy perturbation method for the Fokker-Planck equation,” International Journal for Numerical Methods in Biomedical Engineering, vol. 26, no. 9, pp. 1144-1154, 2010. · Zbl 1200.65085
[31] M. Khan, M. A. Gondal, and S. Kumar, “A novel homotopy perturbation tranform algorithm for linear and nonlinear system of partial differential equations,” World Applied Sciences Journal, vol. 12, no. 12, pp. 2352-2357, 2011.
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