## Analytical approximate solutions for nonlinear fractional differential equations.(English)Zbl 1029.34003

Summary: We consider a class of nonlinear fractional-differential equations bsed on the Caputo fractional derivative and, by extending the application of the Adomian decomposition method, we derive an analytical solution in the form of a series with easily computable terms. For linear equations, the method gives an exact solution, and, for nonlinear equations, it provides an approximate solution with good accuracy. Several examples are discussed.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations
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### References:

 [1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 [2] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002 [3] Samko, G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon & Breach: Gordon & Breach Yverdon · Zbl 0818.26003 [4] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals & Fractional Calculus in Continuum Mechanics (1997), Springer: Springer New York), 223-276 · Zbl 1438.26010 [6] Luchko, Y.; Srivastava, H. M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appl., 29, 73-85 (1995) · Zbl 0824.44011 [8] Shawagfeh, N. T., The decomposition method for fractional differential equations, J. Frac. Calc., 16, 27-33 (1999) · Zbl 0956.34004 [9] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0802.65122 [11] Caputo, M., Linear models of dissipation whose $$Q$$ is almost frequency independent. Part II, J. Roy. Astral. Soc., 13, 529-539 (1967) [12] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals & Fractional Calculus in Continuum Mechanics (1997), Springer: Springer New York), 291-348 · Zbl 0917.73004 [13] Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111, 1, 33-51 (2000) [14] Cherrualult, Y.; Adomian, G., Decomposition method: a new proof of convergence, Math. Comput. Modelling, 18, 103-106 (1993) · Zbl 0805.65057 [15] Abboui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29, 7, 103-105 (1995) · Zbl 0832.47051 [17] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004 [18] Bagley, R. L.; Torvik, P. J., On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, 294-298 (1994) · Zbl 1203.74022 [20] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6 (1997) · Zbl 0890.65071 [22] Wazwaz, A. M., A reliable modification of Adomian’s decomposition method, Appl. Math. Comput., 92, 1-7 (1998) · Zbl 0942.65107
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