## Adomian decomposition: a tool for solving a system of fractional differential equations.(English)Zbl 1061.34003

Summary: Adomian’s decomposition method is employed to obtain solutions of a system of fractional differential equations. The convergence of the method is discussed with some illustrative examples. In particular, for the initial value problem $[D^{\alpha_1} y_1,\dots,D^{\alpha_n}y_n]^t =A(y_1, \dots,y_n)^t, \quad y_i(0)=c_i, \quad i=1,\dots,n,$ where $$A=[a_{ij}]$$ is a real square matrix, the solution turns out to be $\overline y(x)={\mathcal E}_{(\alpha_1,\dots, \alpha_n),1}(x^{\alpha_1}A_1,\dots, x^{\alpha_n}A_n) \overline y(0),$ where $${\mathcal E}_{(\alpha_1,\dots,\alpha_n),1}$$ denotes the multivariate Mittag-Leffler function defined for matrix arguments and $$A_i$$ is the matrix having $$i$$th row as $$[a_i1\dots a_{in}]$$, and all other entries are zero. Fractional oscillation and Bagley-Torvik equations are solved as illustrative examples.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A33 Fractional derivatives and integrals
Full Text:

### References:

 [1] Abboui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. appl. math., 29, 103-105, (1995) · Zbl 0832.47051 [2] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic · Zbl 0802.65122 [3] Atanackovic, T.M.; Stankovic, B., On a system of differential equations with fractional derivatives arising in rod theory, J. phys. A, 37, 1241-1250, (2004) · Zbl 1059.35011 [4] Babolian, E.; Biazar, J.; Islam, R., Solution of the system of ordinary differential equations by Adomian decomposition method, Appl. math. comput., 147, 713-719, (2004) · Zbl 1034.65053 [5] Chechkin, A.V.; Gorenflo, R.; Sokolov, I.M.; Gonchar, V.Yu., Distributed order time fractional diffusion equation, Frac. calc. appl. anal., 6, 259-279, (2003) · Zbl 1089.60046 [6] Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002 [7] 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 [8] Edwards, J.T.; Ford, N.J.; Simpson, A.C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. comput. appl. math., 148, 401-418, (2002) · Zbl 1019.65048 [9] Luchko, Yu.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta math. Vietnam., 24, 207-233, (1999) · Zbl 0931.44003 [10] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 [11] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [12] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003 [13] Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. comput., 131, 517-529, (2002) · Zbl 1029.34003 [14] Torvik, P.J.; Bagley, R.L., On the appearance of the fractional derivative in the behaviour of real materials, J. appl. mech., 51, 294-298, (1984) · Zbl 1203.74022 [15] ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.