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Solution of non-integer order differential equations via the Adomian decomposition method. (English) Zbl 0772.34009
Let $$D^ \alpha_{0+}y$$, $$\text{Re} \alpha>0$$, be the Riemann-Liouville fractional derivative of order $$\alpha$$ [see the book by S. G. Samko, the reviewer and O. I. Marichev, Integrals and derivatives of fractional order and some of their applications (1987; Zbl 0617.26004), §2]. In this paper the decomposition method developed by G. Adomian [Nonlinear stochastic operator equations (1986; Zbl 0609.60072); Math. Comput. Modelling 13, 17-43 (1990; Zbl 0713.65051), etc.] is used to obtain the known solution $$y(x)$$ of the differential equation of order 1/2: $$D^{1/2}_{0+}y+y=0$$. The explicit solution $y(x)=cx^{\alpha-1}E_{\alpha,\alpha}(\lambda x^ \alpha),\;E_{\alpha,\alpha}(z)=\sum^ \infty_{k=0}{z^ k\over\Gamma(\alpha k+\alpha)}$ $$c$$ being an arbitrary real constant, of more general equation of fractional order $$\alpha$$: $$D^ \alpha_{0+}y-\lambda y=0$$, $$0<\alpha<1$$, $$\lambda\in C$$, was found by M. M. Dzherbashyan [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1075-1111 (1968)]. In this connection see also §§40, 43 in the above book of S. G. Samko, the reviewer and O. I. Marichev. We note that the relation for the incomplete Gamma function (page 22 of the paper) is wrong and therefore the expression given in (2.5) is not the solution of the equation $$D^{1/2}_{0+}y+y=0$$.

MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 26A33 Fractional derivatives and integrals
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References:
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