Fractional differential equations: \(\alpha\)-entire solutions, regular and irregular singularities. (English) Zbl 1178.26007

Kochubei, Anatoly N.

Fract. Calc. Appl. Anal. 12, No. 2, 135-158 (2009).

In a recent paper [Appl. Math. Comput. 187, No. 1, 239–249 (2007; Zbl 1121.34008)], A. A. Kilbas, M. Rivero, L. Rodríguez-Germá and J. J. Trujillo initiated a development of the analytical theory of fractional differential equations, which, as the author writes “opens the way for developing a theory of an \(\alpha\)-analytic theory of ordinary differential equations”. The author’s results point into this direction. For the Cauchy problem of the equation
\[ (\mathbb {D}^{(\alpha)}u)(t)=A(t^\alpha)u(t), \quad t>0 \]
with the Caputo-Dzhrbashyan fractional derivative and an entire function \(A(z)\), the author shows that for the solution, known to be of the form \(u(t)=U(t^\alpha)\) with an entire function \(U(z)\), the order of \(U(z)\) does not exceed \(\frac{1}{\alpha}(1+\deg(A))\). This estimation is sharp in a certain sense because it is exact when \(\alpha=1\) or \(\deg(A)=0\).
Further, the author studies systems of fractional equations of the form
\[ t^\alpha(\mathbb {D}^{(\alpha)}u)(t)=A(t^\alpha)u(t),\quad t>0 \]
where \(A(z)\) is a holomorphic matrix function, and also a similar system with the Riemann-Liouville fractional derivative. Under some assumptions he shows that the formal series solution converges near the origin, and he develops an analog to the Frobenius method. In the particular case \(A(z)=\text{const}\) he reveals an interesting effect when the formal series is not a distribution solution, in this case \(\alpha\) is an irrational number slowly approximated by rational ones.

Reviewer: Stefan G. Samko (Faro)

MSC:

26A33	Fractional derivatives and integrals
34M99	Ordinary differential equations in the complex domain

Keywords:

fractional differential equation; Riemann-Liouville derivative; Caputo-Dzhrbashyan fractional derivative; regular singularity; irregular singularity

Citations:

Zbl 1121.34008