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Zbl 1176.34008
Deng, Weihua; Li, Changpin; Guo, Qian
Analysis of fractional differential equations with multi-orders.
(English)
[J] Fractals 15, No. 2, 173-182 (2007). ISSN 0218-348X

Summary: We study two kinds of fractional differential systems with multi-orders. One is a system of fractional differential equations with multi-order, $D^{\bar {\alpha }}_* \bar x (t) = \bar f (t,\bar x), \bar x (0) = \bar x_0$; the other is a multi-order fractional differential equation, $D^{\beta ^*}_* y(t) = g (t,y(t)), D^{\beta _1}_* y(t), \cdots , D^{\beta _n}_* y(t)$. By the derived technique, such two kinds of fractional differential equations can be changed into equations with the same fractional orders providing that the multi-orders are rational numbers, so the known theorems of existence, uniqueness and dependence upon initial conditions are easily applied. And asymptotic stability theorems for their associate linear systems, $D^{\bar {\alpha }}_* \bar x (t) = A\bar x (t),\bar x(0) = \bar x(0)$, and $D^{\beta 1*}_* y(t) = a_0 y(t) + \sum ^n_{i=1} a_i D^{\beta _i}_* y(t), y^{(k)} (0) = y^{(k)}_0, k = 0, 1, \cdots , \lceil \beta ^* \rceil - 1$, are also derived.
MSC 2000:
*34A25 Analytical theory of ODE
34M99 Differential equations in the complex domain

Keywords: Caputo fractional derivative; multi-order fractional differential systems; stability

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