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Mathematics 2015, 3(2), 412-427; doi:10.3390/math3020412

Subordination Principle for a Class of Fractional Order Differential Equations

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria
Academic Editor: Hari M. Srivastava
Received: 27 March 2015 / Accepted: 19 May 2015 / Published: 26 May 2015
(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)
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Abstract

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma>0\) and \(f\) is an \(X\)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \(C_{0}\)-semigroup, generated by the operator \(A\). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered. View Full-Text
Keywords: Riemann–Liouville fractional derivative; \(C_0\)-semigroup of operators; Mittag–Leffler function; completely monotone function; Bernstein function Riemann–Liouville fractional derivative; \(C_0\)-semigroup of operators; Mittag–Leffler function; completely monotone function; Bernstein function
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Bazhlekova, E. Subordination Principle for a Class of Fractional Order Differential Equations. Mathematics 2015, 3, 412-427.

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