Subordination Principle for a Class of Fractional Order Differential Equations

Emilia Bazhlekova
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Emilia Bazhlekova: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria

Mathematics, 2015, vol. 3, issue 2, 1-16

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.

Keywords: Riemann–Liouville fractional derivative; \(C_0\)-semigroup of operators; Mittag–Leffler function; completely monotone function; Bernstein function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2015
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