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Subordination Principle for a Class of Fractional Order Differential Equations

Author

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  • Emilia Bazhlekova

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia 1113, Bulgaria)

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.

Suggested Citation

  • Emilia Bazhlekova, 2015. "Subordination Principle for a Class of Fractional Order Differential Equations," Mathematics, MDPI, vol. 3(2), pages 1-16, May.
  • Handle: RePEc:gam:jmathe:v:3:y:2015:i:2:p:412-427:d:50202
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