Geodesic
Initial-value problem for distributed-order equations with a bounded operator
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 14-22

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Using methods of the theory of the Laplace transform, we prove a theorem on the existence of a unique solution to an initial-value problem for a distributed-order differential equation in a Banach space, which involves a fractional Riemann—Liouville derivative and a bounded operator acting on the unknown function. We find this solution in the form of Dunford–Taylor-type integrals. The results obtained contribute to the theory of resolving operator families for equations in Banach spaces, including fractional-order differential equations and evolutionary integral equations; in particular, we generalize some results of the theory of semigroups of operators to the case of equations of distributed order. Abstract results for equations in Banach spaces are applied to a class of initial-boundary-value problems for distributed-order partial differential equations with polynomials in a self-adjoint elliptic differential operator with respect to the spatial variables.
Keywords: distributed-order equation, fractional Riemann–Liouville derivative, initial-value problem, initial-boundary-value problem.
Mots-clés : Laplace transform 
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     author = {V. E. Fedorov and A. A. Abdrakhmanova},
     title = {Initial-value problem for distributed-order equations with a bounded operator},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {14--22},
     publisher = {mathdoc},
     volume = {188},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_188_a1/}
}
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V. E. Fedorov; A. A. Abdrakhmanova. Initial-value problem for distributed-order equations with a bounded operator. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 14-22. http://geodesic.mathdoc.fr/item/INTO_2020_188_a1/