Abstract
Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.
Highlights
Equations with distributed order derivatives appear in various applied problems concerning certain physical or technical processes—for example, when processes are described by equations with fractional derivatives, the order of which depends on the process parameters: in the theory of viscoelasticity [1], in kinetic theory [2], for modelling diffusion with a logarithmic growth of the mean square displacement [3] and so on (e.g., [4,5,6,7])
In many scientific works equations with distributed fractional derivatives began to be investigated from the mathematical point of view: unique solvability, a qualitative behaviour of solutions
The abstract results are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable
Summary
Equations with distributed order derivatives appear in various applied problems concerning certain physical or technical processes—for example, when processes are described by equations with fractional derivatives, the order of which depends on the process parameters: in the theory of viscoelasticity [1], in kinetic theory [2], for modelling diffusion with a logarithmic growth of the mean square displacement (ultraslow diffusion) [3] and so on (e.g., [4,5,6,7]). Equation (1) were proved, and the obtained results were applied to the study of initial boundary value problems for a class of partial differential equations of distributed order with respect to time. In the present paper we generalised the results of [19] on generators of analytic resolving families of operators and the unique solvability of inhomogeneous Equation (1) for the case of arbitrary b > 0. The obtained results are an extension of the analytic semigroup of operators theory to the case of distributed order equations. The generation theorem for analytic resolving families of operators for the distributed order Equation (2) is proved. In the fourth section the perturbation theorem for generators of the resolving families of operators for Equation (2) is obtained. The abstract results are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable
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