Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

Vladimir E Fedorov,Wei-Shih Du,Aliya A Abdrakhmanova,Marko Kostić

doi:10.3390/math10050681

Abstract

Some new necessary and sufficient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed fractional derivatives in the initial conditions to corresponding inhomogeneous equations. These abstract results are applied to a class of initial boundary value problems for equations with distributed derivatives in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables.

Highlights

The main goal of this work is the study of the unique solvability issues for a special initial value problem to a class of equations with a distributed Riemann–Liouville derivative
Equations with distributed fractional derivatives appear in various fields of investigations applied to the mathematical modelling of some real processes, when an order of a fractional derivative in a model continuously depends on the process parameters: in the kinetic theory [3], in the theory of viscoelasticity [4] and so on [5,6,7]
It is shown that a natural initial value problem for this equation is a problem with given values of special form distributed derivatives of a solution at initial time

Summary

Introduction

The main goal of this work is the study of the unique solvability issues for a special initial value problem to a class of equations with a distributed Riemann–Liouville derivative. The statement of initial value problem (2) for Equation (1) with the Riemann–Liouville derivative is obtained and properties of functions, which arise when applying the Laplace transform to the distributed fractional derivative, are investigated. A theorem on conditions for the operator A, which are necessary and sufficient for the existence of analytic in a sector resolving family of operators of homogeneous Equation (1) is proved in the fourth section This result was applied to studying problem (1), (2) in the fifth section. The last section contains an application of obtained abstract results to a study of a class of initial boundary value problems for equations with a distributed fractional derivative in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables

Equation with Distributed Riemann–Liouville Derivative

Analytic in a Sector Function with a Power Singularity at Zero

Inhomogeneous Equation

Conclusions

Methods