Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations

Fedorov, Vladimir E.

doi:10.3390/math8081306

Cited by 11 publications

(16 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let b∈ (0, 1], ω ∈ C([0, b]; R), ω(α) ≥ 0 at α ∈ [0, b), ω(b) > 0, the spectrum σ(Λ 1 ) do not contain the origin and zeros of the polynomial P n (λ), u 0 ∈ D M . Then, there exists a unique solution of problem (16)-(18). From the proof of Lemma 6, it follows that we can obtain the analogous result for a case b ∈ (1, 2], if we provide the positivity of the value min k∈N,Reλ>a…”

mentioning

confidence: 61%

See 1 more Smart Citation

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

Fedorov

Filin

2021

Fractal Fract

Self Cite

View full text Add to dashboard Cite

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

show abstract

mentioning

confidence: 61%

“…Sufficient and necessary conditions of an unbounded closed densely defined operator A for the existence of an analytic resolving family of operators for Equation (2) are found in [17][18][19]. Thus, an extension of the theorem on generators of analytic semigroups of operators to the case of distributed order equations is obtained.…”

Section: Introductionmentioning

confidence: 97%

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

Fedorov

Filin

2021

Fractal Fract

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note that all the aforementioned studies adopt the assumptions Hypothesis 1–3. Very recently, Fedorov studied linear DODEs that violate Hypothesis 2 resulting in an unbounded operator [ 69 ]. This study expanded the application of DODEs to initial and boundary value problems of ultra-slow diffusion.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

View full text Add to dashboard Cite

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

show abstract

“…By the choice of the space X we take into account Equation 14and condition (12). The function r(•, t) is a gradient, since it belongs to the space H π at t ≥ 0.…”

Section: Function Of the Fluid Velocitymentioning

confidence: 99%

“…Integer or fractional-order degenerate differential equations, i.e., evolution equations not solved with respect to the highest order derivative, are often used to describe various processes in science and engineering: in [9,10] certain classes of the time-fractional order partial differential equations with polynomials differential with respect to the spatial variables elliptic self-adjoint operator, which contain some equations from hydrodynamics and the filtration theory, are studied. In [11] approximate controllability issues for such models are investigated; the unique solvability of similar equations with distributed order time derivatives are researched in [12].…”

Section: Introductionmentioning

confidence: 99%

A Class of Fractional Degenerate Evolution Equations with Delay

Debbouche

Fedorov

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays.

show abstract

Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations

Cited by 11 publications

References 20 publications

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

Applications of Distributed-Order Fractional Operators: A Review

A Class of Fractional Degenerate Evolution Equations with Delay

Contact Info

Product

Resources

About