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

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.

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Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations

Mathematics ◽

10.3390/math8081306 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1306 ◽

Cited By ~ 1

Author(s):

Vladimir E. Fedorov

Keyword(s):

Banach Space ◽

Unbounded Operator ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Perturbation Theorem ◽

Linear Differential ◽

Initial Boundary Value ◽

Families Of Operators ◽

Lower Order Terms ◽

Distributed Order

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.

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Existence, Uniqueness and Asymptotic Behaviour for a Non-Linear Beam Equation

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0242 ◽

1995 ◽

Author(s):

G. Judith Boertjens ◽

Wim T. van Horssen

Keyword(s):

Perturbation Methods ◽

Flexible Structures ◽

Initial Boundary ◽

Beam Equation ◽

Multiple Time ◽

Initial Boundary Value Problems ◽

Initial Boundary Value ◽

The Right ◽

Multiple Time Scale Method ◽

Asymptotic Validity

Abstract The use of perturbation methods for fourth order PDE’s has not yet been examined extensively. Usually approximating power series are applied, which are truncated to one or two modes. Very little — or nothing — is said about the relation between this approximation and the exact solution. In this paper initial boundary value problems for the following equation will be discussed: w t t + w x x x x + ϵ ( u ( π , t ) − u ( 0 , t ) + ∫ 0 π w x 2 d x ) w x x = ϵ g ( x , t , w , w t ) . This equation can be regarded as a model describing wind-induced oscillations of flexible structures like elastic beams, where the small term on the right hand side of the equation represents the windforce acting on the structure. Existence and uniqueness for solutions of these problems will be discussed, as well as finding approximations using a multiple time-scale method. Finally the asymptotic validity of these approximations will be considered.

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The pulsating orb: solving the wave equation outside a ball

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0037 ◽

2016 ◽

Vol 472 (2189) ◽

pp. 20160037 ◽

Cited By ~ 5

Author(s):

P. A. Martin

Keyword(s):

Wave Equation ◽

Laplace Transform ◽

Continuum Mechanics ◽

Weak Solutions ◽

Acoustic Waves ◽

Initial Boundary ◽

Initial Boundary Value Problems ◽

Compatibility Conditions ◽

The Laplace Transform ◽

Initial Boundary Value

Transient acoustic waves are generated by the oscillations of an object or are scattered by the object. This leads to initial-boundary value problems (IBVPs) for the wave equation. Basic properties of this equation are reviewed, with emphasis on characteristics, wavefronts and compatibility conditions. IBVPs are formulated and their properties reviewed, with emphasis on weak solutions and the constraints imposed by the underlying continuum mechanics. The use of the Laplace transform to treat the IBVPs is also reviewed, with emphasis on situations where the solution is discontinuous across wavefronts. All these notions are made explicit by solving simple IBVPs for a sphere in some detail.

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Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0217-x ◽

2014 ◽

Vol 17 (4) ◽

Cited By ~ 28

Author(s):

Zhiyuan Li ◽

Yuri Luchko ◽

Masahiro Yamamoto

Keyword(s):

Asymptotic Behavior ◽

Boundary Value Problems ◽

Boundary Value ◽

Initial Boundary ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Initial Boundary Value Problems ◽

Fractional Diffusion Equations ◽

Initial Boundary Value ◽

Distributed Order

AbstractThis article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1. Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.

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