418 research outputs found
Nonlocal Operational Calculi for Dunkl Operators
The one-dimensional Dunkl operator with a non-negative parameter ,
is considered under an arbitrary nonlocal boundary value condition. The right
inverse operator of , satisfying this condition is studied. An operational
calculus of Mikusinski type is developed. In the frames of this operational
calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy
boundary value problems for Dunkl functional-differential equations
with a given polynomial is proposed. The solution of these equations in
mean-periodic functions reduces to such problems. Necessary and sufficient
condition for existence of unique solution in mean-periodic functions is found
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Time-Dependent Fluid-Structure Interaction
The problem of determining the manner in which an incoming acoustic wave is
scattered by an elastic body immersed in a fluid is one of central importance
in detecting and identifying submerged objects. The problem is generally
referred to as a fluid-structure interaction and is mathematically formulated
as a time-dependent transmission problem. In this paper, we consider a typical
fluid-structure interaction problem by using a coupling procedure which reduces
the problem to a nonlocal initial-boundary problem in the elastic body with a
system of integral equations on the interface between the domains occupied by
the elastic body and the fluid. We analyze this nonlocal problem by the Lubich
approach via the Laplace transform, an essential feature of which is that it
works directly on data in the time domain rather than in the transformed
domain. Our results may serve as a mathematical foundation for treating
time-dependent fluid-structure interaction problems by convolution quadrature
coupling of FEM and BEM
Unraveling Forward and Backward Source Problems for a Nonlocal Integrodifferential Equation: A Journey through Operational Calculus for Dzherbashian-Nersesian Operator
This article primarily aims at introducing a novel operational calculus of
Mikusi\'nski's type for the Dzherbashian-Nersesian operator. Using this
calculus, we are able to derive exact solutions for the forward and backward
source problems (BSPs) of a differential equation that features
Dzherbashian-Nersesian operator in time and intertwined with nonlocal boundary
conditions. The initial condition is expressed in terms of Riemann-Liouville
integral (RLI). Solution is presented using Mittag-Leffler type functions
(MLTFs). The outcomes related to the existence and uniqueness subject to
certain conditions of regularity on the input data are established.Comment: 13 page
Weyl and Marchaud derivatives: a forgotten history
In this paper we recall the contribution given by Hermann Weyl and Andr\'e
Marchaud to the notion of fractional derivative. In addition we discuss some
relationships between the fractional Laplace operator and Marchaud derivative
in the perspective to generalize these objects to different fields of the
mathematics.Comment: arXiv admin note: text overlap with arXiv:1705.00953 by other author
Nonlocal Operational Calculi for Dunkl Operators
The one-dimensional Dunkl operator Dk with a non-negative parameter k, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of Dk, satisfying this condition is studied. An operational calculus of Mikusinski type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations P(Dk)u = f with a given polynomial P is proposed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found
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