1,803 research outputs found
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Optimal L2 estimates for semidiscrete Galerkin methods for\ud parabolic integro-differential equations with nonsmooth data
In this article, we discuss an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth initial data. It is based on energy arguments and on a repeated use of time integration, but without using parabolic type duality technique. Optimal L2-error estimate is derived for the semidiscrete approximation, when the initial data is in L2
Optimal error estimates of a mixed finite element method for\ud parabolic integro-differential equations with non smooth initial data
In this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to mixed methods for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments and without using parabolic type duality technique, optimal L2-error estimates are derived for semidiscrete approximations, when the initial data is in L2. Due to the presence of the integral term, it is, further, observed that estimate in dual of H(div)-space plays a role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof technique used for deriving optimal error estimates of finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, the proposed analysis can be easily extended to other mixed method for PIDE with rough initial data and provides an improved result
Collective Phenonema in the Interaction of Radiation with Matter
The aim of this communication is to present in a concentrated form the main
ideas of a method, developed by the author, for treating strongly
nonequilibrium collective phenomena typical of the interaction of radiation
with matter, as well as to give a survey of several applications of the method.
The latter is called the Scale Separation Approach since its basic techniques
rely on the possibility of separating different space-time scales in
nonequilibrium statistical systems. This approach is rather general and can be
applied to diverse physical problems, several of which are discussed here.
These problems are: Superradiance of nuclear spins, filamentation in resonant
media, semiconfinement of neutral atoms, negative electric current, and
collective liberation of light.Comment: 1 file, 16 pages, LaTe
Time-Fractional Optimal Control of Initial Value Problems on Time Scales
We investigate Optimal Control Problems (OCP) for fractional systems
involving fractional-time derivatives on time scales. The fractional-time
derivatives and integrals are considered, on time scales, in the
Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient
conditions for existence and uniqueness of solution to initial value problems
described by fractional order differential equations on time scales are known.
Here we consider a fractional OCP with a performance index given as a
delta-integral function of both state and control variables, with time evolving
on an arbitrarily given time scale. Interpreting the Euler--Lagrange first
order optimality condition with an adjoint problem, defined by means of right
Riemann--Liouville fractional delta derivatives, we obtain an optimality system
for the considered fractional OCP. For that, we first prove new fractional
integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book
chapter with Springer International Publishing AG. Submitted 23/Jan/2019;
revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial
text overlap with arXiv:1508.0075
Impulsive Nonlocal Neutral Integro-Differential Equations Controllability results on Time Scales
In this work, we studied the controllability results for neutral differential time-varying equation with impulses on time scales & extend these results into nonlocal controllability of neutral functional integro-differential time-varying equation with impulses on time scales. The solutions are obtained employing standard fixed point theorems
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