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Stevan Pilipovic

Publications and source records attributed to Stevan Pilipovic.

50 records · Page 3Linked to original sources
arXiv2011

Micro-local analysis in some spaces of ultradistributions

In this paper we extend some results from our earlier papers on wave-front sets, concerning wave-front sets of Fourier-Lebesgue and modulation space types, to a broader class of spaces of ultradistributions, and relate these wave-front sets with the usual wave-front sets of ultradistributions. Furthermore, we use Gabor frames for the description of discrete wave-front sets, and prove that these wave-front sets coincide with corresponding continuous ones.

math.FA↗
arXiv2011

Frames for weighted shift-invariant spaces

We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence $Φ^{2k+1}$ and the sequence of spaces $V^p_μ(Φ^{2k+1})$, $k\in{\mathbb{N}}$, on $\mathbb{R},$ with the useful properties in sampling, approximations and stability.

math.FA↗
arXiv2011

Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nöther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nöther's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws.

math.FA↗
arXiv2011

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense.

math.FA↗
arXiv2011

Generalized Hamilton's Principle with Fractional Derivatives

We generalize Hamilton's principle with fractional derivatives in Lagrangian $L(t,y(t),{}_0D_t^\al y(t),α)$ so that the function $y$ and the order of fractional derivative $α$ are varied in the minimization procedure. We derive stationarity conditions and discuss them through several examples.

math.FA↗
arXiv2010

Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of fractional distributed-order type for the special choice of weight functions. Prescribing boundary conditions on displacement, we obtain case corresponding to stress relaxation. In solving system of differential and integro-differential equations we use the Laplace transformation in the time domain.

math-ph↗
arXiv2009

Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I

Let $ω,ω_0$ be appropriate weight functions and $q\in [1,\infty ]$. We introduce the wave-front set, $\WF_{\mathscr FL^q_{(ω)}}(f)$ of $f\in \mathscr S'$ with respect to weighted Fourier Lebesgue space $\mathscr FL^q_{(ω)}$. We prove that usual mapping properties for pseudo-differential operators $\op (a)$ with symbols $a$ in $S^{(ω_0)}_{ρ, 0}$ hold for such wave-front sets. Especially we prove \WF_{\mathscr FL^q_{(ω/ω_0)}}(\op (a)f)\subseteq \WF_{\mathscr FL^q_{(ω)}}(f) \subseteq \WF_{\mathscr FL^q_{(ω/ω_0)}}(\op (a)f)\ttbigcup \Char (a). %% Here $\Char (a)$ is the set of characteristic points of $a$.

math.AP↗
arXiv2009

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of viscoelasticity and in the system identfica- tion theory. Also, the existence and uniqueness of a solution to a general linear fractional differential equation in the space of tempered distributions is given.

math.FA↗
arXiv2009

Micro-local analysis in Fourier Lebesgue and modulation spaces. Part II

We consider different types of (local) products $f_1 f_2$ in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$ \qquad P(x,D)f = G(x,J_k f), $$ with appropriate polynomials $P $ and $G$. If the solution locally belongs to appropriate weighted Fourier Lebesgue space ${\mathscr F}L^q_{(ω)} (\rr d)$ and $P$ is non-characteristic at $(x_0,ξ_0),$ then we prove that $(x_0,ξ_0)\not \in WF_{{\mathscr F}L^q_{(\widetilde ω)}} (f)$, where $\widetildeω$ depends on $ω$, $P$ and $G$.

math.FA↗
arXiv2007

Homogeneity in generalized function algebras

We investigate homogeneity in the special Colombeau algebra. It is shown that strongly scaling invariant functions on the d-dimensional space are simply the constants. On the pierced space, strongly homogeneous functions admit tempered representatives, whereas on the whole space, such functions are polynomials with generalized coefficients. We also introduce weak notions of homogeneity and show that these are consistent with the classical notion on the distributional level. Moreover, we investigate the relation between generalized solutions of the Euler differential equation and homogeneity.

math.GM↗
arXiv2004

Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

math.AP↗