Related papers: Schauder estimates for discrete fractional integra…
We consider a nonlinear parabolic equation of fractional order in space and propose its numerical discretization. The fractional derivative is defined through a functional analytic setting, rather than the traditional definition of…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
We survey some results on Lipschitz and Schauder regularity estimates for viscous Hamilton--Jacobi equations with subcritical L\'evy diffusions. The Schauder estimates, along with existence of smooth solutions, are obtained with the help of…
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…
We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…
Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
We investigate discretizations of the integrable discrete nonlinear Schr\"odinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary…
We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete,…
We develop a semi-discrete version of discrete variational mechanics with applications to numerical integration of classical field theories. The geometric preservation properties are studied.
We study integrated semigroups for infinite-dimensional differential-algebraic equations (DAEs) admitting a resolvent index. Building on the notion of integrated semigroups for the abstract Cauchy problem $\frac{d}{d t}x=Ax$, we extend this…
Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the…
This is a first version of a paper concerning abstract evolution equation with fractional time derivatives. Maximal regularity results in spaces of continuous and Hoelder continuous functions are described.
We improve local smoothing estimates for fractional Schr\"{o}dinger equations for $\alpha \in (0,1) \cup (1,\infty)$.
Starting from kicked equations of motion with derivatives of non-integer orders, we obtain "fractional" discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main…
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate…
In an earlier paper we introduced the notion of 'bifurcating continued fractions' in a heuristic manner. In this paper a formal theory is developed for the 'bifurcating continued fractions'.
The paper develops a method for discrete computational Fourier analysis of functions defined on quasicrystals and other almost periodic sets. A key point is to build the analysis around the emerging theory of quasicrystals and diffraction…
We show that discrete quasiprobability distributions defined via the discrete Heisenberg-Weyl group can be obtained as discretizations of the continuous $SU(N)$ quasiprobability distributions. This is done by identifying the phase-point…
This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Gr\"onwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three…