Related papers: Differential Equations for F_q-Linear Functions
We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances.…
In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
Permutation polynomials with explicit constructions over finite fields have long been a topic of great interest in number theory. In recent years, by applying linear translators of functions from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, many…
Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form $Ly = 0$ where $L$ is a linear differential operator of integral order. (Cf., for instance,…
We propose a sufficient condition of the convergence of a power-log series that formally satisfies an algebraic ordinary differential equation (ODE) of arbitrary order. A general form and properties of the functional coefficients of such a…
Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
Invariants of generalized tensor fields on a line are classified using special polynomials P_mk^(-1/lambda) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is…
It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
In view of the role of reaction equations in physical problems, the authors derive the explicit solution of a fractional reaction equation of general character, that unifies and extends earlier results. Further, an alternative shorter…
Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These…
We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations…
In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.
A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be…
In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the…
A class of one-dimensional Fokker-Plank equations having a common stationary solution, which is a power function of the state of the process, was found. We prove that these equations also have generalized self-similar solutions which…
This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of…