Related papers: The integration theory of linear ordinary differen…
The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and…
Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
A symbolic computational algorithm which detects " linear "` solutions of nonlinear polynomial differential equations of single functions, is developed throughout this paper.
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.
Discusses several integrability tests for nonlinear evolution equations.
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
We introduce an elementary argument to the theory of distribution of sequences modulo one.
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative…
In this paper, we establish the theory of nonlinear rough paths. We give the definition of nonlinear rough paths, and develop the integrals. Then, we study differential equations driven by nonlinear rough paths. Afterwards, we compare the…
I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more…
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…
An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…
The wide application of estimation techniques in system analysis enable us to best determine and understand the history of system states. This paper attempts to delineate the theory behind linear and non-linear estimation with a suitable…
We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of…
In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals.