Systems Of Linear Functional Equations Research Articles

ON A BOUNDARY VALUE PROBLEM FOR HIGH-ORDER HYPERBOLIC EQUATION WITH IMPULSE DISCRETE MEMORY

The investigation focuses on a boundary value problem for a high-order hyperbolic equation with impulse discrete memory in a rectangular domain. By introducing new functions, the problem is transformed into a set of boundary value problems for a first-order differential equation with impulse discrete memory, which depends on unknown functions and integral relations. D.S. Dzhumabaev's parametrization method is applied to this equivalent problem. The domain is divided according to the time variable, and functional parameters representing discrete memory values are introduced within the interior domains. As a result, the family of boundary value problems for the first-order differential equation with impulse discrete memory and unknown functions is converted into an equivalent family of integral-multipoint boundary value problems involving functional parameters and unknown functions. These equivalent problems include initial value problems for first-order differential equations related to the new functions. The solutions to the initial value problems are expressed using Volterra integral equations. By substituting these solutions into the boundary and impulse conditions, a system of linear functional equations concerning the functional parameters is derived. An algorithm is developed to solve the equivalent problem, and sufficient conditions for the unique solvability of the family of integral-multipoint boundary value problems with functional parameters and unknown functions are provided. Additionally, sufficient conditions for the unique solvability of the original boundary value problem for the high-order hyperbolic equation with impulse discrete memory are established based on the initial data.

On systems of linear functional equations of the second kind in L 2

We consider the systems of linear functional equations of the third kind with measurecompact operators in L 2 and the general systems of linear integral equations of the third kind in L 2. We propose a method for reducing these systems either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. Various exact and approximate methods for solving the arising integral equations are applicable.

On systems of linear functional equations of the third kind in L 2

On systems of linear functional equations of the third kind in L 2

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gröbner bases

This paper is a sequel to “Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases” (Levandovskyy and Schindelar, 2011). We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of G-algebras (OLGAs). We use the splitting of the computation of a normal form for matrices over Ore localizations into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a given matrix M over an OLGA R, we provide a diagonalization algorithm to compute U,V and D with fraction-free entries such that UMV=D holds and D is diagonal. The fraction-free approach allows to obtain more information on the associated system of linear functional equations and its solutions, than the classical setup of an operator algebra with coefficients in rational functions. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over K[x] and use them in the unimodularity analysis of transformation matrices U,V. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a smaller polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system Singular:Plural follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. In particular, we experience a moderate swell of coefficients and obtain simple transformation matrices. Thus the method we propose is well suited for solving nontrivial practical problems.

A Monomial-by-Monomial Method for Computing Regular Solutions of Systems of Pseudo-Linear Equations

This paper deals with the local analysis of systems of pseudo-linear equations. We define regular solutions and use this as a unifying theoretical framework for discussing the structure and existence of regular solutions of various systems of linear functional equations. We then give a generic and flexible algorithm for the computation of a basis of regular solutions. We have implemented this algorithm in the computer algebra system Maple in order to provide novel functionality for solving systems of linear differential, difference and q-difference equations given in various input formats.

On extension of solutions of a simultaneous system of iterative functional equations

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \[ \varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),\] \[\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541]).

Algebraic splines in locally convex spaces

In a vector space of continuous functions, a variational solution of a finite system of linear functional equations is found. The locally convex topology on the vector space and the properties of the objective functional required for obtaining the solution in the form of a decomposition in the basis dual to the family of functionals of the system are determined. The basis elements are calculated exactly and called basis algebraic splines; their linear span is called the space of algebraic splines in the corresponding locally convex space.

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations μ n (f) = b n for n ∈ N has a real analytic solution f on ω C R d for every right-hand side (b n ) n ∈ N ⊆ C and give a complete characterization of such sequences of analytic functionals (μ n ). We also show that every open set w C R d has a complex neighbourhood Ω C C d such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on Ω.

On systems of linear functional equations

On systems of linear functional equations

On systems of linear functional equations

On systems of linear functional equations

Inverse problem for the reduced wave equation with fixed incident field

The inverse problem for the reduced wave equation Δu+k2n2(x) u=0 is examined when n (x) is assumed to be real continuous and equal to unity outside some prescribed compact domain D0 in R3. The case where a finite set of measurements of the scattered or total field (generated by a single incident wave at a fixed frequency) at points outside D0 is considered. It is shown that the inverse problem can be expressed in terms of a system of linear functional equations, plus a quadratic nonlinear integral equation. By imposing an additional criterion for uniqueness of the functional equations and placing certain restrictions on the size of the measured data, a solution of the system can be generated by successive approximations. It is shown that the iterative scheme yields correction terms to the solution that are obtained by the common linearized approximation where n (x) is assumed to be a very small perturbation of a known quantity and only linear terms are retained.