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.

Over-order multiplicities and their application in controlling delay dynamics. On zeros’ distribution of linear combinations of Kummer hypergeometric functions

A series of recent works have shown that, for a system of linear functional differential equations, a spectral value having a multiplicity exceeding the order of the system tends to correspond to the spectral abscissa of the system, a property called MID for multiplicity-induced-dominancy. In particular, when this multiplicity coincides with the degree of the characteristic quasipolynomial, this property is called generic MID (GMID), in opposition to the intermediate MID (IMID), which corresponds to a multiplicity strictly smaller than the degree. The GMID has been fully characterized for single-delay retarded as well as neutral delay-differential equations thanks to the representation of the corresponding quasipolynomial in terms of a Kummer hypergeometric function. However, apart from partial results, in full generality, no result of the literature enables the characterization of the dominance of a spectral value having an intermediate multiplicity, which is essentially due to the lack of existing results among the open literature pertaining to linear combinations of Kummer functions’ zeros distribution. In this work, we overcome this difficulty and we further investigate the MID to cover the so-called over-order MID, that is, the cases where the multiplicity is larger than the order of the corresponding differential equation.

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

The Dirichlet Problem for an Elliptic Functional Differential Equation with the Compressed, Expanded, and Rotated Argument

The paper is devoted to the Dirichlet problem in a plain bounded domain for a linear divergent-form second-order functional differential equation with the compressed (expanded) and rotated argument of the highest derivatives of the unknown function. Necessary and sufficient conditions for the Gårding-type inequality are obtained in algebraic form. The result may depend not only on the absolute value of the coefficients but also on their signature. Under some restrictions on the structure of the operator and the geometry of the domain, the questions of existence, uniqueness, and smoothness of generalized solutions are studied for all possible values of the coefficients and parameters of transformations in the equation, even when the equation is not strongly elliptic.

Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

The Interplay Between Linear Functional Equations and Inequalities

We study the connections between the equation ∑i=1naif(αix+(1-αi)y)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{i=1}^na_if(\\alpha _ix+(1-\\alpha _i)y)=0 \\end{aligned}$$\\end{document}and the corresponding inequality. ∑i=1naif(αix+(1-αi)y)≥0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{i=1}^na_if(\\alpha _ix+(1-\\alpha _i)y)\\ge 0. \\end{aligned}$$\\end{document}At present, it is clear that linear functional equations should be solved with the use of the results due to L. Székelyhidi. We show that the simplest and most efficient way of dealing with the (continuous) solutions of linear inequalities of the above form is connected with the use of stochastic orderings tools. It will be shown that, in order to solve the inequality, we need to know the continuous solutions of the corresponding equation. In the last part of the paper, we obtain some simple but unexpected connections in the other direction.

Linear Functional Equations and Twisted Polynomials

Linear Functional Equations and Twisted Polynomials

ФУНКЦИОНАЛЬНЫЕ УРАВНЕНИЯ КАК МАТЕМАТИЧЕСКИЕ МОДЕЛИ ЗАДАЧ СОПРЯЖЕНИЯ С ЦИКЛИЧЕСКИМ СДВИГОМ НА СЛОЖНЫХ КРИВЫХ

The paper considers linear functional equations with a shift function having a nonzero derivative satisfying the Helder condition on an arbitrary piecewise smooth curve. Such equations are studied in connection with the theory of boundary value problems for analytical functions, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. The shift function acts cyclically on a set of simple curves forming a given curve, and except the ends of simple curves, there are no periodic points relative to the shift function. The purpose of the study is to find conditions for the existence and uniqueness of a solution (and in the case of non–uniqueness of the cardinality of the set of solutions) of such equations in the classes of Helder and primitive Lebesgue functions with a coefficient and the right part of the same classes.

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

Continuous Solutions of Linear Functional Equations on Piecewise Smooth Curves in Mathematical Models of Boundary Value Problems with a Shift

Boundary-value problem for an elliptic functional differential equation with dilation and rotation of arguments

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.

Robustness of exponential stability of a class of switched positive linear systems with time delays

SummaryThis paper investigates the robustness of exponential stability of a class of positive switched systems described by linear functional differential equations (FDE) under arbitrary switching or average dwell time switching. We will measure the stability robustness of such a system (which is considered as a nominal system) subject to parameter affine perturbations of its constituent subsystems matrices, by introducing the notion of structured stability radius. Some formulas for computing this radius, as well as estimating its lower bounds and upper bounds, are established. In the case of switched linear systems with multiple discrete time‐delays or/and distributed time‐delays, the obtained results yield tractably computable formulas or bounds for the stability radius. The extension of the obtained results to non‐positive systems and the class of multi‐perturbations has been presented. Examples are given to illustrate the proposed method.

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

On the solution of linear functional two-term equations with shift

This work is dedicated to the study of linear functional equations with shift in Hölder space. Previously, for such operators, conditions for invertibility were found and the inverse operator was constructed by the authors. The operators are used in modeling systems with renewable resources. Here we propose another approach to solving functional equations with shift. With the help of an algorithm, the initial equation is reduced to the first iterated equations, then to the second iterated equation. Continuing this process, we obtain the n-th iterated equation and the limit iterated equation. We prove the theorem on the equivalence of the original and the iterated equations. Based on the analysis of the solvability of the limit equation, we find a solution to the original equation. The solution is the sum of an infinite product and a functional series. The results, and the methods for obtaining them, are transparent and not as cumbersome compared to previous works.

Some remarks on the solution of the cell growth equation

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al . 2015 Solutions to an advanced functional partial differential equation of the pantograph type ( Proc. R. Soc. A 471 , 20140947 ( doi:10.1098/rspa.2014.0947 )). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.

Exploring the inhibitory effect of H2O on CO2/CH4 adsorption in coal: Insights from experimental and simulation approaches

To investigate the inhibitory effect of H2O on the adsorption of CH4/CO2 in coal, the anthracite was selected for physical adsorption experiments on CH4/CO2 gases in this study. Combined with Materials Studio software, the effect of moisture on the isothermal adsorption characteristics of CH4/CO2 in coal was investigated by using density functional theory and giant canonical Monte Carlo simulation. The intermolecular interaction energies were obtained, as well as the effects of electrostatic potentials, adsorption sites, and hydrogen bonding of oxygen-containing functional groups. The results show that: ①According to the CH4/CO2 isothermal adsorption experiments, it was found that moisture significantly inhibited the adsorption of CH4/CO2 by coal. For CH4, the relationship between adsorption constants a/b and moisture content can be expressed by the equations y = 43.447-3.348x and y = 3.225-0.751x. For CO2, adsorption constant a follows a power function relationship with moisture content, expressed as y = 29.114x-0.063, while adsorption constant b is represented by the polynomial fitting equation y = 1.672+0.508x+0.083x2. ② The findings have revealed correlations between the moisture influence coefficient and moisture content. The exponential functional equation η = exp(-0.188ω) was found to be more accurate for coal adsorbing CH4, whereas the linear functional equation η = 1/(1 + 0.243ω) was considered more appropriate for CO2. ③ The van der Waals and electrostatic energies between coal and CH4/CO2 decreased with an increase in moisture content, whereas the van der Waals and electrostatic energies between coal and H2O increased. ④ The orders of hydrogen bond formation ability between CH4/CO2 and functional groups were as follows: –CHO > –COOH > –OH > –CH3 and –OH > –COOH > –CHO > –CH3. The ability of H2O to form hydrogen bonds with –COOH, –OH, –CHO, and –CH3 is as follows: –COOH > –OH > –CHO> –CH3. This study provides a theoretical basis for the integrated use of hydraulic fracturing techniques and coalbed gas injection and replacement.

Precise Conditions on the Unique Solvability of the Linear Fractional Functional Differential Equations Related to the ς-Nonpositive Operators

Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to ς-nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are illustrated by an example of the pantograph-type model from electrodynamics.

Computer assisted investigation of alienness of linear functional equations

AbstractIn this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations.

Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part

ABSTRACT In this paper new oscillatory criteria for odd order linear functional differential equations of the type y ( n ) ( t ) + p ( t ) y ( τ ( t ) ) = 0 \[{{y}^{(n)}}(t)+p(t)y(\tau (t))=0\] have been established. Deviating argument τ(t) is supposed to have dominating delay part.

Bohr–Neugebauer Type Result for a Class of Nondensely Nonautonomous Linear Partial Functional Differential Equations with Infinite Delay

Bohr–Neugebauer Type Result for a Class of Nondensely Nonautonomous Linear Partial Functional Differential Equations with Infinite Delay

Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations

In this paper we investigate the mean square asymptotic stability of a perturbed scalar linear stochastic functional differential equation. Specifically, we are able to give necessary and sufficient conditions on the forcing terms for convergence of the mean square, exponential convergence of the mean square, and integrability of the mean square of solutions. It is also essential that the underlying unperturbed SFDE is mean–square asymptotically stable for these results to hold.