Linear Differential-difference Equations Research Articles

Picturing the Growth Order of Solutions in Complex Linear Differential–Difference Equations with Coefficients of φ-Order

Given an unbounded non-decreasing positive function φ, we studied what the relations are between the growth order of any solution of a complex linear differential–difference equation whose coefficients are entire or meromorphic functions of finite φ-order. Our findings extend some earlier well-known results.

Galois groups of linear difference-differential equations

We study the relation between the Galois group G of a linear difference-differential system and two classes C1 and C2 of groups that are the Galois groups of the specializations of the linear difference equation and the linear differential equation in this system respectively. We show that almost all groups in C1∪C2 are algebraic subgroups of G, and there is a nonempty subset of C1 and a nonempty subset of C2 such that G is the product of any pair of groups from these two subsets. These results have potential application to the computation of the Galois group of a linear difference-differential system. We also give a criterion for testing linear dependence of elements in a simple difference-differential ring, which generalizes Kolchin's criterion for partial differential fields.

MODELING STABILITY OF DIFFERENTIAL-DIFFERENCE EQUATIONS WITH DELAY

Differential-difference and differential-functional equations are mathematical models of ma\-ny applied problems in automatic control and management systems, chemical, biological, technical, economic and other processes whose evolution depends on prehistory. In the study of the problems of stability, oscillation, bifurcation, control, and stabilization of solutions of linear differential-difference equations, the location of the roots of the corresponding characteristic equations is very important. Note that there are currently no effective algorithms for finding the zeros of quasipolynomials. When studying the approximation of a system of linear differential-difference equations, it was found that the approximation of nonsymptotic roots of their quasi-polynomials can be found with the help of characteristic polynomials of the corresponding approximating systems of ordinary differential equations . This paper investigates the application of approximation schemes for differential-difference equations to construct algorithms for the approximate finding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with many delays. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer. Computational experiments on special test examples showed the high efficiency of the proposed algorithms for studying the stability of linear differential-difference equations.

Lucas Collocation Method for the Solution of Differential Difference Equations

In order to solve mth-order linear differential difference equations with variable coefficients under mixed conditions, this research suggests a combined operational matrix approach based on Lucas polynomials. The simplicity of the proposed method’s application is a benefit. The technique simplifies the provided problem by turning it into a matrix equation. Absolute errors are used to validate illustrative examples. The solutions are enhanced by residual error estimates. The results, which are displayed in graphs and tables, are contrasted with the accepted approaches in the literature.

Growth properties of meromorphic solutions of some higher-order linear differential-difference equations

Abstract This paper is devoted to the study of the growth of meromorphic solutions of homogeneous and non-homogeneous linear differential-difference equations ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = 0 , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=0, ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = F , \displaystyle\sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F, where A i ⁢ j {A_{ij}} ( i = 0 , … , n {i=0,\ldots,n} , j = 0 , … , m {j=0,\ldots,m} ), F are meromorphic functions and c i {c_{i}} ( 0 , … , n {0,\ldots,n} ) are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng.

Dynamics of fluctuations in quantum simple exclusion processes

We consider the dynamics of fluctuations in the quantum asymmetric simple exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP describes a chain of spinless fermions with random hoppings that are induced by a Markovian environment. We show that fluctuations of the fermionic degrees of freedom obey evolution equations of Lindblad type, and derive the corresponding Lindbladians. We identify the underlying algebraic structure by mapping them to non-Hermitian spin chains and demonstrate that the operator space fragments into exponentially many (in system size) sectors that are invariant under time evolution. At the level of quadratic fluctuations we consider the Lindbladian on the sectors that determine the late time dynamics for the particular case of the quantum symmetric simple exclusion process (Q-SSEP). We show that the corresponding blocks in some cases correspond to known Yang-Baxter integrable models and investigate the level-spacing statistics in others. We carry out a detailed analysis of the steady states and slow modes that govern the late time behaviour and show that the dynamics of fluctuations of observables is described in terms of closed sets of coupled linear differential-difference equations. The behaviour of the solutions to these equations is essentially diffusive but with relevant deviations, that at sufficiently late times and large distances can be described in terms of a continuum scaling limit which we construct. We numerically check the validity of this scaling limit over a significant range of time and space scales. These results are then applied to the study of operator spreading at large scales, focusing on out-of-time ordered correlators and operator entanglement.

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters . It is shown that, when is related to the solutions of a simple linear difference–differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form.

LP Approach to Constrained Stabilization of Linear Differential-Difference Equations with Unbounded Delay

LP Approach to Constrained Stabilization of Linear Differential-Difference Equations with Unbounded Delay

Growth of solutions of linear differential-difference equations with coefficients having the same logarithmic order

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic coefficients of finite logarithmic order. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Growth Properties of Solutions of Complex Linear Differential-difference Equations with Coefficients having the Same Logarithmic Order in the Unit Disc

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic cofficients of finite logarithmic order in the unit disc. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Implicit Linear Differential-Difference Equations in the Module of Formal Generalized Functions over a Commutative Ring

We study an implicit inhomogeneous linear differential-difference equation in the module of formal generalized functions over a commutative ring. We prove the well-posedness of the equation and find the fundamental solution. We obtain a representation of a unique solution to the equation in the form of the convolution of the fundamental solution and a given formal generalized function. We also consider the inhomogeneous second order differential equation over an arbitrary commutative ring.

Availability Analysis of Hybrid Systems Consisting of Main Units and Cold Standby Processors

The present paper studies availability of four hybrid systems configured as series-parallel systems. Each system or configuration consisting of main units and their corresponding processors. Configuration I consist of three processors is a 2-out-of-3 unit connected to 2-out-of-3 processors, Configuration II is a 2-out-of-3 unit connected to 2-out-of-4 processors, Configuration III is a 2-out-of-4 unit connected to 2-out-of-4 processors while Configuration IV is a 2-out-of-4 unit connected to 2-out-of-3 processors. The failure and repair times of units and their processors are assumed to be exponentially distributed. Explicit expressions for steady state availability are developed for each system using first order linear differential difference equations and validated by performing numerical experiments. Analysis of the effect of various system parameters on availability was performed. Graphical illustrations are given to highlight important results. The systems are ranked based on their availability and found that Configuration IV is better. Sensitivity analysis on the model’s outcomes are performed using partial rank correlation coefficients (PRCC) to determine the most critical parameters leading to increase (decrease) in the value of availability.

Stability of delay evolution equations with fading stochastic perturbations

ABSTRACT Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differential equations and linear stochastic difference equations. Here a similar statement is proven for a nonlinear stochastic evolution equation with quickly enough fading stochastic perturbations. More exactly: if the level of the stochastic perturbation is given by a continuous and square integrable function, then the zero solution of the considered exponentially stable deterministic system remains exponentially mean square stable independently on the maximum magnitude of the stochastic perturbations. Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown. Consideration of other types of fading stochastic perturbations (for instance, if stochastic perturbations fade on the infinity not so quickly) is an open problem.

Uniformly convergent numerical method for a singularly perturbed differential difference equation with mixed type

In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.

Моделювання крайових задач для лінійних диференціально-різницевих рівнянь нейтрального типу

In space navigation problems, optimal control of systems with aftereffect, ecology and immunology problems, boundary value problems for differential-difference and integro-differential equations with delays arise, which are an important part of modern theory of differential-functional equations. Finding precise solutions of boundary value problems for differential-difference equations is possible only for the simplest classes of such problems. At present, projection-iterative methods, numerical-analytical method and others are suggested for boundary value problems with delay and of neutral type. The spline-collocation method for solving boundary value problems for differential-difference equations is one of the most efficient algorithms that allows building simple computational schemes. In this paper, we investigate the scheme of modeling boundary value problems for linear differential-difference equations of neutral type with many variable deviations of the argument. A functional space is defined to which the solutions of the considered boundary value problems belong, the properties of the solution smoothness are investigated depending on the structure of the argument deviations. Simple and verifiable sufficient conditions for the boundary value problem solution existence are given. For finding the solution of the boundary value problem, an iterative computational scheme based on the spline approximation method is described. In order to take into account possible discontinuities of the boundary value problem solution derivatives, cubic splines of defect two are used for neutral-type equations. Coefficient conditions for the initial equation which ensure the convergence of the iterative process are obtained. An estimate of the approximate solution error is conducted. A model example of a boundary value problem for a neutral type differential-difference equation is presented on which the iterative scheme is demonstrated. Numerical experiments confirm the obtained theoretical results.

A New Continuous-Discrete Fuzzy Model and Its Application in Finance

In this paper, we propose a fuzzy differential-difference equation for modeling of mixed continuous-discrete phenomena. In the special case, we present the general solution of linear fuzzy differential-difference equations. The dynamical process in the intervals is presented by the corresponding fuzzy differential equation and with impulsive jumps in some points. We illustrate the applicability of the model to study the time value of money.

On the number of linearly independent admissible solutions to linear differential and linear difference equations

AbstractA classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then each solution base of the differential equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least $n-p$ entire functions of infinite order. Here, the transcendental coefficient $A_p$ dominates the growth of the polynomial coefficients $A_{p+1},\ldots ,A_{n-1}$ . By expressing the dominance of $A_p$ in different ways and allowing the coefficients $A_{p+1},\ldots ,A_{n-1}$ to be transcendental, we show that the conclusion of Frei’s theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that $0$ is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich’s theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex q-difference equations.

Cost–benefit analysis of three different series–parallel dynamo configurations

This paper deals with the reliability modelling and performance evaluation of three configurations arranged in series–parallel. Configuration I consist of six units in which four are on operational while two are on standby. Configuration II consist of seven units with three of the units are on standby while the remaining four are on operation. Configuration III comprises of two subsystems C and D with three unit in each subsystem with a unit on standby. Units in each configuration provide 25 MW. The failure and repair times are assumed to be exponentially distributed. Through the transition diagram, system of first-order linear differential difference equations are derived for each configuration and are used to obtain the corresponding explicit expressions of system availability and mean time to failure. Cost–benefit analysis is examined and compared among the configuration through numerical examples to determine the optimal configuration and it was found to be configuration I. This study is important to system designers and developers, maintenance personnel, engineers and plant management in the suitable in designing and analysis of maintenance policy and processes and also the assessment of performance and the safety of the systems in general during and after the burn-in period.

Necessary and Sufficient Conditions of Stability in the Quadratic Mean of Linear Stochastic Partial Differential-Difference Equations Subject to External Perturbations of the Type of Random Variables

We obtain the necessary and sufficient conditions for the stability in the quadratic mean of strong solutions to stochastic partial differential-difference equations with pairwise independent external random perturbations of the type of random variables.

Availability Analysis of a Complex System Consisting of Two Subsystems in Parallel Configuration with Replacement at Failure

In this paper, investigation of the profit of parallel system consisting of two subsystems A and B is carried out. Each of the subsystem consists of two dissimilar components in active parallel. Two different replacement models are considered. Model 1 addresses the availability of the system by considering group replacement of the two subsystems at the system failure. Model 2 addresses the availability of the system by considering individual replacement of subsystem at its failure. In order to examine the validity of the models introduced and conduct sensitivity analysis, some diagrams are drawn for each model. Through the linear differential difference equation, explicit expressions for steady state availability for the two models are derived. Numerical examples are presented to illustrate the obtained results and to analyse the effect of group and individual replacement on the availability.