Solutions Of Functional Differential Equations Research Articles

Dirichlet Solutions of Functional Differential Equations without Delay

Necessary and sufficient conditions for the existence of a valid Dirichlet solution were obtained. A method was developed to find Dirichlet solutions of the functional differential equation with non-delayed linear argument deviation.

Intelligent computing with the knack of Bayesian neural networks for functional differential systems in Quantum calculus model

The purpose behind this research is to utilize the knack of Bayesian solver to determine numerical solution of functional differential equations arising in the quantum calculus models. Functional differential equations having discrete versions are very difficult to solve due to the presence of delay term, here with the implementation of Bayesian solver with means of neural networks, an efficient technique has been developed to overcome the complication in the model. First, the functional differential systems are converted into recurrence relations, then datasets are generated for converted recurrence relations to construct continuous mapping for neural networks. Second, the approximate solutions are determined through employing training and testing steps on generated datasets to learn the neural networks. Furthermore, comprehensive statistical analysis are presented by applying various statistical operators such as, mean squared error (MSE), regression analysis to confirm both accuracy as well as stability of the proposed technique. Moreover, its rapid convergence and reliability is also endorsed by the histogram, training state and correlation plots. Expected level for accuracy of suggested technique is further endorsed with the comparison of attained results with the reference solution. Additionally, accuracy and reliability is also confirmed by absolute error analysis.

Uniqueness and existence of positive periodic solutions of functional differential equations

<abstract><p>In this paper, some new findings on the uniqueness and existence of positive periodic solutions to first-order functional differential equations are presented. These equations have wide applications in a variety of fields. The most important feature of our argument is that we use the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution when $ q=-1 $ and $ -1 < q < 0 $. In addition, we also investigate the existence results of positive periodic solutions by applying a fixed point theorem for completely continuous maps in a cone. Two examples demonstrate our findings.</p></abstract>

On meromorphic solutions of functional-differential equations

AbstractWe consider meromorphic solutions of functional-differential equations \[ f^{(k)}(z)=a(f^{n}\circ g)(z)+bf(z)+c, \]where $n,\,~k$ are two positive integers. Firstly, using an elementary method, we describe the forms of $f$ and $g$ when $f$ is rational and $a(\neq 0)$, $b$, $c$ are constants. In addition, by employing Nevanlinna theory, we show that $g$ must be linear when $f$ is transcendental and $a(\neq 0)$, $b$, $c$ are polynomials in $\mathbb {C}$.

Periodic and bounded solutions of functional differential equations with small delays

We study existence and local uniqueness of periodic solutions of nonlinear functional differential equations of first order with small delays. Bifurcations of periodic and bounded solutions of particular periodically forced second-order equations with small delays are investigated as well.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

English

English

On the Existence of Periodic and Bounded Solutions for Functional Differential Equations of Pointwise Type with a Strongly Nonlinear Right-Hand Side

Solutions of functional differential equation of pointwise type (FDEPT) are in one-to-one correspondence with the traveling-wave type solutions for the canonically induced infinite-dimensional ordinary differential equation and vice versa. In particular, such infinite-dimensional ordinary differential equations are finite difference analogues of equations of mathematical physics. An important class of traveling-wave type solutions is made up of periodic and bounded traveling-wave type solutions. On the other hand, an important class of such systems is systems with strongly nonlinear potentials (polynomial potentials), for which periodic and bounded traveling wave solutions are studied. Such a problem is equivalent to the study of periodic and bounded solutions of the induced FDEPT to which the present work is devoted.

Numerical Methods for Constructing Solutions of Functional Differential Equations of Pointwise Type

The article discusses construction of traveling wave type solutions for the Frenkel-Kontorova model on the propagation of longitudinal waves. For the first time, based on the existence and uniqueness theorem of traveling wave type solutions, as well as the approximation theorem, a complete family of traveling wave type solutions is constructed in the form of subfamilies of bounded solutions (horizontals) and unbounded solutions (verticals).

A new approach to the question of the existence of bounded solutions of functional differential equations of point type

We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions of ordinary differential equations and functional differential equations of point type. These conditions are based on the use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutions and stated in terms of averages over the period on a distinguished sphere in the phase space. The development of this approach enables us to obtain conditions for the existence of bounded solutions for the same classes of functional differential equations.

A new approach to the question of the existence of bounded solutions of functional differential equations of point type

Работа является развитием подхода, ранее использованного автором при выводе условий нового типа для существования периодических решений как для обыкновенных дифференциальных уравнений, так и для функционально-дифференциальных уравнений точечного типа. Такие условия основаны на учете асимптотических свойств решений дифференциальных уравнений, которые могут наблюдаться на сдвигах решений и формулируются в терминах средних по периоду на какой-либо выделенной сфере фазового пространства. Для тех же классов функционально-дифференциальных уравнений развитие отмеченного подхода позволяет получить условия существования ограниченных решений. Библиография: 12 наименований.

On a step method and a propagation of discontinuity

In this paper we analyze how to compute discontinuous solutions for functional differential equations, looking at an approach which allows to study simultaneously continuous and discontinuous solutions. We focus our attention on the integral representation of solutions and we justify the applicability of such an approach. In particular, we improve the step method in such a way to solve a problem of vanishing discontinuity points. Our solutions are considered as regulated functions.

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

We study the existence of monotone wavefronts for a general family of bistable reaction–diffusion equations with delayed reaction term g. Differently from previous works, we do not assume the monotonicity of with respect to the delayed variable that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale–Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov–Schmidt reduction is done in appropriate weighted spaces of C2-smooth functions. This method requires a detailed analysis of associated linear differential operators and their formal adjoints. For two different types of -unimodal functions , we establish the existence of a maximal continuous family of bistable monotone wavefronts. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the obtained bistable waves: (1) independently on the size of delay, each bistable wavefront is monotone; (2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.

A new approach to the question of existence of periodic solutions for functional differential equations of point type

This paper is devoted to a new type of sufficient conditions for the existence of periodic solutions of ordinary differential equations and functional differential equations of point type. These conditions are based on the use of asymptotic properties of solutions of differential equations that have not been used in the study of periodic solutions because of considerations restricting such equations to an interval equal to the period.

On the Asymptotic Properties of Solutions of Functional-Differential Equations with Linearly Transformed Argument

On the Asymptotic Properties of Solutions of Functional-Differential Equations with Linearly Transformed Argument

New Qualitative Results for Solutions of Functional Differential Equations of Second Order

In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.

An Analysis upon the Theory and Elementary Approaches for the Solution of Functional Differential Equations: Special Aspects

An Analysis upon the Theory and Elementary Approaches for the Solution of Functional Differential Equations: Special Aspects

A Perron type theorem for positive solutions of functional differential equations

A Perron type theorem for positive solutions of functional differential equations

Numerical solution of functional differential equations: a Green's function-based iterative approach

ABSTRACTIn this article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential equations. The approach is especially suited for handling boundary value problems (BVPs). The algorithm is illustrated through a number of examples that confirm the high accuracy and efficiency of the strategy. The results of the test examples show excellent agreement with exact solutions and outperforms other existing numerical iterative schemes.

Positive Non-monotone Solutions of Second-Order Delay Differential Equations

In this paper, we study the non-monotonic behavior of solutions of the second-order delay differential equations. We establish the criteria for positive solutions to be non-monotonic on a bounded interval. The results are the first to be reported on the non-monotonic behavior of solutions of functional differential equations. In the final section, two examples are given which illustrate the significance of our results.