Type Of Functional Differential Equations Research Articles

Existence, uniqueness and stability of fuzzy delay differential equations with local Lipschitz and linear growth conditions

Fuzzy delay differential equation driven by Liu's process is a type of functional differential equations. In this paper, we are going to provide and prove a novel existence and uniqueness theorem for the solutions of fuzzy delay differential equation under Local Lipschitz and linear growth conditions. Also the stability of the solutions for fuzzy delay differential is investigated. Finally, to illustrate the main results we give some examples.

Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching

AbstractThis work is mainly concerned with the exponential stability of time-changed stochastic functional differential equations with Markovian switching. By expanding the time-changed Itô formula and the Razumikhin theorem, we obtain the exponential stability results for the time-changed stochastic functional differential equations with Markovian switching. What’s more, we get many useful stability results by applying our new results to several important types of functional differential equations. Finally, an example is given to demonstrate the effectiveness of the main results.

Performance Analysis of Workload Dependent Load Balancing Policies

Load balancing plays a crucial role in achieving low latency in large distributed systems. Recent load balancing strategies often rely on replication or use placeholders to further improve latency. However assessing the performance and stability of these strategies is challenging and is therefore often simulation based. In this paper we introduce a unified approach to analyze the performance and stability of a broad class of workload dependent load balancing strategies. This class includes many replication policies, such as replicate below threshold, delayed replication and replicate only small jobs, as well as strategies for fork-join systems. We consider systems with general job size distributions where jobs may experience server slowdown. We show that the equilibrium workload distribution of the cavity process satisfies a functional differential equation and conjecture that the cavity process captures the limiting behavior of the system as its size tends to infinity. We study this functional differential equation in more detail for a variety of load balancing policies and propose a numerical method to solve it. The numerical method relies on a fixed point iteration or a simple Euler iteration depending on the type of functional differential equation involved. We further show that additional simplifications can be made if certain distributions are assumed to be phase-type. Various numerical examples are included that validate the numerical method and illustrate its strength and flexibility.

A Natural Homotopy Analysis Method for Nonlinear Delay Differential Equations

Delay differential equation (DDEs) is a type of functional differential equation arising in numerous applications from different areas of studies, for example biology, engineering population dynamics, medicine, physics, control theory, and many others. However, determining the solution of delay differential equations has become a difficult task more especially the nonlinear type. Therefore, this work proposes a new analytical method for solving non-linear delay differential equations. The new method is combination of Natural transform and Homotopy analysis method. The approach gives solutions inform of rapid convergence series where the nonlinear terms are simply computed using He's polynomial. Some examples are given, and the results obtained indicate that the approach is efficient in solving different form of nonlinear DDEs which reduces the computational sizes and avoid round-off of errors.

The Collocation Method in the Numerical Solution of Boundary Value Problems for Neutral Functional Differential Equations. Part II: Differential Equations with Deviating Arguments

We consider the numerical solution of boundary value problems for neutral differential equations with deviating arguments by the collocation method, which includes the finite element method and the spectral element method. By using the results of Part I [SIAM J. Numer. Anal., 53 (2015), pp. 2771--2793] of this paper, we obtain concrete convergence theorems for this particular type of functional differential equations.

Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.

More on Simple Iterative Ordinary Differential Equation

Abstract Iterative ordinary differential equation is one type of functional differential equation. The study of this type of functional differential equation was started about 90 years ago. One of the pioneers who published a paper that was presented in 1921 was F.B. Fite, in [1]. Simple iterative ordinary differential equation is one type of such equation. This type of iterative differential equation was introduced in 2002 by M. Podisuk, in [6]. In this paper, nine simple iterative ordinary differential equations with their solutions are presented. The solution of the infinite order simple iterative ordinary differential equation is found.

Entire solutions of a certain type of functional-differential equations

In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.

Monotone‐Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone‐iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Boundary value problems for a class of impulsive functional equations

This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.

An existence theorem for a type of functional differential equation with infinite delay

We prove an existence theorem for a functional differential equation with infinite delay using the Schauder fixpoint theorem. We extend a result in [19] applying the fixed point procedure in an appropriate function space.

A Method to Calculate Periodic Solutions of Functional Differential Equations

We introduce a method to obtain explicitly periodic solutions of some types of functional differential equations whose periods depend rationally on integral delays. Several applications of this method are given.

A reduction of a type of functional-differential equation

By power series methods it is shown how a functional relation between a set of derivatives of an entire function can sometimes be shown to imply the existence of a much simpler functional relation between the same derivatives of this entire function.

On the monotonicity of solutions of certain types of functional-differential equations

On the monotonicity of solutions of certain types of functional-differential equations

Hopf bifurcation with constraints

IN THIS study we give conditions for the existence of periodic orbits of a strongly continuous semiflow which branch off from a steady state for a certain parameter value. The semiflow is defined in a closed subset of a Banach space which contains the steady state. The steady state is kept fixed whereas both the semiflow and its domain may depend strongly continuously on the parameter. It will be shown that an asymptotic @-contraction hypothesis together with a transversality and no-resonance hypothesis for the eigenvalues of the infinitesimal generator of the linearized semiflow imply Hopf bifurcation. The technique consists in a generalized Lyapunov-Schmidt decomposition method using spectral projections which depend strongly continuously on the bifurcation parameter. The existence of a continuum of bifurcating periodic orbits follows from a degree argument. The theory can be applied to partial functional differential equations when the bifurcation parameter changes transformed arguments or the initial or boundary values underly parameter-dependent constraints. In general such semiflows do not exhibit the smoothness hypotheses which are necessary for the application of available Hopf bifurcation theorems. There are. a lot of such theorems either for special types of functional differential equations, partial differential equations, Volterra-integral equations, or for a general semiflow in Banach space but under considerable stronger hypotheses than in the present paper. The reader is referred to [l], [2], [4], [5], [7]-[12], [15] and to the references therein. Of course the stronger hypotheses in these papers allow to get more detailed information on the structure of the branch of the bifurcating periodic orbits such as smootheness with respect to the parameter, local stability, and the direction of bifurcation. In contrast to them the hypotheses in our paper guarantee only local existence and they are too weak to ensure smoothness of the branch nor the existence of a center manifold. But even the local existence problem was still unsolved if the semiflow and its domain depend merely strongly continuously on the parameter. The difficulty is due to the fact that the usual argument basing on the implicit function theorem will not work. Some examples which demonstrate the application of our theory will be given in a subsequent paper. Before we will explain the main hypotheses we shall introduce some notations. B denotes a real Banach space and C C Iw’ a compact neighborhood of 0 E R*, the domain of the bifurcation-parameter. Norms and the modulus of a complex number are always indicated by the symbol 1.1. To every c E C we join a closed subset X’ C B, the domain of the semiflow.