Nonlinear Functional Differential Equations Research Articles

The method of finite differences for nonlinear functional differential equations of the first order

Abstract Nonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

Stability of coupled functional differential-difference equations

General coupled functional differential-difference equations are considered. A new approach to exponential stability of such equations is presented. Consequently, some novel criteria for exponential stability of coupled functional differential-difference equations are given. As applications, the obtained results are used to derive explicit criteria for exponential stability of nonlinear functional differential equations of neutral type and those of singular delay differential equations. Some illustrative examples are provided.

Solving nonlinear functional–differential and functional equations with constant delay via block boundary value methods

This paper deals with the numerical solutions of nonlinear functional-differential and functional equations (FDFEs) with constant delay. The block boundary value methods (BBVMs) are extended to solve the FDFEs. Under the suitable conditions, it is shown that the extended BBVMs are uniquely solvable and globally stable. Moreover, the method can be convergent of order p whenever the Lipschitz condition holds and this method is preconsistent and p-order consistent. With several numerical examples, the theoretical results and computational validity of the extended BBVMs are further confirmed.

Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay

The purpose of this paper is to obtain the error bounds of fully geometric mesh one-leg methods for solving the nonlinear neutral functional differential equation with a vanishing delay. For this purpose, we consider Gq-algebraically stable one-leg methods which include the midpoint rule as a special case. The error of the first-step integration implemented by the midpoint rule on [0,T0] is first estimated. The optimal convergence orders of the fully geometric mesh one-leg methods with respect to T0 and the mesh diameter \(h_{\max }\) are then analyzed and provided for such equation. Numerical studies reported for several test cases confirm our theoretical results and illustrate the effectiveness of the proposed method.

Ulam-Hyers-Stability for nonlinear fractional neutral differential equations

We discuss Ulam-Hyers stability, Ulam-Hyers-Rassias stability and Generalized Ulam-Hyers-Rassias stability for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative by using Picard operator. An example is also given to show the applicability of our results.

Multiple solutions of nonlinear partial functional differential equations and systems

Multiple solutions of nonlinear partial functional differential equations and systems

Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations

Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second Order Hybrid Neutral Functional Differential Equations

Symmetric nonlinear functional differential equations at resonance

Symmetric nonlinear functional differential equations at resonance

A new generalization of Halanay-type inequality and its applications

In this paper, in order to study the dissipativity of nonlinear neutral functional differential equations, a generalization of the Halanay inequality is given. We apply this generalized Halanay inequality to an analysis of the dissipativity of two classes of nonlinear neutral delay integro-differential equations and the sufficient conditions are presented to ensure these systems are dissipative.

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.

Ulam-Hyers-Mittag-Leffler stability for nonlinear fractional neutral differential equations

In this paper, first we discuss two existence and uniqueness results for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivatives with respect to the Chebyshev and Bielecki norms. Second, we use the Picard operator to establish Ulam-Hyers-Mittag-Leffler stability results on a compact interval. Finally, two examples are provided to illustrate our results. Bibliography: 29 titles.

A class of stochastic one-parameter methods for nonlinear SFDEs with piecewise continuous arguments

This paper deals with nonlinear stochastic functional differential equations with piecewise continuous arguments (SFDEPCAs). Based on an adaptation to the underlying one-leg θ-methods for ODEs, a class of new one-parameter methods for nonlinear SFDEPCAs are introduced. The mean-square exponential stability criteria of analytical and numerical solutions are derived. Under the suitable conditions, it is proved that the one-parameter methods are convergent with strong order 1/2. Some numerical experiments are given to illustrate the theoretical results and computational advantages of the induced methods.

ON HYERS-ULAM STABILITY OF NONLINEAR SECOND ORDER ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider the Hyers-Ulam stability of some nonlinear second order ordinary and functional differential equations. As mathematical technique, we use some nonlinear extension of the Gr{\o}nwall integral inequality.

On the oscillation of impulsive vector partial differential equations with distributed deviating arguments

AbstractIn this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for theH-oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result.

Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition

Stability of numerical solutions to stochastic differential equations (SDEs) has received an increasing attention from researchers in applied mathematics and engineering areas, but there has been so far little work on the stability analysis of numerical solutions to highly nonlinear stochastic functional differential equations (SFDEs). The aim of this paper is to study the almost sure exponential stability of the backward Euler scheme for highly nonlinear SFDEs. Firstly, a stability criterion is established for the almost sure exponential stability of the underlying nonlinear SFDE under an extended polynomial growth condition, the global existence of the solutions of the underlying equation is simultaneously shown with a new technique, the Fatou’s Lemma. Secondly, the almost sure exponential stability of the backward Euler scheme is investigated by contrast. The stability criterion shows that the numerical scheme preserves the almost sure exponential stability of the analytic solution under the same growth condition, which improves some related result in literature as a corollary by way. To describe the scheme conveniently, a concept, the interpolation segment, is formally proposed in the paper, which provides a way for constructing a discretized approximation to a continuous function applied in the scheme. To achieve the required results, the classical nonnegative semi-martingale convergence theorem is reformed to a practical version, namely the so called practical nonnegative semi-martingale boundedness lemma in the paper. At the end of the paper, a numerical example is proposed to illustrate the adaption of the growth condition assumed in the paper.

An exponential stability criterion for nonlinear second-order functional differential equations with time-variable delays

This paper is concerned with the exponential stability for nonlinear second-order functional differential equations (FDEs) with time-variable delays. An exponential stability relationship between the FDEs and the corresponding ordinary differential equations (ODEs) is derived. It is proved under some appropriate conditions that the second-order FDEs can preserve the exponential stability of the corresponding ODEs. This stability result is also illustrated with a numerical approach.

Growth Rates of Sublinear Functional and Volterra Differential Equations

This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymptotically so), and the nonlinear dependence grows less rapidly than any linear function. We impose extra regularity properties on a function asymptotic to this nonlinear function, rather than on the nonlinearity itself. The main result of the paper demonstrates that the growth rate of the solution can be found by determining the rate of growth of a trivial functional differential equation (FDE) with the same nonlinearity and all its associated measure concentrated at zero; the trivial FDE is nothing other than an autonomous nonlinear ODE. We also supply direct asymptotic information about the solution of the FDE under additional conditions on the nonlinearity, and exploit the theory of regular variation to sharpen and extend the results.

Oscillation of a perturbed nonlinear third order functional differential equation

Oscillation of a perturbed nonlinear third order functional differential equation

Fractional variational iteration method for time-fractional non-linear functional partial differential equation having proportional delays

In this paper, time-fractional non-linear partial differential equation with proportional delays are solved by fractional variational iteration method taking into account modified Riemann-Liouville fractional derivative. The numerical solutions which are calculated by using this method are better than those obtained by homotopy perturbation method and differential transform method with same data set and approximation order. On the other hand, to improve the solutions obtained by fractional variational iteration method, residual error function is used. With this additional process, the resulting approximate solutions are getting closer to the exact solutions. The results obtained by taking into account different values of variables in the domain are supported by compared tables and graphics in detail.

Multiple solutions of nonlinear elliptic functional differential equations

Multiple solutions of nonlinear elliptic functional differential equations