Integro-differential Equations Of Neutral Type Research Articles

Necessary Conditions of Optimality for Some Stochastic Integrodifferential Equations of Neutral Type on Hilbert Spaces

In this work, we develop the necessary conditions of optimality for partially observed control problems governed by stochastic integrodifferential equations of neutral type on Hilbert spaces driven by relaxed controls. Under suitable conditions, we establish the existence and the uniqueness of mild solution which has a continuous modification. We prove the continuity of the control to the solution map. Then we consider an optimal control problem of Bolza type. Using these results we give necessary conditions of optimality for a class of stochastic integrodifferential equations of neutral type on Hilbert spaces.

A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can also be obtained by using other sigmoidal functions, such as logistic and Gompertz functions. The method allows one to obtain a simultaneous approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. A priori as well as a posteriori estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.

On Spectral Method for Volterra Functional Integro-Differential Equations of Neutral Type

The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence.

Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

Abstract In this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.

Nonlocal cauchy problem for delay fractional integrodifferential equations of neutral type

This article concerns the existence of mild solutions for the fractional integrodifferential equations of neutral type with finite delay and nonlocal conditions in a Banach space X. The existence of mild solutions is proved by means of measure of noncompactness. As an application, the existence of mild solutions for some integrodifferential equation is obtained. Mathematics Subject Classification: 34K05; 34A12.

Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay

We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space . An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski's measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodifferential equation.

The Existence and Exponential Stability for Random Impulsive Integrodifferential Equations of Neutral Type

By applying the Banach fixed point theorem and using an inequality technique, we investigate a kind of random impulsive integrodifferential equations of neutral type. Some sufficient conditions, which can guarantee the existence, uniqueness, and exponential stability in mean square for such systems, are obtained. Compared with the previous works, our method is new and our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the proposed results.

The Existence and Exponential Stability for Random Impulsive Integrodifferential Equations of Neutral Type

The Existence and Exponential Stability for Random Impulsive Integrodifferential Equations of Neutral Type

Finite element and DG analysis for neutral-type Volterra integro-differential equations with weakly singular kernels

In this paper we analyze the numerical solution of Volterra integro-differential equations of neutral type with weakly singular kernels. We establish a priori error estimations for the finite-element-method semi-discretization of the given problem by defining a suitable Ritz–Volterra projection operator: here, the key point in the proof is the fact that the definition of the Ritz–Volterra projection operator that is not related to the neutral term. We then discuss the discontinuous Galerkin time-stepping method for the semi-discretized equation, together with a fully discretized form.

The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays

We analyze the attainable order of convergence of collocation solutions for linear and nonlinear Volterra functional integro-differential equations of neutral type containing weakly singular kernels and nonvanishing delays. The discretization of the initial-value problem is based on a reformulation as a sequence of ODEs with nonsmooth solutions. The paper concludes with a brief description of possible alternative numerical approaches for solving various classes of such functional integro-differential equations.

Extended backward differentiation methods in the numerical solution of neutral Volterra integro-differential equations

The numerical solution of first order Volterra integro-differential equations of neutral type with continuous kernel is considered. Preliminary results from the application of extended backward differentiation methods are given and comparisons are made with Adams-Bashforth / Adams-Moulton predictor-corrector methods and collocation methods.

On well-posedness of integro-differential equations in weighted $L^2$-spaces

In this paper we consider the problem of constructing a well-posed state space model for a class of singular integro-differential equations of neutral type. The work is motivated by the need to develop a framework for the analysis of numerical methods for designing control laws for aeroelastic systems. Semigroup theory is used to establish existence and well-posedness results for initial data in weighted $L^{2}-$spaces. It is shown that these spaces lead naturally to the dissipativeness of the basic dynamic operator. The dissipativeness of the solution generator combined with the Hilbert space structure of these weighted spaces make this choice of a state space more suitable for use in the design of computational methods for control than previously used product spaces.

Existence of periodic solutions to integro-differential equations of neutral type via limiting equations

AbstractIn this paper, we present some results on the existence of periodic solutions to Volterra integro-differential equations of neutral type. The main idea is to show the convergence of an equibounded sequence of periodic solutions of certain limiting equations which are of finite delay. This makes it possible to apply the existing Liapunov–Razumikhin technique for neutral equations with finite delay to obtain existence of periodic solutions of Volterra neutral integro-differential equations (of infinite delay). Some comparisons between our results and the existing ideas are also provided.

Numerical Methods for a Class of Singular Integro-Differential Equations Based on Semigroup Approximation

In this paper computational methods for singular integro-differential equations of neutral type are discussed. The study is based on a well-posed state-space formulation and uses semigroup theoretical framework to analyze the convergence of semi- and fully discrete numerical schemes. Feasibility of the fully discrete scheme is demonstrated by applying it to Abel–Volterra equations of the first kind and to a singular neutral functional differential equation.

On the oscillatory properties of the solutions of a class of integro‐differential equations of neutral type

In the present paper the oscillatory properties of the solutions of the equation urn:x-wiley:01611712:media:ijmm163126:ijmm163126-math-0001 are investigated where n ≥ 1, L is an operator of the difference type, It ⊂ ℝ, K : DK → ℝ, DK⫅ℝ3, x : [αx, ∞] → ℝ. Under natural conditions imposed on L, It and K it is proved that for n even all ultimately nonzero solutions oscillate and for n odd they either oscillate or tend to zero as t → ∞.