Value Problems For Integro-differential Equations Research Articles

Boundary value problems of integrodifferential equations under boundary conditions taking into account physical nonlinearity

When solving integrodifferential equations under boundary conditions, taking into account physical nonlinearity, a broad class of boundary-value problems of oscillations arises associated with various boundary conditions at the edges of a flat element. When taking into account non-stationary external influences, the main parameters is the frequency of natural vibrations of a flat component, taking into account temperature, prestressing, and other factors. The study of such problems, taking into account complicating factors, reduces to solving rather complex problems. The difficulty of solving these problems is due to both the type of equations and the variety. We analyze the results of previous works on the boundary problems of vibrations of plane elements. Possible boundary conditions at the edges of a flat element and the necessary initial conditions for solving particular problems of self-oscillation and forced vibrations, and other problems are considered. The set of equations, boundaries, and initial conditions make it possible to formulate and solve various boundary value problems of vibrations for a flat element. The oscillation equations for a flat element in the form of a plate given in this paper contain viscoelastic operators that describe the viscous behavior of the materials of a flat component. In studying oscillations and wave processes, it is advisable to take the kernels of viscoelastic operators regularly, since only such operators describe instantaneous elasticity and then viscous flow.

Approximation of solutions of boundary value problems for integro-differential equations of the neutral type using a spline function method

Boundary value problems for nonlinear integro-differential equations of the neutral type are investigated. A scheme for approximating the boundary value problem solution using cubic splines of defect two is proposed and substantiated. A model example illustrating the proposed approximation scheme is considered.

Homogenized Models with Memory Effect for Heterogeneous Periodic Media

The homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems.

APPROXIMATION OF BOUNDARY VALUE PROBLEMS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY

In mathematical modeling of physical and technical processes, the evolution of which depends on prehistory, we arrive at differential equations with a delay. With the help of such equations it was possible to identify and describe new effects and phenomena in physics, biology, technology. An important task for differential-functional equations is to construct and substantiate finding approximate solutions, since there are currently no universal methods for finding their precise solutions. Of particular interest are studies that allow the use of methods of the theory of ordinary differential equations for the analysis of delay differential equations. Schemes for approximating differential-difference equations by special schemes of ordinary differential equations are proposed in the works N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy in various functional spaces. The purpose of this paper is to apply approximation schemes of differential-difference equations to approximation of solutions of boundary-value problems for integro-differential equations with a delay. The paper presents sufficient conditions for the existence of a solution of the boundary value problem for integro-differential equations with many delays. The scheme of its approximation by a sequence of boundary value problems for ordinary integro-differential equations is proposed and the conditions of its convergence are investigated. A model example is considered to demonstrate the given approximation scheme.

Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

<abstract><p>Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.</p></abstract>

On the Unique Solvability of a Family of Boundary Value Problems for Integro-Differential Equations of Mixed Type

On the Unique Solvability of a Family of Boundary Value Problems for Integro-Differential Equations of Mixed Type

Solution of third order linear and nonlinear boundary value problems of integro-differential equations using Haar Wavelet method

In this paper, numerical solution of third order integro-differential equation with boundary conditions is given utilizing Haar collocation technique. Both nonlinear and linear integro-differential equations are solved using this method. The third order derivative is approximated using Haar functions in both nonlinear and linear integro-differential equations. Integration is used to obtain the expression of lower order derivatives as well as the solution for the unknown function. The Gauss elimination approach is utilized for linear systems and Broyden approach is adopted for nonlinear systems. Validation and convergence of the proposed approach are illustrated using some examples. At various collocation and gauss points, the maximum absolute and root mean square errors are compared to the exact solution. The convergence rate is also measured using different numbers of nodal points, and it is nearly equal to 2.

Well-Posedness and Spectral Analysis of Integrodifferential Equations of Hereditary Mechanics

The well-posedness of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces is studied, and spectral analysis of the operator functions that are the symbols of these equations is performed. The equations under consideration are an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory, which have a number of other important applications. Results concerning the well-posedness of these integrodifferential equations in weighted Sobolev spaces of vector functions defined on the positive half-line with values in a Hilbert space are obtained. The localization and structure of the spectrum of the operator functions that are the symbols of these equations are established.

Boundary value problem solution existence for linear integro-differential equations with many delays

For the study of boundary value problems for delay differential equations, the contraction mapping principle and topological methods are used to obtain sufficient conditions for the existence of a solution of differential equations with a constant delay. In this paper, the ideas of the contraction mapping principle are used to obtain sufficient conditions for the existence of a solution of linear boundary value problems for integro-differential equations with many variable delays.
 Smoothness properties of the solutions of such equations are studied and the definition of the boundary value problem solution is proposed. Properties of the variable delays are analyzed and functional space is obtained in which the boundary value problem is equivalent to a special integral equation. Sufficient, simple for practical verification coefficient conditions for the original equation are found under which there exists a unique solution of the boundary value problem.

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.

A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm–Volterra type

In this paper, the numerical solution of periodic Fredholm–Volterra integro–differential equations of first-order is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic condition of the problem is satisfied. The exact solution ux is represented in the form of series in the space W22. In the mean time, the n-term approximate solution unx is obtained and is proved to converge to the exact solution ux. Furthermore, we present an iterative method for obtaining the solution in the space W22. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear equations.

A new efficient technique for solving two-point boundary value problems for integro-differential equations

In this paper, we propose a new efficient method based on a combination of Adomian decomposition method (ADM) and Green’s function for solving second-order boundary value problems (BVPs) for integro-differential equations (IDEs). The proposed method depends on constructing Green’s function before establishing the recursive scheme for the solution components. Unlike the ADM or modified ADM , the proposed method avoids solving a sequence of difficult nonlinear equations (transcendental equations) for the unknown parameters. The proposed method provides a direct recursive scheme for obtaining the series solution with easily calculable components. We also provide a sufficient condition that guarantees a unique solution to the second-order BVPs for IDEs. Convergence and error analysis of the proposed method are also discussed. Convergence analysis is reliable enough to estimate the error bound of the series solution. Some numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.

Spectral Analysis of Integro–Differential Equations in Viscoelasticity Theory

We study spectral properties of boundary value problems for integro-differential equations in viscoelasticity theory. The sum of N exponential functions with negative exponents is taken for the convolution kernel. We prove that the spectrum of the boundary value problem is the union of N sequences of real numbers and two sequences all elements, except for finitely many ones, are not real. In particular, we obtain conditions under which these two sequences contains no real numbers. Bibliography: 8 titles.

Singular initial value problem for a system of integro-differential equations unsolved with respect to the derivative

A singular initial value problem for a system of integro-differential equations of the first order unsolved with respect to the derivative of the unknown functions is considered. It is shown that under certain assumptions there is a solution lying in a region, having the vertex coinciding with the initial point, in which the graph of a solution of given singular problem is located. The main result gives asymptotic estimates of a solution and its derivative with respect to a general solution of an auxiliary differential equation. An approach which combines topological method of T. Ważewski and Schauder’s fixed point theorem is used. The results generalize some previous ones, where the singular initial value problems for integro-differential equations were studied.

On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem (0.1) x ′ ( t ) = p ( t ) ∫ 0 σ ( t ) q ( s ) x ( τ ( s ) ) d s , (0.2) x ( t ) = φ ( t ) for t ∈ [ τ 0 , 0 ) , lim inf t → + ∞ | x ( t ) | < + ∞ is considered, where p ∈ C ( R + ; ( 0 , + ∞ ) ) , q ∈ L loc ( R + ; R + ) , 0 ≤ σ ( t ) ≤ t , τ ( t ) < t for t ∈ R + , lim t → + ∞ σ ( t ) = lim t → + ∞ τ ( t ) = + ∞ , τ 0 = inf { τ ( t ) : t ∈ R + } , φ ∈ C ( [ τ 0 , 0 ) ) and sup { ∣ φ ( t ) ∣ : t ∈ [ τ 0 , 0 ) } < + ∞ . Sufficient conditions are obtained for problem (0.1) and (0.2) to have a solution, a unique solution and a unique oscillatory solution.

On nonlinear integro-differential equations of hyperbolic type

This paper considers unified initial-boundary value problems for integro-differential equations of hyperbolic type. For such problems where the forcing functions satisfy certain monotonicity conditions, monotone sequences converging to coupled minimal–maximal solutions may be constructed from a single pair of coupled lower–upper solutions. The coupled minimal–maximal solutions coincide in a unique solution provided the problem satisfies certain one-sided Lipschitz conditions.

On stability and boundary value problems for integro-differential equations

The basis of the Fourier method of variables’ separation and a special reduction procedure for the stability of solutions of partial boundary value problems is studied.

Solution of boundary value problems for integro-differential equations by using differential transform method

In this study, we extended the differential transform method (DTM) to solve the integro-differential equations. New theorems for the transformation of integrals are introduced and prooved. The commonly encountered linear and nonlinear integro-differential equations that appear in literature are solved as an illustration for the efficiency of the method. Some numerical results are also given to demonstrate the superiority of the method to other common techniques. The results we obtained by using this technique is much more accurate compared to the existing ones.

Remarks on the periodic boundary value problems for integrodifferential equations of volterra type

This paper is concerned with the monotone method on the periodic boundary value problems for integrodifferential equations of Volterra type. This problem has been studied by several authors[1–6]. The purpose of this paper is to relax the restrictions as much as possible on the coefficients M and N that appear in the one-sided Lipschitz condition.

Integro-differential inequalities with initial time difference and applications

Integro-differential inequalities with initial time difference arediscussed. They play an important role in the investigation of initial value problems of integro-differential equations where the initial time differs. The existence of extremal solutions is investigated by the monotone iterative technique.