Second-order Integro-differential Equations Research Articles

On stabilization of solutions to incomplete second-order integrodifferential equations

We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.

Boundary Observability of Semi-Discrete Second-Order Integro-Differential Equations Derived from Piecewise Hermite Cubic Orthogonal Spline Collocation Method

In this paper we consider space semi-discretization of some integro-differential equations using the harmonic analysis method. We study the problem of boundary observability, i. e., the problem of whether the initial data of solutions can be estimated uniformly in terms of the boundary observation as the net-spacing \(h\rightarrow 0\). When \(h\rightarrow 0\) these finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We shall consider the piecewise Hermite cubic orthogonal spline collocation semi-discretization.

О решении методом регуляризации А.Н. Тихонова одной обратной задачи физики твердого тела и оценка погрешности этого метода

The paper considers one-dimensional Fredholm integral equation of the first kind with a closed core. It is known that the equation has a unique solution in the space $W_2^1[a,b]$. We use Tikhonov’s regularization method of the first-order to solve the equation. The method allows us to reduce the equation to a variational problem. Solving the variational problem we get integro-differential equation of second order. We apply the finite-difference approximation method to reduce the original problem to a system of algebraic equations. regularization parameter. We obtain an error estimate for the proposed algorithm taking into account the error of finitedifference approximation and state the relation between the approximation with the error and the regularization parameter and the error of the initial data. This algorithm is used to solve the problem of determining the phonon spectrum of the crystal given its heat capacity.

Global existence and boundedness of solutions of a certain nonlinear integro-differential equation of second order with multiple deviating arguments

In this paper, we consider the global existence and boundedness of solutions for a certain nonlinear integro-differential equation of second order with multiple constant delays. We obtain some new sufficient conditions which guarantee the global existence and boundedness of solutions to the considered equation. The obtained result complements some recent ones in the literature. An example is given of the applicability of the obtained result. The main tool employed is an appropriate Lyapunov-Krasovskii type functional.

The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces

By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).

Electrogravitational stability of streaming compound jets

The electrohydrodynamic stability of a self-gravitating streaming compound jet has been investigated for all modes of perturbation. The jets are immersed in a dielectric motionless tenuous medium pervaded by varying electric field. A second-order integro-differential equation of Mathieu type has been derived and some reported works are recovered as limiting cases from it.

Global existence and blow-Ups for certain ordinary integro-differential equations

We will study ordinary integro-differential equations of second order with nonlinearity given as a convolution, but differently from the widely investigated cases. In addition, the kernel depends on the solution. Such equations play a key role in the theory of glass-forming liquids, and we will establish results on global existence and investigate the long-term behavior. In contrast, we give examples where blow-ups occur.

Maximum Principles for Second-Order Impulsive Integro-Differential Equations with Integral Jump Conditions

We are concerned with two new maximum principles for second-order impulsive integro-differential equations with integral jump conditions. The impulse effects or jump conditions of this paper are involved in terms of integral of past states. As an application, we introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear second-order impulsive functional integro-differential equation with integral jump conditions.

Volterra second-order integro-differential equations unresolved for the higher derivative. The case of semibounded operator coefficients

The theorems of existence and uniqueness of strong solutions to Volterra second-order integrodifferential equations in a Hilbert space are proved in the case where the energy pumping to a system occurs (the energy dissipation operator is bounded from below), and the system can be unstable (the potential energy operator is bounded from below as well).

On the global existence and boundedness of solutions to a nonlinear integro-differential equations of second order

The purpose of this paper is to study the global existence and boundedness of solutions to a kind of nonlinear integro-differential equations of second order by integral inequalities. Some illustrative examples are also provided. Our results complement and improve some ones in the literature.

Numerical Solution of Second-Order Linear Fredholm Integro-Differetial Equations by Trigonometric Scaling Functions

The main aim of this paper is to apply the Hermite trigonometric scaling function on [0, 2π] which is constructed for Hermite interpolation for the linear Fredholm integro-differential equation of second order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of algebraic linear equations by expanding the approximate solution. Some numerical example is included to demonstrate the validity and applicability of the presented technique, the method produces very accurate results, and a comparison is made with exiting results. An estimation of error bound for this method is presented.

Multiplicity of Periodic Solutions for Third-Order Nonlinear Differential Equations

We study the existence of periodic solutions for third-order nonlinear differential equations. The method of proof relies on Schauder’s fixed point theorem applied in a novel way, where the original equation is transformed into second-order integrodifferential equation through a linear integral operator. Finally, examples are presented to illustrate applications of the main results.

Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.

Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces

Abstract This paper studies the existence and uniqueness of solutions for a nonlocal singular boundary value problem of second-order integro-differential equations in weighted spaces. The method of quasilinearization is applied to obtain monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of the problem at hand. An illustrative example is presented. MSC:34A45, 34B16, 45J05.

Nontrivial solvability of a class of nonlinear integro-differential equations of second order

The article is devoted to the study of nontrivial solvability and the asymptotic behavior of solutions for some classes of nonlinear integro-dofferential equations with a noncompact operator in a special case. Combining special factorization methods with the methods of the theory of linear integral equations of convolution type, we prove existence theorems for these classes of equations. With the help of a priori estimates, we calculate the limits of solutions obtained at infinity. The examples exhibited in the article are of mathematical interest in their own right. They are particular cases of the equations considered and have important applications in quantum mechanics.

Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM

Aims: In this paper an approximate method for the solution of third-order differential equation with two and three point boundary condition is developed using iterative reproducing kernel method. Methods: The third order boundary value problem is converted into integro-differential equation of second order two point boundary value problem. The reproducing kernel method which takes the form of a convergent series with easily computable components is used for the solution of second order two point boundary value problem. Results: Six numerical examples are given to demonstrate the efficiency of the present method. The results obtained are better than the existing methods developed in [19,20,21,22,23]. Conclusions: In this paper, the solution of linear and nonlinear third order (two and three point) boundary value problem is determined. For the solution of third order three point boundary value problem reproducing kernel method is proposed and obtained a good accuracy in absolute errors. As the reproducing kernel method cannot solve the third order three-point boundary value problems directly, so the third order boundary value problem is converted to second order two point boundary value problem after absorbing the nonlocal condition at . The method developed is compared with those developed by Li et al. [19], El-Salam et al. [20], Khan and Aziz [21], Li and Wu [22] and Wu and Li [23]. As observed in Example 3, 4, 5 and 6 that the method obtained in this paper is better than [19]-[23]. Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for third-order linear as well as nonlinear boundary value problem.

Approximate solution of second-order integrodifferential equation of Volterra type in RKHS method

In this paper, an application of reproducing kernel Hilbert space (RKHS) method is applied to solve second-order integrodifferential equation of Volterra type. The analytical solution is represented in the form of series in the reproducing kernel space. The n−truncation approximation un(x) is obtained and proved to converge to the analytical solution u(x). Moreover, the presented method has an advantages that it is possible to pick any point in the interval domain and as well the approximate solution and its derivatives will be applicable Numerical experiments are displayed to illustrate the validity, accuracy, efficiency and applicability of the proposed method. Results indicates that our technique is simple, straightforward and effective.

Periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions

This paper is concerned with the existence of extremal solutions of periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions. We introduce a new definition of lower and upper solutions with integral jump conditions and prove some new maximum principles. The method of lower and upper solutions and the monotone iterative technique are used. MSC: 34B37; 34K10; 34K45

On the solvability of a nonlinear second-order integro-differential equation with sum-difference kernel on a semiaxis

We consider a class of nonlinear second-order integro-differential equations with sum-difference kernel on a positive semiaxis. By constructing a special factorization of the initial linear integro-differential operator, we prove the existence of a nonnegative, nontrivial, and monotonically increasing solution and determine its asymptotic behavior at infinity. The relevant examples are presented.

Homotopy Analysis Method for Second‐Order Boundary Value Problems of Integrodifferential Equations

In this paper, series solution of second‐order integrodifferential equations with boundary conditions of the Fredholm and Volterra types by means of the homotopy analysis method is considered. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The homotopy analysis method provides us with a simple way to adjust and control the convergence region of the infinite series solution by introducing an auxiliary parameter. The proposed technique is applied to a few test examples to illustrate the accuracy, efficiency, and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.