A New Representation for Second Order Stochastic Integral-differential Operators and Its Applications

In this paper, we＇ll prove new representation theorems for a kind of second order stochastic integral- differential operator by stochastic differential equations （SDEs） and backward stochastic differential equations （BSDEs） with jumps, and give some applications.

Non-local Problem for the Loaded Integral-differential Equation in Double-connected Domain

In this work an existence and uniqueness of solution of the non-local boundary value problem for the loaded elliptic-hyperbolic type equation with integral-differential operations in double-connected domain have been investigated. The uniqueness of solution is proved by the method of integral energy using an extremum principle for the mixed type equations, and the existence is proved by the method of integral equations.

INITIAL VALUE PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH SINGULAR INTEGRAL-DIFFERENTIAL TERM

We study the initial value problem for a nonlinear parabolic equation with singular integral-differential term. By means of a series of a priori estimations of the solutions to the problem andLeray-Schauder fixed point principle, we demonstrate the existence and uniqueness theorems ofthe generalized and classical global solutions. Lastly, we discuss the asymptotic properties of thesolution as t tends to infinity.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

THE STABILITY AND APPROXIMATION PROPERTIES OF RITZ-VOLTERRA PROJECTION AND APPLICATIONS,Ⅱ

The object of this paper is to investigate the convergence of semidiscrete finite element approximations to the parabolic and hyperbolic integral-differential equations ,Sobolev equations and visco-elasticy equations. The Ritz-Volterra projection will be used to unity much of the anal- ysis for the different types of problems.Optimal order error estimates are obtained in Lp and W1p spaces for 2≤p≤∞ .

INTERFACIAL CRACK ANALYSIS IN THREE-DIMENSIONAL TRANSVERSELY ISOTROPIC BI-MATERIALS BY BOUNDARY INTEGRAL EQUATION METHOD

The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropic materials.

Operator Equation and Application of Variation Iterative Method

In this paper, we study some semi-closed 1-set-contractive operators A and investigate the boundary conditions under which the topological degrees of 1-set contractive fields, deg (I-A, Ω, p) are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which extend and improve many famous theorems such as the Leray-Schauder theorem, and operator equation, etc. Lemma 2.1 generalizes the famous theorem. The calculation of topological degrees and index are important things, which combine the existence of solution of for integration and differential equation and or approximation by iteration technique. So, we apply the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are proceed to examine some a class of integral-differential equations, to illustrate the effectiveness and convenience of this method.

Some Approximation in Cone Metric Space and Variational Iterative Method

In this paper, we give some new results of common fixed point theorems and coincidence point case for some iterative method. By using of variation iteration method and an effective modification of He’s variation iteration method discusses some integral and differential equations, we give out some new conclusion and more new examples.

Mathematical Modeling Algorithms for Creating New Materials with Desired Properties Using Nano-Hierarchical Structures

In the enormous and still poorly mastered gap between the macro level, where well developed continuum theories of continuous media and engineering methods of calculation and design operate, and atomic, subordinate to the laws of quantum mechanics, there is an extensive meso-hierarchical level of the structure of matter. At this level unprecedented previously products and technologies can be artificially created. Nano technology is a qualitatively new strategy in technology: it creates objects in exactly the opposite way—large objects are created from small ones [1]. We have developed a new method for modeling acoustic monitoring of a layered-block elastic medium with several inclusions of various physical and mechanical hierarchical structures [2]. An iterative process is developed for solving the direct problem for the case of three hierarchical inclusions of l, m, s-th ranks based on the use of 2D integro-differential equations. The degree of hierarchy of inclusions is determined by the values of their ranks, which may be different, while the first rank is associated with the atomic structure, the following ranks are associated with increasing geometric sizes, which contain inclusions of lower ranks and sizes. Hierarchical inclusions are located in different layers one above the other: the upper one is abnormally plastic, the second is abnormally elastic and the third is abnormally dense. The degree of filling with inclusions of each rank for all three hierarchical inclusions is different. Modeling is carried out from smaller sizes to large inclusions;as a result, it becomes possible to determine the necessary parameters of the formed material from acoustic monitoring data.

SINGULAR INTEGRAL DIFFERENTIAL EQUATIONS WITH BOTH CAUCHY KERNEL AND CONVOLUTION KERNEL AND RIEMANN BOUNDARY VALUE PROBLEM

In this paper, we set up and discuss a kind of singular integral differential equation with convolution kernel and Canchy kernel. By Fourier transform and some lemmas, we turn this class of equations into Riemann boundary value problems, and obtain the general solution and the condition of solvability in class {0}.

Travelling Wave Solutions of Integro-differential Equations of One-dimensional Neuronal Networks

Travelling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing（decreasing） Travelling wave solutions are established. Some faults in previous studies are corrected.

Uniform asymptotic stability in functional differential equations with infinite delay

The fading memory space is used as phace space, and Liapunov methods are used to give conditions ensuring that the zero solution of functional differential equations with infinite delay is uniformly asymptotically stable.