Existence of solutions for nonlinear impulsive Hammerstein integral equations in Banach spaces

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SOLUTIONS FOR SECOND ORDER IMPULSIVE INTEGRO- DIFFERENTIAL EQUATION ON UNBOUNDED DOMAINS IN BANACH SPACES

Under loose conditions, the existence of solutions to initial value problem are studied for second order impulsive integro-differential equation with infinite moments of impulse effect on the positive half real axis in Banach spaces. By the use of recurrence method, Tonelii sequence and the locally convex topology, the new existence theorems are achieved, which improve the related results obtained by Guo Da-jun.     

三维间断位移法及强奇异和超奇异积分的处理方法

从积分方程Somigliana等式出发,导出三维状态下单位位错集度的基本解.在此基础上,建立了边界积分方程,并给出了其离散形式.对强奇异和超奇异积分,采用了Hadamard定义的有限部分积分来处理.最后,给出了计算裂纹应力强度因子的算例,并与解析解进行了比较,证实了该方法的有效性.     

冲击动力系统的鲁棒稳定性分析

考虑冲击动力系统的ｋ－ｐ周期运动的鲁棒稳定性问题。首先，根据微分方程的解、冲击条件和衔接条件，应用迭代法给出了系统存在ｋ－ｐ周期运动的充分必要条件，并利用稳定性的等价原理，通过周期运动的扰动差分方程导出其稳定条件；然后，着重对含有不确定参数的冲击动力系统的ｋ－ｐ周期运动的稳定性进行了分析，得出了鲁棒稳定的充分条件，文末给出了用于阐明理论结果的算例。     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Fundamental solutions of Laplace's and Navier's differential operators defined on Riemann spaces with a circular branch line

Fundamental solutions of the differential operators for the potential problem and the elastostatic problem are established. They are not defined on the ordinary three-dimensional space as the classical 1/R solution and Kelvin's solution but on Riemann spaces with circular branch lines and a finite as well as an infinite number of sheets. The solutions can be used as the kernels of boundary integral equations. Equations of this type should be useful for the determination of displacements and stresses in elastic bodies with slits and cracks of certain shapes.     

Localization of Shear Waves Near Layers Separating Two Regularly Laminated Half-Spaces

The existence conditions for surface and normal shear waves in finite and infinite periodically laminated structures with broken translational symmetry are studied theoretically and numerically. The problems posed are reduced to systems of linear algebraic equations. The existence condition for their nontrivial solutions yield dispersion relations. Conditions for the existence of surface and normal shear waves are established. Some results are plotted. The dependence of the dispersion spectrum on the physical and geometrical properties of the symmetry breaker is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 128–134, January 2005.     

Method of Integral Equations in 2D and 3D Problems of Plate Impact on a Fluid of Finite Depth

This paper describes a method of integral equations for solution of the 2D and 3D problems of plate impact on an incompressible fluid of finite depth. The solutions of the equations obtained are investigated analytically and numerically. The behavior of the impact impulse is studied for various fluid depths and aspect ratios of the plate.     

Input-to-state stability of impulsive reaction–diffusion neural networks with infinite distributed delays

In this paper, we focus on the global existence–uniqueness and input-to-state stability of the mild solution of impulsive reaction–diffusion neural networks with infinite distributed delays. First, the model of the impulsive reaction–diffusion neural networks with infinite distributed delays is reformulated in terms of an abstract impulsive functional differential equation in Hilbert space and the local existence–uniqueness of the mild solution on impulsive time interval is proven by the Picard sequence and semigroup theory. Then, the diffusion–dependent conditions for the global existence–uniqueness and input-to-state stability are established by the vector Lyapunov function and M-matrix where the infinite distributed delays are handled by a novel vector inequality. It shows that the ISS properties can be retained for the destabilizing impulses if there are no too short intervals between the impulses. Finally, three numerical examples verify the effectiveness of the theoretical results and that the reaction–diffusion benefits the input-to-state stability of the neural-network system.

     

On the nature of multitude scalings in decaying isotropic turbulence

We report multitude scaling laws for isotropic fully developed decaying turbulence through group theoretic method employing on the spectral equations both for modelling and without any modelling of nonlinear energy transfer. For modelling, besides the existence of classical power law scalings, an exponential decay of turbulent energy in time is obtained subject to exponentially decaying integral length scale at infinite Reynolds number limit. For the transfer without modelling, at finite Reynolds number, in addition to general power law decay of turbulence intensity with integral length scale growing as a square root of time, an exponential decay of energy in time is explored when integral length scale remains constant. Both the power and exponential decaying laws of energy agree to the theoretical results of George (1992), George and Wang (2009) and experimental results of fractal grid generated turbulence by Hurst and Vassilicos (2007). At infinite Reynolds number limit, a general power law scaling is obtained from which all classical scaling laws are recovered. Further, in this limit, turbulence exhibits a general exponential decaying law of energy with exponential decaying integral length scale depending on two scaling group parameters. The role of symmetry group parameters on turbulence dynamics is discussed in this study.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

On the elastic strip with an internal crack

The paper presents a method to deal with an inclined crack in an elastic strip. No assumptions of symmetry are made. The method involves the solutions for a cracked plane and an uncracked strip and results in two coupled singular integral equations with finite interval of integration. A crack in a half-plane arises as a limiting case. For internal cracks the integral equations are of a standard type and do not present any numerical difficulties. Results are presented for loads according to the technical beam theory.     

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.     

Existence and Integral Representations of Weak Solutions for Elastic Plates with Cracks

The existence and continuous dependence on the data are investigated in Sobolev spaces for the problem of bending of a Reissner-Mindlin-type plate weakened by a crack when the displacements or the moments and force are prescribed along the two sides of the crack. The cases of both an infinite and a finite plate are considered, and representations are sought for the solutions in terms of single layer and double layer potentials with distributional densities. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.     

Evaluation of Green’s function for vertical point-load excitation applied to the surface of a layered semi-infinite elastic medium

The topic of this paper is to show that the integrals of infinite extent representing the surface displacements of a layered half-space loaded by a harmonic, vertical point load can be reduced to integrals with finite integration range. The displacements are first expressed through wave potentials and the Hankel integral transform in the radial coordinate is applied to the governing equations and boundary conditions, leading to the solutions in the transformed domain. After the application of the inverse Hankel transform it is shown that the inversion integrands are symmetric/antimetric in the transformation parameter and that this characteristic is preserved for any number of layers. Based on this fact the infinite inversion integrals are reduced to integrals with finite range by choosing the suitable representation of the Bessel function and use of the fundamental rules of contour integration, permitting simpler analytical or numerical evaluation. A numerical example is presented and the results are compared to those obtained by the CLASSI program.     

Application of complex stochastic averaging to non-linear random vibration problems

The stochastic averaging procedure in a complex-variable setting, used previously by Ariaratnam and Tarn to analyze a linear system under random parametric excitation, is extended to non-linear systems under both parametric and external random excitations. It is shown that equations for the moments of the system response, while still constituting an infinite hierarchy, form a simpler pattern compared with the corresponding equations obtained from the usual amplitude and phase formulation. From this simpler pattern, it is possible to identify those moments which tend to zero at the stationary state. Furthermore, a much smaller number of equations needs to be solved when the infinite hierarchy is truncated to calculate approximately the non-zero moments.     

Integral transform solution of developing laminar duct flow in Navier-Stokes formulation

The generalized integral transform technique is employed in the hybrid numerical-analytical solution of the Navier-Stokes equations in streamfunction-only formulation, which govern the incompressible laminar flow of a Newtonian fluid within a parallel plate channel. Owing to the analytic nature of this approach, the outflow boundary condition for an infinite duct is handled exactly, and the error involved in considering finite duct lengths is investigated. The present error-controlled solutions are used to inspect the relative accuracy of previously reported purely numerical schemes and to compare Navier-Stokes and boundary layer formulations for various combinations of inlet conditions and Reynolds number.     

混合边界多裂隙体在简谐变温场作用下的断裂问题（张开型）

给出了无限平面作用有简谐变化的点热源时的位移场、应力场基本解、用间接法构造出混合边值多裂隙体在简谐变温场作用下的热断裂问题的边界积分方程,并离散求解.数值结果表明,该方法求解多裂隙体的简谐热断裂问题精度好,计算工作量少.文中计算了含边界裂缝的平板、含三条平行裂缝的平板在简谐变温场作用下缝端应力强度因子的变化过程,并与实验结果进行了比较,两者吻合良好.     

Approximate kinetic equations in rarefied gas theory

The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals.It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation.A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5].In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations.To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.The author wishes to thank A. A. Nikol'skii for discussions of the study and V. A. Rykov for the numerical results presented for the exact solution.