Difference Potentials Method and its Applications

The purpose of this article is to acquaint the reader with the general concepts and capabilities of the Difference Potentials Method (DPM). DPM is used for the numerical solution of boundary-value and some other problems in difference and differential formulations. Difference potentials and DPM play the same role in the theory of solutions of linear systems of difference equations on multi-dimensional non-regular meshes as the classical Cauchy integral and the method of singular integral equations do in the theory of analytical functions (solutions Cauchy-Riemann system). The application of DPM to the solution of problems in difference formulation forms the first aspect of the method. The second aspect of the DPM implementation is the discretization and numerical solution of the Calderon-Seeley boundary pseudo-differential equations. The latter are equivalent to elliptical differential equations with variable coefficients in the domain; they are written making no use of fundamental solutions and integrals. Because of this fact ordinary methods for discretization of integral equations cannot be applied in this case. Calderon-Seeley equations have probably not been used for computations before the theory of DPM appeared. This second aspect for the implementation of DPM is that it does not require difference approximation on the boundary conditions of the original problem. The latter circumstance is just the main advantage of the second aspect in comparison with the first one. To begin with, we put forward and justify the main constructions and applications of DPM for problems connected with the Laplace equation. Further, we also outline the general theory and applications: both those already realized and anticipated.     

The b and h functions for integral equations with displacement kernels: A computational method and an application to radiative transfer

Many problems in the theory of radiative transfer reduce to the solution of Fredholm integral equations with displacement kernels. Frequently, we are interested in the solutions of the Fredholm integral equations as well as certain functionals on the solution (reflection and transmission coefficients, etc.). Earlier it was shown that these functionals can be expressed algebraically in terms of the basic functions b and h. Normally, these functions are computed as solutions of an initial-value problem. Since they represent internal interactions due to isotropic illuminations, they are also solutions of a linear two-point boundary-value problem, which, unfortunately, s unstable. The purpose of this paper is to show that this unstable problem can be solved using a Gram-Schmidt orthogonalization scheme. This is demonstrated by making comparisons against earlier calculations using the initial-value method. In addition, the process is ideally suited to take advantage of multitasking on parallel processors.     

The method of volterra integral equations in contact problems for thin-walled structural elements

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 96–103. Original article submitted March 15, 1997.     

Diffraction by a wedge at skew incidence: integral representations of Cauchy-Carleman for the electromagnetic fields

An electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional difference equations by means of Sommerfeld integrals and Malyuzhinets theorem. By introducing an integral operator it is shown that the solutions of this system of functional equations can be defined in terms of integral representations whose kernels are solutions of a singular integral equation of Cauchy-Carleman type for which an explicit solution is given.     

Approximation of Stochastic Nonlinear Equations of Schrödinger Type by the Splitting Method

In this article, we construct a splitting method for nonlinear stochastic equations of Schrödinger type. We approximate the solution of our problem by the sequence of solutions of two types of equations: one without stochastic integral term, but containing the Laplace operator and the other one containing only the stochastic integral term. The two types of equations are connected to each other by their initial values. We prove that the solutions of these equations both converge strongly to the solution of the Schrödinger type equation.     

Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type “admissible strategy” against “admissible control”, and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman–Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadène, Hassani (Probab Theory Relat Fields 132:237–264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier. Partially supported by the NSF of P.R.China (No. 10701050; 10671112), Shandong Province (No. Q2007A04), and National Basic Research Program of China (973 Program) (No. 2007CB814904).     

Simplified renormalization group method for ordinary differential equations

The renormalization group (RG) method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.     

Stokes flow of a micropolar fluid exterior to several non-intersecting closed surfaces,but contained by an exterior contour

The problem of determining the Stokes flow of a micropolar fluid exterior to several closed surfaces but contained by an exterior contour that encloses all the interior surfaces, is formulated as a system of linear Fredholm integral equations of the second kind. These integral equations are obtained when the velocity and microrotation vector fields are represented by a double-layer potential with unknown density, and certain singular solutions of the Stokes' micropolar equations. This double-layer potential is defined over the union of all the surfaces involved including the exterior contour. The singularities, corresponding to a concentrated force and concentrated couple located within each interior surface, give rise to force and torque whose magnitudes are linearly dependent on the unknown density of the double layer. It is shown that the system possesses a unique continuous solution when the boundaries are Lyapunov surfaces and the boundary data is continuous.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

On an iterative method for a class of integral equations of the first kind

In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non-negativity of density functions, the proposed iterative scheme was defined in such a way that only non-negative solutions can be approximated. The first part of the paper presents a motivation for the iterative method and discusses its convergence. In particular, it is shown that there is a connection between the iterative scheme and a certain concave functional associated with integral equations of this type. This functional can be interpreted as a generalization of the log-likelihood function of a model from emission tomography. The second part of the paper investigates the convergence properties of the discrete analogue of the iterative method associated with the discretized equation. Sufficient conditions for local convergence are given; and it is shown that, in general, convergence is very slow. Two numerical examples are presented.     

Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

On solutions set of a multivalued stochastic differential equation

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.     

Nonlocal diffusion,a Mittag-Leffler function and a two-dimensional Volterra integral equation

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

周期手性结构电磁散射问题的积分方程方法

In this paper, we consider the electromagnetic scattering by a periodic chiral structure. The media is homogeneous and the structure is periodic in one direction and invariant in another direction. The electromagnetic fields inside the chiral medium are governed by Maxwell equations together with the Drude-BornFedorov equations. We simplify the problem to a two-dimensional scattering problem and discuss the existence and the uniqueness of solutions by an integral equation approach. We show that for all but possibly a discrete set of wave numbers, the integral equation has a unique solution.