Showing papers on "Summation equation published in 2003"

Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations

Abstract: The known (2+1)-dimensional breaking soliton equation, the coupled KP equation with three potentials and a new (3+1)-dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the (1+1)-dimensional AKNS equations. The Abel–Jacobi coordinates are introduced to straighten out the associated flows, from which algebraic-geometrical solutions of the (2+1)-dimensional breaking soliton equation, the coupled KP equation and the (3+1)-dimensional evolution equation are explicitly given in terms of the Riemann theta functions.

The electric field integral equation on Lipschitz screens: definitions and numerical approximation

Abstract: We use the integral equation approach to study electromagnetic scattering by perfectly conducting (non-orientable) Lipschitz screens. The well-posedness of the electric field integral equation is derived. The Galerkin method for this problem is analysed in a general setting and optimal error bounds are proved for conforming finite elements in natural norms.

Some Non-Linear S.P.D.E's That Are Second Order In Time

Abstract: We extend J.B. Walsh's theory of martingale measures in order to deal with stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.

A complex boundary integral method for multiple circular holes in an infinite plane

Abstract: A complex boundary integral equation method, combined with series expansion technique, is presented for the problem of an infinite, isotropic elastic plane containing multiple circular holes. Loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The analysis procedure is based on the use of a complex hypersingular integral equation that expresses a direct relationship between all the boundary tractions and displacements. The unknown displacements on each circular boundary are represented by truncated complex Fourier series, and all of the integrals involved in the complex integral equation are evaluated analytically. A system of linear algebraic equations is obtained by using a Taylor series expansion, and the block Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the accuracy, versatility, and efficiency of the approach.

Numerical solution of linear Fredholm and volterra integral equation of the second kind by using Legendre wavelets

Abstract: Uses the continuous Legendre wavelets on the interval [0,1) in the manner of M. Razzaghi and S. Yousefi, to solve the linear second kind integral equations. We use quadrature formula for the calculation of inner products of any functions, which are required in the approximation for the integral equations. Then, we reduced the integral equation to the solution of linear algebraic equations.

Fractional Diffusion, Irreversibility and Entropy

Abstract: Three types of equations linking the diffusion equation and the wave equation are studied: the time fractional diffusion equation, the space fractional diffusion equation and the telegrapher's equation. For each type, theentropy production is calculated and compared. It is found that the two fractional diffusions, considered as linking bridges between reversible and irreversible processes, possess counter-intuitive properties: as the equation becomes more reversible, the entropy production increases The telegrapher's equation does not have the same counter-intuitive behavior. It is suggested that the different behaviors of these equations might be related to the velocities of the corresponding random walkers.

On BV-Solutions of Some Nonlinear Integral Equations

Abstract: In this paper we prove uniqueness theorems for bounded variation (shortly: BV) solutions and continuous BV-solutions of the Hammerstein and the Volterra–Hammerstein integral equations. We investigate real-valued functions and functions with values in a Banach space.

Solvability and Spectral Properties of Integral Equations on the Real Line: II. $L^p$-Spaces and Applications

Abstract: We consider the solvability of linear integral equations on the real line, in operator form (λ − K)φ = ψ, where λ ∈ C and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on Lp(R), 1 ≤ p ≤ ∞, and on BC(R). We establish conditions on families of operators, {Kk : k ∈ W}, which ensure that if λ = 0 and λφ = Kkφ has only the trivial solution in BC(R), for all k ∈ W , then for 1 ≤ p ≤ ∞, (λ − K)φ = ψ has exactly one solution φ ∈ Lp(R) for every k ∈ W and ψ ∈ Lp(R). The results of considerable generality apply in particular to kernels of the form k(s, t) = κ(s − t)z(t) and k(s, t) = κ(s − t)z(s, t), where κ, κ ∈ L1(R), z ∈ L∞(R), z ∈ BC(R2) and κ(s) = O(s−b) as |s| → ∞, for some b > 1. As a significant application we consider the problem of acoustic scattering by a sound-soft, unbounded one-dimensional rough surface which we reformulate as a second kind boundary integral equation. Combining the general results of earlier sections with a uniqueness result for the boundary value problem, we establish that the integral equation is well-posed as an equation on Lp(R), 1 ≤ p ≤ ∞, and on weighted spaces of continuous functions.

Oscillation of an Euler-Cauchy dynamic equation

Abstract: The Euler-Cauchy differential equation and difference equation are well known. Here we study a more general Euler-Cauchy dynamic equation. For this more general equation when we have complex roots of the corresponding characteristic equation we for the first time write solutions of this dynamic equation in terms of a generalized exponential function and generalized sine and cosine functions. This result is even new in the difference equation case. We then spend most of our time studying the oscillation properties of the Euler-Cauchy dynamic equation. Several oscillation results are given and an open problem is posed.

A solution procedure for second–order difference equations and its application to electromagnetic–wave diffraction in a wedge–shaped region

Abstract: This paper proposes an efficient solution procedure for second–order functional difference equations, and outlines this procedure through investigating electromagnetic–wave diffraction by a canonical structure comprising an impedance wedge and an impedance sheet bisecting the exterior region of the wedge. Applying the Sommerfeld–Malyuzhinets technique to the original boundary–value problem yields a linear system of equations for the two coupled spectral functions. Eliminating one spectral function leads to a second-order difference equation for the other. The chief steps in this work consist of transforming the second–order equation into a simpler one by making use of a generalized Malyuzhinets function χϕ(α), and in expressing the solution to the latter in an integral form with help of the so–called S–integrals. From this integral expression one immediately obtains a Fredholm equation of the second kind for points on the imaginary axis of the complex plane. Solving this integral equation by means of the well–known quadrature method enables us to calculate the sought–for spectral function inside the basic strip via an interpolation formula and outside it via an analytic extension. The second spectral function is obtained through its dependence upon the first. The uniform asymptotic solution, which is of particular interest in the geometrical theory of diffraction, follows, by evaluating the Sommerfeld integrals in the far field from the exact one. Several examples demonstrate the efficiency and accuracy of the proposed procedure as well as typical behaviour of the far–field solutions for such a canonical problem of diffraction theory.

A meshless method for large deflection of plates

Abstract: The nonlinear integro-differential Berger equation is used for description of large deflections of thin plates. An iterative solution of Berger equation by the local boundary integral equation method with meshless approximation of physical quantities is proposed. In each iterative step the Berger equation can be considered as a partial differential equation of the fourth order. The governing equation is decomposed into two coupled partial differential equations of the second order. One of them is Poisson's equation whereas the other one is Helmholtz's equation. The local boundary integral equation method is applied to both these equations. Numerical results for a square plate with simply supported and/or clamped edges as well as a circular clamped plate are presented to prove the efficiency of the proposed formulation.

Direct excess entropy calculation for a Lennard-Jones fluid by the integral equation method

Abstract: The present work is devoted to the calculation of excess entropy by means of correlation functions, in the framework of integral equation theory. The tangent linear method is set up to get exact thermodynamic derivatives of the pair-correlation function, essential for the calculation of the physical quantities, as well as to carry out an optimization process for the achievement of thermodynamic consistency. The two-body entropy of the Lennard-Jones fluid is in very good agreement with the available molecular dynamics results, attesting the high degree of accuracy of the integral equation scheme. It is shown that an accurate prediction of the excess entropy and the resulting residual multiparticle entropy relies on the correct evaluation of the excess chemical potential, especially at high density. Two independent routes to calculate the latter are compared, and the consequences are discussed.

High-order solution for the electromagnetic scattering by inhomogeneous dielectric bodies

Abstract: [1] A high-order method of moment solution with quadrature-point-based sampling is presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. It is demonstrated that the method leads to exponential convergence in both the radar cross section (RCS) and the volume current density. It is also demonstrated that the method can be more efficient than surface field integral equation formulations for thin-material scattering.

A Walsh function method for a non-linear Volterra integral equation

Abstract: A new method for the numerical solution of linear and non-linear Volterra integral equations, using the discontinuous wavelet packets known as the Walsh functions, is proposed and investigated. Sufficient conditions for the method to converge are derived and a priori error estimates are obtained. Given sufficient regularity conditions on the integral equation, the method is shown to be locally of order two.

Approximations of solutions to nonlinear sobolev type evolution equations

Abstract: In the present work we study the approximations of solutions to a class of nonlinear Sobolev type evolution equations in a Hilbert space. These equations arise in the analysis of the partial neutral functional dierential equations with unbounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next we consider the Faedo-Galerkin approximations of the solutions and prove some convergence results. Finally we demonstrate some of the applications of the results established.

Boundary value problem with integral conditions for a linear third-order equation

Abstract: We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the generated operator.

Thermodynamically consistent equation of state of hard sphere fluids

Abstract: The Wiener–Hopf technique has been been applied to solve the Ornstein–Zernike equation for hard sphere fluids and to calculate thereby a thermodynamically consistent equation of state. An analytic form of a thermodynamically consistent equation of state for hard sphere fluids is obtained in which the correlation range is treated as an adjustable parameter. With a suitable choice of the range parameter the equation of state presented is found to be numerically comparable to the Carnahan–Starling equation of state in accuracy.

Coupled-channel integral equations for quasi-one-dimensional systems

Abstract: An integral equation approach is developed for the propagation of electrons in two-dimensional quantum waveguides. The original two-dimensional problem is transformed into a set of one-dimensional coupled equations by expanding the full wave function in terms of simple transverse basis functions. The equivalence of the Schrodinger equation with suitable boundary conditions in configuration space to an integral equation approach in momentum space can thus be illustrated in a coupled channel situation with a minimum of geometrical complications. The application to scattering from a point defect embedded in a waveguide is considered. In this case the scattering integral equations reduce to a set of algebraic equations, and typical coupled channel phenomena can be discussed through straightforward mathematical techniques. The convergence problems due to a singular perturbation are briefly considered, and the differences between genuine one-dimensional problems and the present two-dimensional case are discussed.