Random integral equation formulation of a generalized Langevin equation

In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.     

Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

Integration of the Tomonaga-Schwinger equation

Some integrations of the Tomonaga-Schwinger equation with a non-local interaction are studied with mathematical rigor. It is proved that the related initial value problem has a unique solution in any finite region of the space-time corresponding to each set of space-like surfaces which covers the region. Such an analysis can be extended to the case of quantum electrodynamics by the aid of a Lorentz-invariant topology introduced in the *-algebra of electromagnetic field operators.     

Higher order solutions of Lieb-Liniger integral equation

We study higher order solutions of Lieb-Liniger integral equation for a one-dimensional δ-function Bose gas. By use of the power series expansion method, the integral equation is solved and the correction terms which improve the Bogoliubov theory are calculated analytically in the weak coupling regime. Physical quantities such as the ground state energy and the chemical potential are represented by a dimensionless parameter γ=c/ρ, where c is the interaction strength and ρ is the number density of particles while the quasi-momentum distribution function is expressed in terms of a dimensionless parameter λ=c/K, where K is the cut-off momentum.     

Multiple scattering using the Foldy-Twersky integral equation

The importance of multiple scattering processes for the propagation of ultrasound through dispersions is assessed. Under certain well defined conditions, the Foldy-Twersky integral equation can be used to find a value for the amplitude and complex wavenumber of the multiply scattered plane wave collected by either the receiving or the emitting transducer. Expressions for the amplitude and wavenumber are evaluated in terms of particle number density, wavelength of ultrasound and the forward and backward scattering amplitudes. In the case where the wavelength is much greater than the particle size, explicit expressions for the real and imaginary parts of the wavenumber are provided, in terms of the scattering and absorption cross-sections, as far as terms in the square of the number density of particles.     

An integral equation approach to acoustic cloaking

The paper presents an original boundary integral equation (BIE) formulation for the analysis of the acoustic cloaking of a scatterer. The advantage of such an approach is the lower computational burden, especially when the analysis of a large portion of the hosting domain is required. The partial differential equation governing the propagation inside the cloak is recast in the form of non-homogeneous wave equation, with field sources depending on the mechanical properties of the cloak. The boundary integral formulation is derived using the standard procedure. The boundary element method (BEM) is used to derive the matrix transfer function of the cloak. The latter is applied to the incident field at the cloak's outer boundary to obtain the total field at arbitrary locations in the host. The formulation is applied to the simple case of a radially symmetric cloak embedding a circular obstacle. Numerical results are presented for sound-hard and sound-soft obstacles, including a study of the cloaking efficiency as a function of the frequency.     

麦克斯韦积分方程的一种数值解法

直接从麦克斯韦积分方程出发,采用FIT算法,模拟电磁场的分布情况。     

Admissibility of transition from the Smolukhovskii integral equation to the Einstein-Fokker-Planck differential equation

For Markov processes we consider limits for the growth rate of the probability density w(t_o, x_o; t, x) of the transition from state x_o at time to to states for which x lies between x and x + dx at time t. These limits follow from the condition of admissibility of transition from the Smolukhovskii integral equation to the Einstein-Fokker-Planck differential equation (EFP). We examine the influence of these limits on the form of the coefficients in EFT. The indicated examination is performed for the prelimit region as well, that is, the region at the end of which we have w(t_o, x_o; t, x) 0.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 99–104, October, 1986.The authors are deeply grateful to D. M. Sedrakyan and A. O. Melikyan for their interest in this work and useful comments.     

Helmholtz边界积分方程的多频计算

提出了利用无穷级数展开的方法,将波数从Helmholtz边界积分方程的特解中分离出来,使随波数变化的系统矩阵变为波数的矩阵级数形式,同时证明了级数截断时的收敛性。数值结果表明,结合CHIEF方法,用级数展开的方法不仅能有效地克服频域内非唯一现象,节省计算时间;而且当频率较高时,在单元粗剖分下也能得到满意的结果。     

The integral equation approach to four-nucleon bound states

Within the framework of the Yakubovsky four-body equations the 0⁺ bound states of ⁴He are determined. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The problem is reduced to the solution of four coupled integral equations in two variables. The separable approximation of the kernels makes it possible to reduce the problem to a set of single variable integral equations. The separable approximation method employed is based on the Hilbert-Schmidt expansion applied to the kernels of four-body equations. The ground state energy of ⁴He is found to be ?45.73 MeV, the excited 0⁺ level lies at ?11.69 MeV. In conclusion we discuss the accuracy of various approximate methods in the four-nucleon problem.     

Integration method for the Dirac equation

An integration method for the Dirac equation is proposed. The method, based on diagonalization, reduces the problem to one of integration of independent second-order differential equations.     

Solution of the transport integral equation with anisotropic scattering

The paper deals with the solution of the integral equation for particle transport in homogeneous material systems having plane and spherical symmetry. Emphasis is put on the explicit inclusion of anisotropic scattering (higher Legendre components of the scattering kernel). The present approach is based on a generalization of the Integral Transform method. The solution is represented as an expansion with respect to analytical basis functions with coefficients satisfying a certain linear system. The determination of this linear system and its matrix elements in a form convenient for numerical purposes is the central point of the paper.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

Derivation of lattice Boltzmann equation via analytical characteristic integral

A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.     

Structure of fluid interfaces: an integral equation study

We develop a method for treating an interface between coexisting fluid phases to study its structure and thermodynamical properties. Our approach includes calculated correlation functions. Approximations are intrinsically optimized. The method is successfully applied to the liquid vapour surface of a Lennard-Jones fluid, to the liquid–liquid interface of a demixing system and to the surface of a Stockmayer liquid towards vapour. In all cases we compare our results to simulations. The agreement with simulation results demonstrates the reliability of our approximations and shows that the technique we applied provides a powerful and robust method for studying inhomogeneous fluids.     

On the path integral solutions of the master equation

The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation we reproduce the result obtained by integrating over Fourier series coefficients and by other methods.