The Generalized Laguerre Matrix Method or Solving Linear Differential-Difference Equations with Variable Coefficients

In this paper, a new and efficient approach based on the generalized Laguerre matrix method for numerical approximation of the linear differential-difference equations (DDEs) with variable coefficients is introduced. Explicit formulae which express the generalized Laguerre expansion coefficients for the moments of the derivatives of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. In the scheme, by using this approach we reduce solving the linear differential equations to solving a system of linear algebraic equations, thus greatly simplify the problem. In addition, several numerical experiments are given to demonstrate the validity and applicability of the method

Multidomain Solutions of Incompressible Flows with Complex Geometry by Generalized Differential Quadrature. G.U. Aero Report 9118

A multi-domain generalized differential quadrature method for the solution of two-dimensional, steady, incompressible Navier-Stokes equations in the stream function-vorticity formulation around an arbitrary geometry is presented, and applied to the flows past a backward facing step and a square step in a channel. In each subdomain, the spatial derivatives are discretized by local generalized differential quadrature. The resultant set of ordinary differential equations for vorticity are solved by the 4-stage Runge-Kutta scheme, and the set of algebraic equations for the stream function are solved by LU decomposition. Patching conditions at the interface of subdomains are used. A residual averaging technique is applied to accelerate the convergence to steady state resolution. Good agreement is obtained, compared with available experimental data and other numerical results even though only a few grid points are used

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors

First order parent formulation for generic gauge field theories

We show how a generic gauge field theory described by a BRST differential can systematically be reformulated as a first order parent system whose spacetime part is determined by the de Rham differential. In the spirit of Vasiliev's unfolded approach, this is done by extending the original space of fields so as to include their derivatives as new independent fields together with associated form fields. Through the inclusion of the antifield dependent part of the BRST differential, the parent formulation can be used both for on and off-shell formulations. For diffeomorphism invariant models, the parent formulation can be reformulated as an AKSZ-type sigma model. Several examples, such as the relativistic particle, parametrized theories, Yang-Mills theory, general relativity and the two dimensional sigma model are worked out in details.Comment: 36 pages, additional sections and minor correction