An explicit finite difference solution to the convection-dispersion equation

An explicit finite difference equation has been development for the solution of the convection-dispersion equation. This equation has been over the entire range of 2D/vΔx between zero and one, region where no completely satisfactory method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.     

Numerical solution of the impact problem for a rigid mass and a viscoelastic rod of finite length

The problem of the propagation and interaction of longitudinal waves in a rod struck by a rigid mass is considered. A numerical computer solution is obtained for the case when the mechanical behavior of the rod material is expressed by a rectangular relaxation spectrum. A numerical method of integral transform inversion is developed and applied to problems of polymer dynamics.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 450–456, May–June, 1971.     

Global bifurcations of a taut string with 1:2 internal resonance

The global bifurcations of a taut string are investigated with the case of 1:2 internal resonance. The method of multiple scales is applied to obtain a system of autonomous ordinary differential equations. Based on the normal form theory, the desired form for the global perturbation method is obtained. Then the method developed by Kovacic and Wiggins is used to find explicit sufficient conditions for chaos to occur by identifying the existence of a Silnikov-type homoclinic orbit. Finally, numerical results obtained by using fourth-order Runge–Kutta method agree with the theoretical analysis at least qualitatively.     

An explicit formula for the action of a finite group on a commutative ring

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map is defined. is onto if and only if there exists an element x_H such that . We will show that the existence of x_P for every subgroup P of prime order determines the existence of x_G by exhibiting an explicit formula for x_G in terms of the x_P, where P varies over prime order subgroups. Since is onto if and only if is, where g∈G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x_P’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.     

An explicit linear solution for the quadratic dynamic programming problem

For a given vectorx ₀, the sequence {x _t} which optimizes the sum of discounted rewardsr(x _t, x_t+1), wherer is a quadratic function, is shown to be generated by a linear decision rulex _t+1=Sx _t +R. Moreover, the coefficientsR,S are given by explicit formulas in terms of the coefficients of the reward functionr. A unique steady-state is shown to exist (except for a degenerate case), and its stability is discussed.     

An explicit formula for the solution of the Fisher-Wright selection model in population genetics

An explicit formula for the solution of the discrete version of the Fisher-Wright selection model is given. Mathematically, the equations of this model form a system of non-linear difference equations of first order.     

A new explicit solution method for the diffraction through a slit

The two-dimensional time harmonic problem of diffraction through a slit is considered for arbitrary Dirichlet data, prescribed in the slit; on the screen itself the wavefield satisfies a homogeneous Neumann boundary condition. First of all, a sequence of special Dirichlet data is constructed, for each of these data the resulting wavefield can be expressed in closed form. The sequence can be constructed in a way which yields an orthonormal basis of Dirichlet data. After this has been done, arbitrary Dirichlet data can be expanded into a series of the orthonormal basis functions; this results in a representation for the searched wavefield. The presented method gives a deep insight into the mathematical structure of the problem.     

An approach to studying nonsteady-state vibrations of a finite length cylinder

A method is given for analyzing the nonsteady-state vibrations of a finite length cylinder based on using Laplace transforms and approaches developed previously for determining the vibration natural frequencies and forms. A numerical method is demonstrated for determining residues with reference to expressions for displacements and stresses. Quantitative estimates are given for stresses in a cylinder for a specific principle of the change in load with time and features of the time-dependent behavior of displacements in relation to the magnitude of the area of loading at the end are studied.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 129–133, 1987.     

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of excursions of Markov processes. Our setting allows us to solve the Skorokhod embedding problem, in particular, for the age process of excursions of a Markov process, for diffusions and their signed age processes, for Azéma’s martingale and for Bessel processes of dimension smaller than 2.This work is a continuation and an important generalization of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110]. Our methodology is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Itô measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110] for measures with atoms.     

An analytical solution of the diffusion- convection equation over a finite domain

Numerical solutions to the diffusion-convection equation are usually evaluated through comparison with analytical solutions in one dimension. Literature survey indicates the most frequently used analytical solution is one derived for asemi-infinite domain. This paper presents an analytical solution to this problem over a finite domain. Comparison is made with a solution to a mathematically similar problem in heat conduction with radiation.     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Summary. A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed. Received August 31, 1998 / Published online July 12, 2000