ON RITZ METHOD OF ANINTEGRO-DIFFERENTIAL EQUAITON

This paper deals with the Ritz method of an integro-differential equation related with Riemann zeta-function.     

Explicit Representations for Local Lagrangian Numerical Differentiation

It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.     

Bifurcations of travelling wave solutions for two generalized Boussinesq systems

Using the methods of dynamical systems for two generalized Boussinesq systems, the existence of all possible solitary wave solutions and many uncountably infinite periodic wave solutions is obtained. Exact explicit parametric representations of the travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

Exact Pollard-like internal water waves

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates above the thermocline. The solution is provided in Lagrangian coordinates. In the process we derive the appropriate dispersion relation for the internal water waves in a stable stratification and discuss the particles paths. An analysis of the dispersion relation for the constructed model identifies one mode of the internal water waves.     

On Meeting Energy Balance Errors in Co-Simulations

The term co-simulation denotes the coupling of some simulation tools for dynamical systems into one big system by having them exchange data at points of a fixed time grid and extrapolating the received data into the interval, while none of the steps is repeated for iteration. From the global perspective, the simulation thus has a strong explicit component. Frequently, among the data passed across subsystem boundaries there are flows of conserved quantities, and as there is no iteration of steps, system-wide balances may not be fulfilled: the system is not solved as one monolithic equation system. If these balance errors accumulate, simulation results become inaccurate. Balance correction methods which compensate these errors by adding corrections for the balances to the signal in the next coupling time step have been considered in past research. But establishing the balance of one quantity a posteriori due to the time delay in general cannot establish the balances of quantities that depend on the exchanged quantities, usually energy. In most applications from physics, the balance of energy is equivalent to stability. In this paper, a method is presented which allows users to choose the quantity that should be balanced to be that energy, and to accurately balance it. This establishes also numerical stability for many classes of stable problems.     

High‐resolution methods for preserving the sum of mass fractions: improved χ‐scheme and an alternative

When high resolution convection schemes are used for discretizing chemical species mass balance equations, the mass fractions are not guaranteed to add to one. We show that a proposed remedy called χ —scheme (Darwish and Moukalled, Comput. Methods Appl. Mech. Engrg. 192 (2003): 1711) will degrade to a diffusive first‐order scheme when a chemical species vanishes from the mixture, for example, because of chemical reactions. We propose an improvement to the χ ‐scheme to overcome this problem. Furthermore, a computationally efficient alternative scheme is proposed and evaluated with several examples, to quantify the improvements in the accuracy and the computational time. Published 2013. This article is a U.S. Government work and is in the public domain in the USA.     

Galerkin domain decomposition procedures for parabolic equations on rectangular domain

Galerkin domain decomposition procedures for parabolic equations with three cases of boundary conditions on rectangular domain are discussed. These procedures are non‐iterative and non‐overlapping ones. They rely on implicit Galerkin method in the sub‐domains and integral mean method on the inter‐domain boundaries to present explicit flux calculation. Thus, the parallelism can be achieved by the use of these procedures. Two kinds of approximating schemes are presented. Because of the explicit nature of the flux calculation, a less severe time‐step constraint is derived to preserve stability. To bound L²‐norm error estimates, new elliptic projections are established and analyzed. Numerical experiments are provided to confirm theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

NONCLASSICAL POTENTIAL SYMMETRIES AND INVARIANT SOLUTIONS OF HEAT EQUATION

Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conserved form can be constructed by using the nonclassical potential symmetry generators which are derived from their adjoint system. These explicit solutions cannot be obtained by using the Lie or Lie-Backlund symmetry group generators of differential equations.     

Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions

Consider the optimal stopping problem of a one-dimensional diffusion with positive discount. Based on Dynkin's characterization of the value as the minimal excessive majorant of the reward and considering its Riesz representation, we give an explicit equation to find the optimal stopping threshold for problems with one-sided stopping regions, and an explicit formula for the value function of the problem. This representation also gives light on the validity of the smooth-fit (SF) principle. The results are illustrated by solving some classical problems, and also through the solution of: optimal stopping of the skew Brownian motion and optimal stopping of the sticky Brownian motion, including cases in which the SF principle fails.     

Solution of one class of systems of stochastic differential equations

In this paper we prove that instead of solving a class of systems of stochastic differential equations with a multidimensional Wiener process one can solve a system of total differential equations. The latter system admits an application of classical methods. This fact enables one to solve the initial system explicitly.