Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.

Approximate solutions for nonequilibrium airflow in hypersonic nozzles,

Contract no. AF40(600)-804. AEDC TN-60-181. Arnold Engineering Development Center, Arnold Air Force Station, Tennessee. Air Research and Development Command, United States Air Force."

Approximate solutions to the optimum climbing trajectory for a rocket-powered aircraft,

"Purdue Research Foundation. Research project no.1255. Project Ae-25. This research was supported by the National Advisory Committee for Aeronautics, Washington, D.C., under Contract no. NAW-6465."

Approximate solutions to optimum flight trajectories for a turbo-jet powered aircraft,

"Purdue Research Foundation. Research project no. 1255. Project Ae-25. This research was supported by the National Advisory Committee for Aeronautics, Washington, D. C., under Contract no. NAW-6465."

Neural fields with sigmoidal firing rates: approximate solutions

Many tissue level models of neural networks are written in the language of nonlinear integro-differential equations. Analytical solutions have only been obtained for the special case that the nonlinearity is a Heaviside function. Thus the pursuit of even approximate solutions to such models is of interest to the broad mathematical neuroscience community. Here we develop one such scheme, for stationary and travelling wave solutions, that can deal with a certain class of smoothed Heaviside functions. The distribution that smoothes the Heaviside is viewed as a fundamental object, and all expressions describing the scheme are constructed in terms of integrals over this distribution. The comparison of our scheme and results from direct numerical simulations is used to highlight the very good levels of approximation that can be achieved by iterating the process only a small number of times.

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.

Lagrangian approximations and weak solutions of the Navier-Stokes equations

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations.

Solutions of a two-dimensional dam-break problem

Solutions of a two-dimensional dam break problem are presented for two tailwater/reservoir height ratios. The numerical scheme used is an extension of one previously given by the author [J. Hyd. Res. 26(3), 293–306 (1988)], and is based on numerical characteristic decomposition. Thus approximate solutions are obtained via linearised problems, and the method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids non-physical, spurious oscillations.

On Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear PDEs: Case of Elliptic Simplest Equation

2010 Mathematics Subject Classification: 74J30, 34L30.

Solutions of Tikhonov functional equations and applications to multiplication operators on Szegö spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert-Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.

Hybrid modeling of convective laminar flow in a permeable tube associated with the cross-flow process

The confined flows in tubes with permeable surfaces arc associated to tangential filtration processes (microfiltration or ultrafiltration). The complexity of the phenomena do not allow for the development of exact analytical solutions, however, approximate solutions are of great interest for the calculation of the transmembrane outflow and estimate of the concentration, polarization phenomenon. In the present work, the generalized integral transform technique (GITT) was employed in solving the laminar and permanent flow in permeable tubes of Newtonian and incompressible fluid. The mathematical formulation employed the parabolic differential equation of chemical species conservation (convective-diffusive equation). The velocity profiles for the entrance region flow, which are found in the connective terms of the equation, were assessed by solutions obtained from literature. The velocity at the permeable wall was considered uniform, with the concentration at the tube wall regarded as variable with an axial position. A computational methodology using global error control was applied to determine the concentration in the wall and concentration boundary layer thickness. The results obtained for the local transmembrane flux and the concentration boundary layer thickness were compared against others in literature. (C) 2007 Elsevier B.V. All rights reserved.

An efficient heuristic for total flowtime minimisation in no-wait flowshops

In this paper, we address the problem of scheduling jobs in a no-wait flowshop with the objective of minimising the total completion time. This problem is well-known for being nondeterministic polynomial-time hard, and therefore, most contributions to the topic focus on developing algorithms able to obtain good approximate solutions for the problem in a short CPU time. More specifically, there are various constructive heuristics available for the problem [such as the ones by Rajendran and Chaudhuri (Nav Res Logist 37: 695-705, 1990); Bertolissi (J Mater Process Technol 107: 459-465, 2000), Aldowaisan and Allahverdi (Omega 32: 345-352, 2004) and the Chins heuristic by Fink and Voa (Eur J Operat Res 151: 400-414, 2003)], as well as a successful local search procedure (Pilot-1-Chins). We propose a new constructive heuristic based on an analogy with the two-machine problem in order to select the candidate to be appended in the partial schedule. The myopic behaviour of the heuristic is tempered by exploring the neighbourhood of the so-obtained partial schedules. The computational results indicate that the proposed heuristic outperforms existing ones in terms of quality of the solution obtained and equals the performance of the time-consuming Pilot-1-Chins.

Bending of stochastic Kirchhoff plates on Winkler foundations via the Galerkin method and the Askey-Wiener scheme

In this paper, the method of Galerkin and the Askey-Wiener scheme are used to obtain approximate solutions to the stochastic displacement response of Kirchhoff plates with uncertain parameters. Theoretical and numerical results are presented. The Lax-Milgram lemma is used to express the conditions for existence and uniqueness of the solution. Uncertainties in plate and foundation stiffness are modeled by respecting these conditions, hence using Legendre polynomials indexed in uniform random variables. The space of approximate solutions is built using results of density between the space of continuous functions and Sobolev spaces. Approximate Galerkin solutions are compared with results of Monte Carlo simulation, in terms of first and second order moments and in terms of histograms of the displacement response. Numerical results for two example problems show very fast convergence to the exact solution, at excellent accuracies. The Askey-Wiener Galerkin scheme developed herein is able to reproduce the histogram of the displacement response. The scheme is shown to be a theoretically sound and efficient method for the solution of stochastic problems in engineering. (C) 2009 Elsevier Ltd. All rights reserved.