Construction Of Analytical Solutions Research Articles

Analytical-numerical solution of coupled singular mixed diffusion type problems

The aim of this paper is to construct analytical approximate solutions for systems of partial differentiable equations of the form Eu t ( x, t) − = Au xx ( x, t) = 0 for x ϵ [0, 1] and t > 0, subject to u(0, t) = 0and Bu(1, t) + Cu x (1, t) = 0 for t > 0 and u( x, 0) = f( x), x ϵ [0, 1]. Here, u( x, t), f( x) are m-component vectors and E, A, B, C are m th -order complex matrices where B and C are not simultaneously singular and E is singular. Scalar regular Sturm-Liouville problems are used to find exact series solution; the series is truncated, then its matrix exponential terms replaced by Taylor series approximation. With appropriate selections, the resulting approximate solution will be arbitrarily close (uniformly) on a domain [0, 1] × [ t o , t 1] where t o > 0.

Analytical/numerical description of sound field in irregular waveguides

The problems of the harmonic wave propagation in waveguides with varying cross sections and sound radiation from open ends of ones are considered. The main attention gives to development method of complete analytical description of the sound fields in both cases. This gives a basis to get accurate quantitative and qualitative description of the fields near the junctions of the waveguide cross sections and open ends. Numerical implementation of the general analytical formulas requires special procedures to be adequate the local singularities in vicinities of corner points. The approach is based on the introduction of the idea of the general solution of the acoustical boundary problem. The idea gives a way to construct analytical solutions for the problems under consideration and can be extended to a broad spectrum of radiation and scattering problems. The well known solutions for regular waveguides in the form of normal waves are used as important part of the general solutions. Numerical data for concrete waveguides demonstrate the accuracy and convergence rate of the method. Analysis of the data shows interesting physical properties of propagating waves, flows of energy, and local structure of sound field at the juncture of waveguide parts.

On construction of analytic solutions to the three body problem by use of computer experiments

New analytic approximations of the general three-body problem are given. They are obtained by making use of approximate analytic theory as well as 26000 computed orbits. These solutions give the energy of the binary after the three-body system has broken up, or alternatively the terminal velocity of the escaping body. It is shown that the hard binary scattering and the break-up of bound three-body systems are described by the same analytic expression.

Suction through Broad Oceanic Gaps

Abstract This paper supplements an earlier article, by Nof and Olson, on the nonlinear flow through broad gaps. In that paper, the steady inviscid transport through a gap located on the left-hand side of a boundary current has been computed. When the gap is located on the right-hand side, instead of the left-hand side, the Nof and Olson solution breaks down because of a singularity at the upstream edge of the gap. The singularity is associated with an infinitely negative pressure resulting from the fact that particles make a “U” turn as they pass through the gap. The present study focuses on these special singular cases. To avoid the singularity, the use of the integrated moment of momentum is adopted because with an appropriate choice of a coordinate system, the contribution of the unknown force (associated with the singularity) to the integrated torque vanishes. This enables one to construct analytical solutions which give the transport through the gap as well as the force associated with the singularit...

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.