Iterative solution of a class of linear equations with application to reconstruction of three-dimensional object arrays

An iterative algorithm baaed on probabilistic estimation is described for obtaining the minimum-norm solution of a very large, consistent, linear system of equations AX = g where A is an (m times n) matrix with non-negative elements, x and g are respectively (n times 1) and (m times 1) vectors with positive components.

Intrinsic equations of steady rotating, incompressible viscous fluid flow

The equations governing the flow of a steady rotating incompressible viscous fluid are expressed in intrinsic form along the vortex lines and their normals. Using these equations the effects of rotation on the geometric properties of viscous fluid flows are studied. A particular flow in which the vortex lines are right circular helices is discussed.

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x1,…, xm) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.

An appendix to the paper "approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic"

The theory of Varley and Cumberbatch [l] giving the intensity of discontinuities in the normal derivatives of the dependent variables at a wave front can be deduced from the more general results of Prasad which give the complete history of a disturbance not only at the wave front but also within a short distance behind the wave front. In what follows we omit the index M in Eq. (2.25) of Prasad [2].

Finite field computation technique for exact solution of systems of linear equations and interval linear programming problems

An error-free computational approach is employed for finding the integer solution to a system of linear equations, using finite-field arithmetic. This approach is also extended to find the optimum solution for linear inequalities such as those arising in interval linear programming probloms.

A Finite Variable Difference Relaxation Scheme for hyperbolic-parabolic equations

Using the framework of a new relaxation system, which converts a nonlinear viscous conservation law into a system of linear convection-diffusion equations with nonlinear source terms, a finite variable difference method is developed for nonlinear hyperbolic-parabolic equations. The basic idea is to formulate a finite volume method with an optimum spatial difference, using the Locally Exact Numerical Scheme (LENS), leading to a Finite Variable Difference Method as introduced by Sakai [Katsuhiro Sakai, A new finite variable difference method with application to locally exact numerical scheme, journal of Computational Physics, 124 (1996) pp. 301-308.], for the linear convection-diffusion equations obtained by using a relaxation system. Source terms are treated with the well-balanced scheme of Jin [Shi Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, Mathematical Modeling Numerical Analysis, 35 (4) (2001) pp. 631-645]. Bench-mark test problems for scalar and vector conservation laws in one and two dimensions are solved using this new algorithm and the results demonstrate the efficiency of the scheme in capturing the flow features accurately.

The non-linear response of a gyrostabilized platform: Effects of correction torque time delay and of random inputs

First, the non-linear response of a gyrostabilized platform to a small constant input torque is analyzed in respect to the effect of the time delay (inherent or deliberately introduced) in the correction torque supplied by the servomotor, which itself may be non-linear to a certain extent. The equation of motion of the platform system is a third order nonlinear non-homogeneous differential equation. An approximate analytical method of solution of this equation is utilized. The value of the delay at which the platform response becomes unstable has been calculated by using this approximate analytical method. The procedure is illustrated by means of a numerical example. Second, the non-linear response of the platform to a random input has been obtained. The effects of several types of non-linearity on reducing the level of the mean square response have been investigated, by applying the technique of equivalent linearization and solving the resulting integral equations by using laguerre or Gaussian integration techniques. The mean square responses to white noise and band limited white noise, for various values of the non-linear parameter and for different types of non-linearity function, have been obtained. For positive values of the non-linear parameter the levels of the non-linear mean square responses to both white noise and band-limited white noise are low as compared to the linear mean square response. For negative values of the non-linear parameter the level of the non-linear mean square response at first increases slowly with increasing values of the non-linear parameter and then suddenly jumps to a high level, at a certain value of the non-linearity parameter.

The fully implicit stochastic-alpha method for stiff stochastic differential equations

A fully implicit integration method for stochastic differential equations with significant multiplicative noise and stiffness in both the drift and diffusion coefficients has been constructed, analyzed and illustrated with numerical examples in this work. The method has strong order 1.0 consistency and has user-selectable parameters that allow the user to expand the stability region of the method to cover almost the entire drift-diffusion stability plane. The large stability region enables the method to take computationally efficient time steps. A system of chemical Langevin equations simulated with the method illustrates its computational efficiency.

Effect of surface radiation on conjugate natural convection in a horizontal annulus driven by inner heat generating solid cylinder

This paper reports a numerical study of the laminar conjugate natural convection heat transfer with and without the interaction of the surface radiation in a horizontal cylindrical annulus formed between an inner heat generating solid circular cylinder and an outer isothermal circular boundary. Numerical solutions are obtained by solving the governing equations with a pressure correction method on a collocated (non-staggered) mesh. Steady-state results are presented for the flow and temperature distributions and Nusselt numbers for the heat generation based Grashof number ranging from 10(7) to 10(10), solid-to-fluid thermal conductivity ratios of 1, 5, 10, 50 and 100, radius ratios of 0.226 and 0.452 and surface emissivities of 0-0.8 with air as the working medium. It is observed that surface radiation reduces the convective heat transfer in the annulus compared to the pure natural convection case and enhances the overall Nusselt number.

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier-Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e(vertical bar k vertical bar/kd) at high wavenumbers vertical bar k vertical bar. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e-(C(k/kd) ln(vertical bar k vertical bar/kd)) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C-* = 1/ ln 2. The same behavior with a universal constant C-* is conjectured for the Navier-Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier-Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

Extending temporal simulations

Direct numerical simulations (DNS) of spatially growing turbulent shear layers may be performed as temporal simulations by solving the governing equations with some additional terms while imposing streamwise periodicity. These terms are functions of the means whose spatial growth is calculated easily and accurately from statistics of the temporal DNS. Equations for such simulations are derived.

Two-temperature accretion around rotating black holes: a description of the general advective flow paradigm in the presence of various cooling processes to explain low to high luminous sources

We investigate viscous two-temperature accretion disc flows around rotating black holes. We describe the global solution of accretion flows with a sub-Keplerian angular momentum profile, by solving the underlying conservation equations including explicit cooling processes self-consistently. Bremsstrahlung, synchrotron and inverse Comptonization of soft photons are considered as possible cooling mechanisms. We focus on the set of solutions for sub-Eddington, Eddington and super-Eddington mass accretion rates around Schwarzschild and Kerr black holes with a Kerr parameter of 0.998. It is found that the flow, during its infall from the Keplerian to sub-Kepleria transition region to the black hole event horizon, passes through various phases of advection: the general advective paradigm to the radiatively inefficient phase, and vice versa. Hence, the flow governs a much lower electron temperature similar to 10(8)-10(9.5) K, in the range of accretion rate in Eddington units 0.01 less than or similar to (M) over dot less than or similar to 100, compared to the hot protons of temperature similar to 10(10.2)-10(11.8) K. Therefore, the solution may potentially explain the hard X-rays and gamma-rays emitted from active galactic nuclei (AGNs) and X-ray binaries. We then compare the solutions for two different regimes of viscosity. We conclude that a weakly viscous flow is expected to be cooling dominated, particularly at the inner region of the disc, compared to its highly viscous counterpart, which is radiatively inefficient. With all the solutions in hand, we finally reproduce the observed luminosities of the underfed AGNs and quasars (e. g. Sgr A*) to ultraluminous X-ray sources (e. g. SS433), at different combinations of input parameters, such as the mass accretion rate and the ratio of specific heats. The set of solutions also predicts appropriately the luminosity observed in highly luminous AGNs and ultraluminous quasars (e. g. PKS 0743-67).