Approximate Linear Equation Research Articles

Power series method and approximate linear differential equations of second order

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

A Utility Function to Solve Approximate Linear Equations for Decision Making

Suppose there are a number of decision variables linearly related to a set of outcome variables. There are at least as many outcome variables as the number of decision variables since all decisions are outcomes by themselves. The quality of outcome is evaluated by a utility function. Given desired values for all outcome variables, decision making reduces to “solving” the system of linear equations with respect to the decision variables; the solution being defined as decision variable values such that maximize the utility function. This paper proposes a family of additively separable utility functions which can be defined by setting four intuitive parameters for each outcome variable: the desired value of the outcome, the lower and the upper limits of its admissible interval, and its importance weight. The utility function takes a nonnegative value within the admissible domain and negative outside; permits gradient methods for maximization; is designed to have a small dynamic range for numerical computation. Small examples are presented to illustrate the proposed method.

Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion.

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen’s system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA’s performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license.

Approximate Solution of Some Classes of Integral Equations Using Bernstein Polynomials of Two-Variables

The research aims to find approximate solutions for two dimensions Fredholm linear integral equation. Using the two-variables of the Bernstein polynomials we find a solution to the approximate linear integral equation of the type two dimensions. Two examples have been discussed in detail.

Approximate Solution of Some Classes of Integral Equations Using Bernstein Polynomials of Two-Variables

The research aims to find approximate solutions for two dimensions Fredholm linear integral equation. Using the two-variables of the Bernstein polynomials we find a solution to the approximate linear integral equation of the type two dimensions. Two examples have been discussed in detail.

A model for the energy bands of an “open”-type periodic structure: a periodic viaduct coupled with the half-space

The periodic viaduct in this study was defined as a new kind of periodic structure—the “open”-type periodic structure—for the first time. Knowledge about the energy bands of this kind of “open” periodic viaduct is important for its aseismic design. Using the transfer matrix method and the compliances for the pile foundations, the impedances for the piers were obtained. Based on the Bloch theorem and the transfer matrix method, the nonlinear polynomial eigenvalue equation for the energy bands of the periodic viaduct undergoing in-plane motion was derived using the impedance of the piers. Based on the obtained nonlinear eigenvalue equation, the approximate linear eigenvalue equation for the periodic viaduct was obtained, and numerical results for the energy bands of the periodic viaduct were presented. The numerical results in this paper demonstrate that when the periodic viaduct is undergoing in-plane motion, there exist three lattice waves: the first kind of wave is a highly decaying wave; the second kind of lattice wave can propagate only at some frequency ranges; and the third kind of lattice wave can propagate at most frequencies.

Energy bands of a periodic viaduct in out-of-plane vibration: Coupling with a half-space

Abstract In this study, the periodic viaduct is used as an example to introduce a new type of phononic crystal structure: the “open”-type phononic crystal structure. A numerical model for analysis of the energy band of a periodic viaduct undergoing out-of-plane vibration is developed in this study. The most remarkable characteristic of the proposed model for the energy band of the periodic viaduct is that it can take into account the coupling between the periodic viaduct and the half-space soil. The viaduct considered in this study is assumed to be a regularly periodic arrangement of unit cells along its longitudinal direction. For simplicity, each unit cell is assumed to be consisted of a pile foundation, a pier and a horizontal beam. To obtain the compliances for the pile foundations, the pile-soil interaction problem is solved by the fictitious pile method first. Using the transfer matrix method and the obtained compliances for the pile foundations, the impedances for the piers are obtained. Based on the Bloch theorem and the transfer matrix method, the nonlinear polynomial eigenvalue equation for the periodic viaduct is derived using the impedance of the piers. Utilizing the obtained nonlinear eigenvalue equation, the approximate linear eigenvalue equation for the periodic viaduct is obtained and numerical results for the energy bands of the periodic viaduct are presented. Numerical results of this paper show that for the out-of-plane vibration of the periodic viaduct, there are three kinds of lattice waves propagating in the periodic viaduct. Moreover, in a low frequency range, all three lattice waves are evanescent, which will lead to the localization of lattice waves in the periodic viaduct.

Saturation in asymmetric structures under internal resonance

In this paper, a novel model for a single-storey irregular-plan building, subjected to earthquake and harmonic translational excitation, is presented to consider non-linear inertial coupling terms, which are ignored in conventional linear models. Furthermore, the effect of internal resonance is investigated. The governing equations of motion for a model of this type of structure are achieved in a non-inertial rotational system of reference, attached to the centre of mass of the rigid diaphragm of its floor. Non-linearities are caused by non-linear inertial coupled lateral and torsional responses. Using the method of multiple scales, the non-linear equations are reduced to approximate linear equations, and subsequently the dynamic stability of the steady-state responses near a two-to-one ratio of internal resonant conditions is studied. The difference between the responses of the proposed non-linear model and the conventional linear one is shown in time history and frequency domain analyses. By inspecting the effects of amplitude and frequency detuning parameters in the responses, some non-linear phenomena, such as saturation, hysteretic and jumping, are observed.

Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynamically compensate the solution using real flow information

Reproduction of the exact structure of real turbulent flows is crucial in many applications. Four Dimensional variation (4D-VAR) is widely used in numerical weather forecasting, but it requires huge computational power to repeatedly solve flow dynamics and its adjoint, and, therefore, is not suitable to apply to problems of real-time flow reproduction such as feedback flow control. Kalman filter and observer, in which numerical solution converges to the real state asymptotically by means of the feedback signal proportional to the difference between the calculated state and the real state, requiring much less computational load than the variational method, are potential candidates to solve the problem. By comparing Kalman filter and observer, the latter has simpler structure retaining essential part of the state estimation. This study deals with a special type of observer, or measurement-integrated simulation (MI simulation), in which a SIMPLER-based flow solver is used as the mathematical model of the system in place of approximate small dimensional linear differential equations usually used in observers. Reproduction of the exact structure of a turbulent flow was investigated by a MI simulation. A numerical experiment was performed for a fully developed turbulent flow in a pipe with a square cross section. The MI simulation was performed with the feedback from the standard solution in the flow domain for the cases using: (1) all velocity components at all grid points, (2) partial velocity components at all grid points, or (3) all velocity components at partial grid points. Convergence of the MI simulation to the standard solution was investigated using the steady error norm for the convergent state and the time constant for the transient state. The result of the MI simulation using all the velocity information exponentially converges to the standard solution with a steady state error reduced from that of the ordinary simulation in a range of the feedback gain. Decreasing the feedback gain reduces the effect of feedback, and a feedback gain which is too large destabilizes the closed loop system, resulting in large error. The time constant decreases almost inversely proportional to the feedback gain as long as the feedback system is stable. For the MI simulation with the feedback using limited information, feedback using two velocity components by omitting one transverse velocity component showed a good result, although the other results were not satisfactory. For the MI simulation with the feedback using limited grid points, the result of the MI simulation applying the feedback at the grid points on every 20th plane in the x 1 direction was almost the same as that using all grid points at some feedback gain, while the result with the feedback on the planes skipped in the x 2 direction requires 10 times more planes to achieve the same reduction rate.

Counterion condensation and shape within Poisson–Boltzmann theory

An analytical approximation to the nonlinear Poisson-Boltzmann (PB) equation is applied to charged macromolecules that possess one-dimensional symmetry and can be modeled by a plane, infinite cylinder, or sphere. A functional substitution allows the nonlinear PB equation subject to linear boundary conditions to be transformed into an approximate linear (Debye-Hückel-type) equation subject to nonlinear boundary conditions. A simple analytical result for the surface potential of such polyelectrolytes follows, leading to expressions for the amount of condensed (or renormalized) charge and the electrostatic Helmholtz energy for polyelectrolytes. Analytical high-charge/low-salt and low-charge/high-salt limits are shown to be similar to results obtained by others based on PB or counterion condensation theory. Several important general observations concerning polyelectrolytes treated within the context of PB theory can be made including: (1) all charged surfaces display some counterion condensation for finite electrolyte concentration, (2) the effect of surface geometry is described primarily by the sum of the Debye constant and the mean curvature of the surface, (3) two surfaces with the same surface charge density and mean curvature condense approximately identical fractions of counterions, (4) the amount of condensation is not determined by a predefined "condensation distance" although such a distance can be determined uniquely from it, and (5) substantial condensation occurs if the Debye constant of the electrolyte is much less than the mean curvature of a highly charged polyelectrolyte.

New Learning Algorithm for High-Quality Velocity Measurement and Control When Using Low-Cost Optical Encoders

A new compensation method that can greatly reduce the slit errors (i.e., transition location errors) due to nonidealities in optical incremental encoders caused by manufacturing limitations in the code wheel, optical components, and analog circuitry is presented. An M/T-type constant sample-time digital tachometer (CSDT), which involves pulse-count and high-frequency timer measurement, effectively time-stamps the encoder transitions. Using CSDT-based data, encoder compensation techniques that improve velocity measurement accuracy are presented. These do not require precise knowledge of the shaft velocity, thereby eliminating the need for high-accuracy reference equipment. During the initial learning stage (possibly performed in situ ), slit errors are calculated through pseudoinverse-based solutions of simple approximate linear equations or an iterative method that requires very little memory storage. Subsequent operation of the motion system utilizes adjusted slit positions for more accurate velocity calculation. The performance improvement in velocity measurement is experimentally demonstrated using motor drive systems, each of which includes a field-programmable gate array (FPGA) and a digital signal processor (DSP). Results from open-loop velocity measurement and closed-loop servo control applications are given, with the latter highlighting the resultant reduction in high-frequency motor torque. Slit error reductions in the range of 60%-86% are obtained (typically approximately 80%) with a similar improvement in velocity measurement error.

Controllability and synthesis of control for nonlinear systems

We solve motion control problems with feedback under the linear approximation of equations of the considered system. Due to the smallness of the nonlinear part and the controllability of the linear approximating system this property is preserved under a nonlinear perturbation. We perform the study by the method of Lyapunov functions and by the comparison method within finite and infinite time intervals. We illustrate the obtained results by an example of controlling a simple mechanical system.

On the Concept of Electrical Impedance for an Electrolytic Cell

The analysis of the electrical impedance of an electrolytic cell in the shape of a slab is performed. We have solved, numerically, the differential equations governing the phenomenon of the redistribution of the ions in the presence of an external electric field, and compared the results with the ones obtained by solving the linear approximation of these equations. The control parameters in our study are the amplitude and the frequency of the applied voltage, assumed a simple harmonic function of the time. We show that for the large amplitudes of the applied voltage, the actual current is no longer harmonic at low frequencies. From this result it follows that the concept of electrical impedance of a cell is a useful quantity only in the case where the linear approximation of the fundamental equations of problem work well.

Eigenwaves in a two-component system of particles with nonzero magnetic moments

Two-component systems of particles with nonzero intrinsic magnetic moments are shown to support not only the transverse electromagnetic and longitudinal waves, but also self-consistent spin waves and waves that propagate independently in each component and degenerate into oscillations for spin-zero systems. For all types of waves, the dispersion relations are obtained by solving the quantum hydrodynamics equations and field equations in linear approximation.

A PARALLEL TO THE LEAST SQUARES FOR POSITIVE INVERSE PROBLEMS

A new method is proposed to solve systems of linear approximate equations Xθ ≈ y where the unknowns θ and the data y are positive and the matrix X consists of nonnegative elements. Writing the i-th near-equality Xi ·θ/yi ≈ 1 the assumed model is Xi ·θ/yi = ζi with mutually independent positive errors ζi . The loss function is defined bywi(ζi − 1)log ζi in which wi is the importance weight for the i-th near-equality. A reparameterization reduces the method to unconstrained minimization of a smooth strictly convex function implying the unique existence of positive solution and the applicability of Newton's method that converges quadratically. The solution stability is controlled by weighting prior guesses of the unknowns θ. The method matches the maximum likelihood estimation if all weights wi are equal and ζi independently follow the probability density function ∝ t ω(1−t) ,0 <ω .

Estimation of radius and thickness of a thin spherical shell in water using the midfrequency enhancement of a short tone burst response

The midfrequency enhancement phenomenon for tone burst backscattering by thin spherical shells in water has been discussed by several investigators. In their works, it is found that the earliest elastic tone burst echo is enhanced relative to the specular reflection, and this enhancement is mainly due to the lowest subsonic antisymmetric Lamb wave. In this paper, the tone burst backscattering obtained from the convolution integral of the incident tone burst and the impulse response by a submerged spherical shell is investigated to display the midfrequency enhancement. The modified ray approximations are used to calculate the echo contributions from different Lamb waves. The numerical results show that the ratio of a∕h and the dimensionless echo delay have nearly linear relationship with the frequency of greatest enhancement. Based on this property, two linear approximate equations are formulated to evaluate the radius and thickness of a thin spherical shell. A simple method is developed to estimate the frequency of greatest enhancement and the corresponding echo delay from a short tone burst echo with a higher carrier frequency. The evaluated results show that the present method is effective on determination of the radius and thickness of a thin spherical shell in water.

Transverse vibrations of rotating shafts: Probability density and first-passage time of whirl radius

Transverse vibrations are considered for a single mass/two-degrees-of-freedom rotating shaft with linear internal or “rotating” damping and nonlinear external damping. The shaft is excited by external random forces. Analysis of resulting random vibrations is based on stochastic averaging method which yields separated (in the linear approximation) equations for complex amplitudes of forward and backward whirling motions. The former of these motions is shown to be dominant at rotation speeds in the vicinity of the instability threshold. Using this approximation an analytical solution is obtained for probability density of squared radius of the shaft's whirl. This solution can be used to detect on-line shaft's instability from its observed response. Solution is also obtained for expected time for reaching given level by the squared whirl radius of the shaft.

Diffusion and memory effects for stochastic processes and fractional Langevin equations

We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker–Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the non-linear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker–Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative (defined according to Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969). In the generic linear or non-linear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution.

Nonsymmetric unified theory of gravitation, electromagnetism and Yang–Mills field

The recently studied new variation of Einstein's metric nonsymmetric unified field theory of gravitation and electromagnetism is enlarged to include the Yang–Mills field theory. It is shown that the antisymmetric part of the metric tensor, now a 2 × 2 matrix, can be made to describe both a field obeying Maxwell's equations and a field obeying Yang–Mills's field equations in the flat space linear approximation, thereby making its identification with the sum of a generalized electromagnetic and isotopic field strength tensors a possibly consistent procedure. The theory is shown to be free of unphysical ghost-negative energy radiative modes even when expanded on a curved Riemannian background. The Einstein–Maxwell–Yang–Mills theory is contained in the first approximation of the field equations on a curved general relativity background.