An efficient algorithm for solving Zakharov-Shabat inversescattering problem

An efficient numerical method for solving Zakharov-Shabat (ZS) inverse scattering problem is presented. In this method, instead of equivalent second-order differential equations to the Gel'fand-Levitan-Marchenko (GLM)-type integral equations, equivalent first-order differential equations are adopted and sufficiently accurate solutions to Zakharov-Shabat inverse problem can be achieved without iterations. Examples for applying it to design nonuniform transmission line (NTL) filters are also provided     

Time-Domain Analysis of a Lossy Nonuniform Transmission Line

An analytical solution of the coupled Telegrapher's equations for the voltage and current on a homogeneous lossy transmission line is presented. The resulting expression is obtained in the form of an exact time-domain propagator operating on the line voltage and current. It is shown that an application of Simpson's rule yields a simple accurate numerical representation of the propagator that can be used to analyze both homogeneous and inhomogeneous transmission lines. Numerical dispersion in lossy media is examined proving that this method has no numerical dispersion.      

High precision approximate analytical solutions to ODE using LS-SVM

The problem of solving differential equations and the properties of solutions have always been an important content of differential equation the study. In practical application and scientific research, it is difficult to obtain analytical solutions for most differential equations. In recent years, with the development of computer technology, some new intelligent algorithms have been used to solve differential equations. They overcomes the drawback of traditional methods and provide the approximate solution in closed form (i.e., continuous and differentiable). The least squares support vector machine (LS-SVM) has nice properties in solving differential equations. In order to further improve the accuracy of approximate analytical solutions and facilitative calculation, a novel method based on numerical methods and LS-SVM methods is presented to solve linear ordinary differential equations (ODEs). In our approach, a high precise of the numerical solution is added as a constraint to the nonlinear LS-SVM regression model, and the optimal parameters of the model are adjusted to minimize an appropriate error function. Finally, the approximate solution in closed form is obtained by solving a system of linear equations. The numerical experiments demonstrate that our proposed method can improve the accuracy of approximate solutions.     

A Note on the Generation of Phase Plane Plots on a Digital Computer

A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated as a pair of coupled differential equations with phase error and phase error derivative as dependent variables and arc length as the independent variable, which is uniformly distributed along the phase plane trajectory. The sinusoidal phase-locked loop and polarity-type Costas loop are used as illustrations of the proposed method of solution.     

An Analytic Model to Account for Quantum–Mechanical Effects of MOSFETs Using a Parabolic Potential Well Approximation

An analytic model to account for the quantum–mechanical effects (QMEs) of the MOSFETs using a parabolic potential well approximation is presented in this paper. Based on the solution of the coupled SchrÖdinger and Poisson equations following the Wentzel–Kramer–Brillouin method, a transcendental equation of the subband energy level has been rigorously derived to obtain an approximate analytic solution for the subband energy levels and the inversion charge centroid. Calculated results from the obtained analytical solution are compared with the previous approximate solutions reported in the literature and the numerically simulated data. A good agreement between the analytical and numerical is obtained, proving the validity of the analytic modeling of QMEs.     

Time-Domain Perturbational Analysis of Nonuniformly Coupled Transmission Lines

A method of analyzing the time-domain behavior of a pair of nonuniformly coupled, dispersionless transmission lines is presented. The coupling coefficient of the system is assumed to be slowly varying with position. The set of coupled equations is transformed into a form for which the method of characteristics applies. Instead of numerically integrating the coupled equations, we decouple the equations by writing the solution in the form of a perturbational series. The resulting zeroth-order term corresponds to the inverse transform of the WKB approximation in the frequency domain, which contains only the wavefront and amplitude information. The higher order terms can be directly interpreted as reflections along the lines. Causality is satisfied to all orders. This method has the advantage of easier implementation, and is more versatile than frequency-domain methods as well as the brute-force numerical integration of the coupled partial differential equations.     

An analysis of gain-switched semiconductor lasers generating pulse-code-modulated light with a high bit rate

A theoretical analysis of gain-switched semiconductor lasers is described. Results of the numerical solution of the coupled rate equations for photon and electron densities are presented, along with analytical expressions which have been derived by using certain approximations to solve these nonlinear differential equations. The two sets of results are seen to be in good agreement. The design requirements to be met in order to use the pulse-code-modulated output in an optical communications system are discussed. It is shown theoretically that bit rates, of the order of 7 Gbits/s without time-division multiplexing, and 35 Gbits/s with multiplexing can be obtained.     

Electromagnetic scattering by stratified inhomogeneous anisotropic media

An analytical formulation is presented for the computation of scattering and transmission by general anisotropic stratified material. This method employs a first-order state-vector differential equation representation of Maxwell's equations whose solution is given in terms of a4 times 4transition matrix relating the tangential field components at the input and output planes of the anisotropic region. The complete diffraction problem is solved by combining impedance boundary conditions at these interfaces with the transition matrix relationship. A numerical algorithm is described which solves the state-vector equation using finite differences. The validation of the resultant computer program is discussed along with example calculations.     

The discrete Fourier transform method of solvingdifferential-integral equations in scattering theory

An accurate and efficient numerical method is presented for solving many differential-integral equations arising from electromagnetic scattering theory. It uses the discrete Fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations. As a consequence, this method is extremely simple to implement, uses less computer memory than comparable methods, and yields accurate predictions. The differential-integral equation is recast into a periodic form conducive to application of the discrete Fourier convolution theorem. The differential operators are approximated by appropriate finite-difference and discrete-convolution operators. All these quantities are computed by using the fast Fourier transform. An approximate solution is obtained by using the conjugate gradient method. Results are compared to experimental data or analytical solutions for a 3λ×3λ metal plate (where λ is the wavelength), a homogeneous and a layered infinite circular dielectric cylinder, and a dielectric sphere. The accuracy of the method is further illustrated by comparing predictions with independent measurements by R.A. Ross (1966) on a 2λ×1λ metal plate at grazing incidence. In all cases, agreement is excellent     

An analytical solution for the coupled stripline-like microstripline problem

An analytical method is presented for determining the Maxwell's capacitance matrix of multiconductor shielded microstrips with coupled conductors of arbitrary widths, spacings, and inhomogeneous media, in terms of elliptic integrals. The method is based on conformal mapping and the theory of singular integral equations. Two kinds of problems are presented as examples: the first consists of simple structures with one or two coupled striplines; the other type concerns the general case of multiconductor coupled structures for which simple analytical expressions are not available     

Numerical Solution for Fractional-Order Differential Systems with Time-Domain and Frequency-Domain Methods

For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional-order differential equations based on the definition of Grunwald-Letnikov is presented. The results of numerical solution by using the novel method and the frequency-domain method are compared, and the limitations of frequency-domain method are discussed.     

An analytical solution to the microstrip line problem

An analytical method for determining the line capacitance of a microstrip line is presented. The solution is exact, but it is expressed by means of the solution of an infinite system of linear equations whose coefficients are the result of certain numerical quadratures. The analysis is carried out for the case of two dielectric substrates. Changes to include additional stratified layers are readily available using the transfer matrix method described by P.M. van Berg et al. (1985). Comparison of the results obtained using the proposed formula with those obtained using exact formulas (available in particular cases) shows that, in cases of practical interest, it is sufficient to consider only the first two equations in the above-mentioned infinite set of linear equations     

Study on Time-Dependent Coupled Rate Equations in Non-LTE Plasmas*

The matrix-block method presented in this paper greatly simplifies and quickens the numerical solution of numerous time-dependent coupled rate equations. The solution of the coupled rate equations in the quasi-steady state is also presended. The physical meaning of the matrices defined in the solution is discussed. By using the matrix-block method to solve the coupled rate equations of 1409states in Ta plasmas in the steady state and considering the effect of the reabsorption of resonance lines, the optimum region in the Ne/Te plane for gain of the 4.483 nm laser line in Ni-like Ta collisional x-ray lasers is presented. The effects of autoionization and dielectronic capture on the population of Ni-like Ta ions are also discussed.     

A new method of analysis for PWM switching power converters

A new method of analysis for pulse-width modulation (PWM) switching power converters is presented. It allows one to find an approximate periodic solution for the converter vector state variable. The converter is modelled by a differential equation with periodic coefficients. This equation is substituted by an equivalent system of linear differential equations with constant coefficients. Only the forced (steady-state) solutions should be found for each equation of this system. The equations are solved in sequence. The final steady-state solution of the PWM differential equation is obtained as the sum of these forced solutions. The method allows one to find the converter dc transfer function and efficiency, to evaluate their frequency dependences, and to find the critical frequency and ripple. The first three equations of the equivalent system are usually adequate for practical purposes, and these equations are obtained by an easy formal procedure. One can also obtain the dynamic equation of the state variable dc component, and calculate the converter line to output and duty cycle to output transfer functions. A boost converter is used as an example to confirm the analytical results by numerical simulation.     

An analytical approach to the analysis of dispersioncharacteristics of microstrip lines

An analytical method for determining the dispersion characteristics of microstrip lines is given. The method uses dual integral equations. The analysis is rigorous, and it expresses the solution of the dispersion equation in terms of the solution of a double infinite system of linear equations. The system coefficients are given by certain quadratures. The numerical examples reveal the high convergence order of the method     

A Second-Order Algorithm for Solving Dynamic Cell Membrane Equations

This paper describes an extension of the so-called Rush-Larsen scheme, which is a widely used numerical method for solving dynamic models of cardiac cell electrophysiology. The proposed method applies a local linearization of nonlinear terms in combination with the analytical solution of linear ordinary differential equations to obtain a second-order accurate numerical scheme. We compare the error and computational load of the second-order scheme to the original Rush-Larsen method and a second-order Runge-Kutta (RK) method. The numerical results indicate that the new method outperforms the original Rush-Larsen scheme for all the test cases. The comparison with the RK solver reveals that the new method is more efficient for stiff problems.     

Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution

This investigation is concerned with the numerical solution of time-harmonic electromagnetic scattering by axisymmetric penetrable bodies having arbitrary cross-sectional profiles and even continuously inhomogeneous consistency. The initiation of this effort involved the discovery and development of the coupled azimuthal potential (CAP) formulation, which is valid in generally lossy isotropic inhomogeneous rotationally symmetric media. Electromagnetic fields in such regions can be represented, using the CAP formulation, in terms of two continuous potentials which satisfy a self-adjoint system of partial differential equations or, equivalently, a variational criterion. Using an optimized variational finite-element algorithm in conjunction with a triregional unimoment method, a versatile computer program is described that provides scattering solutions for each of multiple incident fields impinging upon an arbitrarily shaped inhomogeneous penetrable body of revolution. An extensive evaluation of the accuracy and convergence of the algorithm is presented, which includes comparison of scattering computations and experimental measurements atX-band for several solid and hollow plexiglas bodies of revolution with maximum interior dimensions of over 4 wavelengths.     

Numerical solution of time domain integral equations for arbitrarily shaped conductor/dielectric composite bodies

In this work, we present a numerical solution of the coupled time domain integral equations to obtain induced currents and scattered far fields on a three-dimensional, arbitrary shaped conducting/dielectric composite body illuminated by a Gaussian electromagnetic plane wave pulse. The coupled integral equations are derived utilizing the equivalence principle. The solution method is based on the method of moments and involves the triangular patch modeling of the composite body, in conjunction with the patch basis functions. Detailed mathematical steps along with several numerical results are presented to illustrate the efficacy of this approach.     

Edge diffraction in the vicinity of the tip of a composite wedge

The electromagnetic field due to a line source radiating in the presence of a two-dimensional composite wedge composed of a number of conducting and dielectric materials is obtained. The Fourier transform path integral method (FTPI) is described and used to perform the numerical analysis. An important feature of the FTPI method is that it is based on a global solution to the Helmholtz scalar wave equation. As such the method avoids numerical enforcement of boundary conditions and the necessity of reformulating the analytical/numerical equations for each geometric configuration. The total scattered field is presented for several cases where one of the dielectric wedge sections is lossy, including examples of microwave scattering from a crested ocean surface and an air-ocean-sea ice interface     

Planar Diffraction Analysis of Homogeneous and Longitudinally Inhomogeneous Gratings Based on Legendre Expansion of Electromagnetic Fields

Planar grating diffraction analysis based on Legendre expansion of electromagnetic fields is reported. In contrast to conventional RCWA in which the solution is obtained using state variables representation of the coupled wave amplitudes; here, the solution is expanded in terms of Legendre polynomials. This approach, without facing the problem of numerical instability and inevitable round off errors, yields well-behaved algebraic equations for deriving diffraction efficiencies, and can be employed for analysis of different types of gratings. Thanks to the recursive properties of Legendre polynomials, for longitudinally inhomogeneous gratings, wherein differential equations with non-constant coefficients are encountered, it can also be used to analyze the whole structure at one stroke. Although this is the only case for which the presented approach is efficient from both aspects of stability and computation load, the presented approach is applied to different test cases, and justified by comparison of the results to those obtained using previously reported methods. The method is general, and can handle many different cases like thick gratings, non-Bragg incidence, and cases in which higher diffracted orders or evanescent orders corresponding to real eigenvalues, have to be included in the solution of the Maxwell's equations. In deriving the formulation, a rigorous approach is followed