Pade Method Research Articles

Differential Quadrature Method for Fully Intrinsic Equations of Geometrically Exact Beams

In this paper, a differential quadrature method of high-order precision (DQ−Pade), which is equivalent to the generalized Pade approximation for approximating the end of a time or spatial interval, is used to solve nonlinear fully intrinsic equations of beams. The equations are a set of first-order differential equations with respect to time and space, and the explicit unknowns of the equations involve only forces, moments, velocity and angular velocity, without displacements and rotations. Based on the DQ−Pade method, the spatial and temporal discrete forms of fully intrinsic equations were derived. To verify the effectiveness and applicability of the proposed method for discretizing the fully intrinsic equations, different examples available in the literatures were considered. It was found that when using the DQ−Pade method, the solutions of the intrinsic beam equations are obviously superior to those found by some other usual algorithms in efficiency and computational accuracy.

An Efficient Delay Estimation Model for High Speed VLSI Interconnects

In this paper a closed-form matrix rational model for the computation of step and finite ramp responses of Resistance Inductance Capacitance (RLC) interconnects in VLSI circuits is presented. This model allows the numerical estimation of delay and overshoot in lossy VLSI interconnects. The proposed method is based on the U-transform, which provides rational function approximation for obtaining passive interconnect model. With the reduced order lossy interconnect transfer function, step and finite ramp responses are obtained and line delay and signal overshoot are estimated. The estimated delay and overshoot values are compared with the Euder method, Pade method and HSPICE W- element model. The 50% delay results are in good agreement with those of HSPICE within 0.5% error while the overshoot error is within 1% for a 2 mm long interconnect. For global lines of length more than 5 mm in SOC (system on chip) applications, the proposed method is found to be nearly four times more accurate than existing methods.

Asymptotic Decomposition Method for Fractional Order Riccati Differential Equation

In this study, the asymptotic Adomian decomposition method (AADM) is implemented to solve fractional order Riccati differential equations. The product integration method is used to solve the singular integrals, resulted from fractional derivative. Some fractional order Riccati differential equations are presented as examples to illustrate the ability and efficiency of the proposed approach. The approximate solutions of AADM are compared with the results of the Laplace Adomian Pade method (LAPM). Generalizing AADM for solving fractional Riccati differential equations by the far-field approximation indicates the novelty of the paper.

Pade Method for Construction of Numerical Algorithms for Fractional Initial Value Problem

In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. We regard the Grunwald–Letnikov fractional derivative as a kind of Taylor series and get the approximation equation of the Taylor series by Pade approximation. Based on the approximation equation, we construct the corresponding numerical algorithms for the fractional initial value problem. Finally, we use some examples to illustrate the applicability and efficiency of the proposed technique.

Solution of Fuzzy Initial Value Problems Using Homotopy Analysis Method and Padè Approximate

In this paper, we have studied and developed some analytical methods for solving some linear and fuzzy differential equations of the first degree with their fuzzy elementary conditions using the homotopy analytical method, using the approximate Pade method, and comparing the results with the exact solution of the equation, which indicates that the error decreases exponentially with the N degree of approximation. Also, we made some improvements to the solution results by using Pade enhancements as shown in the two examples with tables and figures.

Natural resonance frequency identification for sea waves and EEG epileptic patients using Pade approximation and neural network

Pade method is used in the spectral domain estimation to classify both the remote sensing and the biomedical signals. The first example presented from remote sensing is the sea wave classification while the second example depicted from biomedical engineering field is the Epilepsy seizure type classification. Feature extractions of both the Global Navigation Satellite Systems (GNSS) signal and the epilepsy seizure from a human Electroencephalograph (EEG) signal are based on the poles location of the signal.

Self-energy Padé approach for analytic continuation: Application to the zero-gap Kondo lattice model

A modified Pade approach is presented as analytic continuation for numerical methods that evaluate correlation functions in imaginary time. Instead of the direct analytic continuation of the correlation functions, the Pade method is applied for the self-energy that is then used for deriving the Green's functions at real energies. We find that this modified, self-energy Pade approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the modified Pade approach are analyzed by actual calculations for the illustrative cases of the non-interacting Anderson lattice and Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied, where we use the modified Pade approach as analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities, that are associated as a direct and an indirect gap in a band hybridization picture.

Choice of the order of the regression model for forecasting of random non-stationary economic processes

The object of research is heteroskedastic processes that affect the production of military goods of exporting countries. Today, armed conflicts are the most significant factor affecting the volume of production and export of weapons, since it assumes that the parties have the necessary quantity of weapons and is, in a sense, a stochastic process. The work is devoted to forecasting stochastic effects on the production processes of military goods of exporting countries. As an example, an economic system with stochastic effects and bottleneck problems in production units is considered. The model of the output process is presented as a random process with slow non-stationarity (heteroscedastic process). The methods for predicting non-stationary random processes are used. The problem of choosing and substantiating a mathematical model for predicting a heteroskedastic process is investigated, and considered. It is proved that the most capable short-term forecasting method is the Pade approximation method. It is shown that the Pade method, in fact, is a method of approximation by analytical (finely rational) functions, therefore it can be interpreted as a method of constructing a model of autoregression and moving average (ARIMA). Modifications of the ARIMA model, such as a model of autoregression and integrated moving average or autoregression and fractal integrated moving average, are considered. A modified method is developed for choosing the order of the autoregressive model according to the Akaike information criterion and beyond the Bayesian information criterion. The model problems and examples of experimental dependencies are analyzed. An effective technique is proposed for choosing the order of regression models used in the practical forecasting of stochastic processes, based on the canonical layouts of a random function. To partition the distribution function into non-equidistant intervals with constant flow intensities, an economic recurrence algorithm is used. The calculation results can be used to optimally select the order of the regression model, which approximates the real production process in the form of a time series with random external influences.

Analytic Pade-like approximations of exp(–sqrt(s)) for simulations of diffusion processes in the semi-infinite geometry

Detailed transient behaviors of mass and heat transfer processes are required to solve partial differential equations. When those partial differential equations are coupled, they are still difficult to solve in time domain. For linear mass and heat transfer processes, their Laplace-domain solutions are obtainable and, when they are approximated by rational polynomials in the Laplace variable s, the problems can be transformed to a set of ordinary differential equations solved easily in time domain for various initial conditions. In this approximation, the conventional Pade method based on the Tayler series expansion of the Laplace-domain solutions has been well developed and effective. However, for some mass and heat transfer processes in the semi-infinite geometry, the Pade approximation is not applicable because the Laplace-domain solutions involving exp(–sqrt(s)) are not analytic at s = 0. Here, for such processes, analytical methods to approximate exp(–sqrt(s)) by rational polynomials are proposed. First it is expanded in series in terms of cosh–1(2ksqrt(s)) which converges fast. This series, when truncated, is analytic at s = 0 and its Pade approximations are available. The proposed method enables partial differential equations be replaced to a set of ordinary differential equations, reducing computations considerably for coupled partial differential equations. Performances of the proposed method are illustrated with several realistic mass and heat transfer processes.

Comment on: “On the computation of eigenvalues of the anharmonic Coulombic potential”

In this comment we show that the claims by Cassidy et al (J Math Chem 56: 477, 2018) about the limitations of the Riccati Pade method (RPM) are unfounded. To this end we compare the performance of the RPM and their DESCM by means of the calculation of the eigenvalues of a class of one-dimensional potentials. We show that the RPM is simple and extremely accurate. In addition to the exponential convergence of its approximate eigenvalues the RPM does not require any variable transformation, adjustable parameters or scaling that are necessary for the improvement of the performance of the DESCM. An additional remarkable feature of the RPM is that it provides upper and lower bounds to the eigenvalues of the perturbed Coulomb potentials chosen for this test.

Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace- Padé Method

In this study, the solutions of random partial differential equations are examined. The parameters and the initial conditions of the random component partial differential equations are investigated with Beta distribution. A few examples are given to illustrate the efficiency of the solutions obtained with the random Differential Transformation Method (rDTM). Functions for the expected values and the variances of the approximate analytical solutions of the random equations are obtained. Random Differential Transformation Method is applied to examine the solutions of these partial differential equations and MAPLE software is used for the finding the solutions and drawing the figures. Also the Laplace- Pade Method is used to improve the convergence of the solutions. The results for the random component partial differential equations with Beta distribution are analysed to investigate effects of this distribution on the results. Random characteristics of the equations are compared with the results of the deterministic partial differential equations. The efficiency of the method for the random component partial differential equations is investigated by comparing the formulas for the expected values and variances with results from the simulations of the random equations.

A new method to obtain analytic approximations applied to the J1(x) function

In this work an analytic approximation for J1(x) has been found, which is simple and precise, and good for most of the applications of this functions in Physics. The technique here used remarks a little the Pade method, since rational functions are used, but now this type of function are combined in a efficient way with elementary functions, furthermore series power and asymptotic expansions are used simultaneously, as in the Multipoint Quasi-rational Approximation, MPQA method. However here important improvements have been introduced. Though the form of the approximate is built considering the above two expansions, however the parameters of the approximations are determinates by two methods, one similar to the minimum square error method and the other using the coefficients of two expansions, power and asymptotic. Better results are obtained with the first procedure.

A simple approximation for the modified Bessel function of zero order I0(x)

New efficient analytic approximations have been found for the modified zero-order Bessel functions I0(x). The method, used herein, improves prior techniques, such as multipoint quasi-rational approximations, MPQA. The present work is also an improvement of previous works, in the sense that the form of the approach is more efficient, that is, a smaller relative error is obtained using a lower number of parameters. As in the Pade method rational functions are used with coefficients determined from the powers series, but now also asymptotic expansions are also used simultaneously with that series, and consequently the rational functions have to be combined with elementary functions, in such a way, that the approach is a bridge between both expansions. The approximation now found is a rational function combined with a hyperbolic function. Only three parameters are needed to obtain a relative error smaller than 0.6 percent.

The WCS‐FDTD method for simulating graphene

AbstractA weakly conditionally stable‐finite difference time‐domain (WCS‐FDTD) scheme to simulate graphene is developed. The Pade fit method is used to approximate the interband conductivity of graphene and introduced into the WCS‐FDTD method directly by using the auxiliary differential equation. The accuracy of the Pade method is verified, and the effect of chemical potential on the accuracy is discussed. The time step size in proposed method is only determined by one spatial cell size, so the computational efficiency of this method is considerably larger than those of the conventional FDTD method and implicit explicit‐finite (HIE)‐FDTD method. An infinite graphene sheet and a graphene‐based metal sheet are simulated by using presented method, and the results are compared with theoretical value and the numerical results of the conventional FDTD method and HIE‐FDTD method. Simulated results show that proposed WCS‐FDTD method has high accuracy, and compared with the FDTD method and HIE‐FDTD method, its computational time is largely shorted. Besides, the numerical example presents that the interband conductivity has important effect on the performance of the graphene at high‐frequency range, especially when the chemical potential is small.

Radioprotection de l’entourage des patients traités à l’iode 131 pour des cancers différenciés de la thyroïde : comparaison de deux méthodes d’estimation des temps de restriction de contact

In France, it is required to give patients some written radiation safety advice following an iodine-131 therapy of thyroid. However, there is no regulatory methodology proposed to assess the restriction times required to limit radiation dose received by their family and by third persons. In this study, we compared two methods to perform the calculation of restriction times for differentiated thyroid cancer: the first one, called Global method, follows general recommendations and the second one, called PaDe method, provides customized recommendations based on the effective half-life measured for each patient during the 48h of hospitalization. For these simulations, the input parameters for both methods were dose constraints, contact patterns, final dose rate measurement at discharge of the patient, effective half-life and dose rate variation with distance. For the Global method, these parameters were set and were not changed. For the PaDe method, the effective half-life and the final dose rate were adjusted for each patient. Comparison between the two methods was carried out for 103 patients. For 68.0 % of the patients, no restrictions were necessary according to the PaDe method whereas the Global method required restrictions to be applied during up to 2 days. For almost 16.5 % of them, restriction times estimated with the PaDe method were systematically lower. For the remaining 15.5 %, at least one contact pattern restriction had to be increased according to the PaDe method. Furthermore, with the Global method, the same instructions were given to 97.0 % of the patients. These results showed the limits of the Global method which does not consider individually tailored biological data. Eventually, the use of the personalized PaDe method optimizes the radiation protection by improving the well being of the patient or reinforcing the family and third person's protection.

20. Radiation protection following iodine-131 therapy for thyroid cancers: Comparison of two methods to calculate restriction times

Introduction In France, it is required to give patients some radiation safety advice following an iodine-131 therapy of thyroid. However, there is no regulatory methodology proposed to assess the restriction times required to limit radiation dose received by their family and by third persons. In this study, we compared two methods to perform the calculation of restriction times for differentiated thyroid cancer: the first one followed the recommendations of the Conseil Superieur d’Hygiene Publique de France (CSHPF) [1] and the second one, called the “patient-dependent” method (PaDe), used the effective half-life measured for each patient during the 48h-of hospitalization. Material and methods For these simulations, the input parameters were dose constraints, contact patterns, dose rate measurements and dose rate variation with distance. For the CSHPF method, these parameters were fixed and were not changed. For the PaDe method, the input parameters were mostly defined according to the literature [2] . The dose constraint for members of the public was set to 1 mSv. The effective half-life was calculated for each individual patient from a mono-exponential fit of dose rate measurements performed during the hospitalization. Using this effective half-life and the dose rate measured at the time of discharge, restriction times were computed for each contact pattern. Comparison between the two methods was carried out for 59 patients. Results For 63% of the patients (37/59), no restrictions were necessary according to the PaDe method whereas the CSHPF method required the restrictions to be applied during up to 2 days. For almost 20% of them (12/59), restriction times estimated with the PaDe method were systematically lower. For the remaining 17% (10/59), the CSHPF method underestimated the restrictions for at least one contact pattern. Furthermore, with the CSHPF method, the same instructions were given to 97% of the patients (57/59). These results showed the limits of the CSHPF method which does not consider individually tailored biological data and applies a theoretical inverse square law to derive dose rates at different distances. Conclusion In order to improve the well-being of the patient and to optimize the radiation protection, this study made us change our method to follow the PaDe computed recommendations.

On two different kinds of resonances in one-dimensional quantum-mechanical models

We apply the Riccati–Pade method and the Rayleigh–Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh–Ritz method reveals each set at a particular interval of rotation angles the Riccati–Pade method yields both of them as roots of the same Hankel determinants.

Notes on the Padé Approximation for an Anharmonic Oscillator

Pade approximation is studied taking as an example the ground-state energy E0(1, β) of the anharmonic oscillator with x2 + βx4 potential. The convergence and the other properties of the Pade approximant have so far been deduced from its relation to the Stieltjes function. However, E0(1, β) cannot be related to a Stieltjes function, but to the so-called once-subtracted Stieltjes function, so that we modify the theory of the Pade approximation, first showing that the same formula as the one for the Stieltjes function can be used also for the once-subtracted Stieltjes function. The properties of the modified Pade approximants are deduced and confirmed by numerical examples, establishing their error bounds at the same time. The Pade approximation is found to be superior in precision even for β = 0.1 in lower orders than the perturbation theory of any orders. The Pade method is suggested to be applicable to a wider class of problems.

Improvements in the visual evaluation of soil structure

AbstractSpade methods to visually evaluate soil structural quality (Sq) are simple, quick, cheap and can be used by farmers, gardeners, consultants and the scientific community. However, European and Brazilian users of one such method, viz. visual evaluation of soil structure (VESS) which is a development of the Peerlkamp test, have been concerned about its subjectivity. The method of soil slice break‐up and operator influence on scores have been questioned. Thus, our aim was to make soil scoring by the VESS technique more objective and thus to revise the scoring guide. We compared scoring with normal breaking up of the soil slice by hand with scoring after breaking up the slice by dropping (drop shatter) to make the soil break‐up more operator independent. After slice break‐up, aggregates were split by hand and their internal porosity was evaluated to develop the use of visible porosity as an aid to scoring. This proved inconclusive on its own, so a method of reducing larger aggregates to 1.5–2.0 cm core fragments and describing their shape and porosity was developed to score soil Sq. Breaking up a spadeful of soil by hand or by dropping resulted in the same Sq score. The method of reducing aggregates and evaluation of their shape improved VESS, particularly in the middle range of soil quality and the revised chart is shown. VESS was sensitive to changes in Sq in layers within the profile and its use for diagnosing Sq in different layers allows targeted soil improvement by tillage.

Homotopy analysis and Homotopy Pad$acute{e}$ methods for two-dimensional coupled Burgers' equations

In this paper, analytic solutions of two-dimensional coupled Burgers' equations are obtained by the Homotopy analysis and the Ho- motopy Pade methods. The obtained approximation by using Homotopy method contains an auxiliary parameter which is a simple way to con- trol and adjust the convergence region and rate of solution series. The approximation solutions by (m, m) Homotopy Pade technique is often independent of auxiliary parameter and this technique accelerate the convergence of the related series.