Pade Approximants Research Articles

Analysis of transmission lines considering frequency-dependent parameters with linear and nonlinear loads

The primary objective of this paper is to model transmission lines with frequency-dependent parameters loaded with linear and nonlinear terminations using an approach based on Pade approximations. To validate our algorithm in the case of lines with constant losses and frequency-dependent parameters, examples of applications in the time and frequency domains are presented. The simulations and obtained results are reported and analysed.

Heat Transfer Analysis over the Boundary Layer Stagnation-Point Flow of Couple Stress Fluid over an Exponentially Stretching Sheet

The present study provides a comprehensive discussion for the problem of boundary layer flow and heat transfer analysis of stagnation point flow of couple stress fluid over an exponentially stretching surface. The governing equations of couple stress fluid model are assumed under boundary layer approach. The nonlinear partial differential equations are simplified by using similar similarity transformations. The analytical solutions of abridged equations are computed with the help of homotopy analysis method (HAM). The convergence of the HAM solutions have been deliberated by plotting <i>ћ </i>- curves and also through homotopy pade approximation. The physical features of pertinent parameters have been discussed through graphs.

Adaptive NN Control of Input-Delayed Stochastic Nonlinear Systems With States and Input Quantization

In this article, the adaptive neural control problem is studied for the input-delayed stochastic nonlinear system that has both states and input quantization. The main difficulty is to deal with the discontinuity of the virtual control laws caused by states quantization, and to compensate the influence of the input quantization by building the relation between the control input and quantized input. Moreover, this paper considers a sector bounded quantizer and proposes a new adaptive neural network (NN) control algorithm to handle the problem of quantization errors cannot be automatically bounded. For the unknown function in stochastic strict-feedback systems, the radial basis function neural network (RBFNN) is utilized to approximate it. The Pade approximation technique is employed to tackle the issue of input delay. All the closed-loop signals can maintain semi-globally uniformly ultimately bounded (SGUUB) in probability under the constructed controller and quantizer. Lastly, the efficiency of the developed control method is evaluated by a simulation experiment.

Numerical comparisons for solving fractional order integro-differential equations with non-local boundary conditions

In this paper, univariate Pade approximation is applied to fractional power sries solutions of fractional integro-differential equations with non-local boundary conditions. As it is seen from comparisons, univariate Pade approximation gives reliable solutions and numerical results.

The Laplace transform decomposition method applied to the numerical solution of the food chain ecoepidemic model

The aim of this study is to theoretically investigate the model parameters of the food chain ecoepidemic model using the Laplace transform decomposition method. The first order autonomous differential equations governing the model were reduced via the Laplace transform and Adomian decomposition to an algebraic equation, where the inverse Laplace transform gives the pertinent parameters. The analytically obtained solution which is the partial sum of the infinite series is then treated with Pade approximation to improve its convergence. To validate the results obtained, we compare the numerical solution via LADM with other semi-analytical methods viz LADM-Pade and Daftardar-Jafari (DJM. The results obtained are presented in tables and figures. The comparison showed, there is excellent agreement between the methods. It is observed the method is valid, efficient, applicable, robust, and computationally convenient to handle both linear and nonlinear differential as well partial differential equations.

Thermal distribution through a moving longitudinal trapezoidal fin with variable temperature-dependent thermal properties using DTM-Pade approximant

The present inspection elaborates the thermal distribution through a convective-radiative longitudinal trapezoidal moving fin with an internal heat source by considering the linear and nonlinear temperature-dependent thermal conductivities and heat transfer coefficient. The non-dimensionalization of the energy equation is carried out using dimensionless terms, and the resulting nonlinear ordinary differential equation (ODE) is solved analytically using the differential transform method (DTM) with Pade approximant approach. Furthermore, the consequence of several non-dimensional parameters, namely thermal conductivity parameter, convection-conduction parameter, Peclet number, radiation-conduction parameter, fin taper ratio, temperature ratio parameter, and internal heat generation parameter on temperature gradient and fin efficiency is explained with the help of graphs. The significant primary results of the investigation illustrate that the thermal distribution through the fin gradually decreases as the magnitude of the convection-conduction parameter, temperature ratio parameter, and radiation-conduction parameter rise. In contrast, it upsurges for heat generation and thermal conductivity parameters.

Robust output regulation for unknown nonlinear strict‐feedback systems with input delay

AbstractThis article investigates the robust output regulation problem for unknown nonlinear strict‐feedback systems with input delay. Nonlinear fuzzy internal model and adaptive fuzzy backstepping control are integrated to obtain a feedforward‐feedback composite controller. As a distinctive feature, Pade approximation technique is introduced to deal with the input delay of nonlinear output regulation problem for the first time. The process of control strategy can be divided into two steps. In the first step, a novel nonlinear fuzzy internal model is designed to get the feedforward compensator. In the second step, an adaptive fuzzy backstepping control scheme is proposed to get the feedback controller, which can guarantee semiglobally uniformly ultimately bounded of the closed‐loop system. Simulation results of an interconnected power system show the validity of the theoretical results.

Four-dimensional Integral Model of Dry Friction on the Example of Wheel Movement

A four-dimensional model of dry friction in the interaction of a solid wheel and a horizontal rough surface is investigated. It is assumed that there is no separation between the wheel and the horizontal surface. The movement of the body occurs in conditions of combined dynamics, when in addition to the sliding movement, the body participates in spinning and rolling. The equation of motion of the wheel is compiled using the Appel equation.The resulting model of sliding, spinning, and rolling friction is given for the case where the contact area is a circle. The cumbersome integral expressions were replaced by fractional-linear Pade approximations. Pade approximations accurately describe the behavior of the components of the friction model. A mathematical model is proposed that describes the simultaneous sliding, spinning and rolling of a solid wheel. The dependences of the parallel and perpendicular components of the friction force and the torque of the spinning friction were ploted with respect to the parameter that characterizes the movement of the wheel. Comparisons of the integral friction model and the model based on Pade approximations are presented. The results of the comparison showed a qualitative correspondence of the models. After obtaining the equation of motion, the simulation of motion at a constant control torque of the wheel is carried out. The graphs allow you to match the logical behavior of the wheelmovement.

On Multipoint Padé Approximants whose Poles Accumulate on Contours that Separate the Plane

In this note we consider asymptotics of the multipoint Pade approximants to Cauchy integrals of analytic non-vanishing densities defined on a Jordan arc connecting $ -1 $ and $ 1 $. We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.

Elastic and absorption electron collisions with acetaldehyde

We report a theoretical–experimental investigation on elastic electron scattering by acetaldehyde (CH $$_3$$ CHO) in the 0.5–800 eV energy range. Theoretically, the Pade approximation and the single-center partial-wave expansion method were applied to solve the Lippmann–Schwinger scattering equation, whereas a molecular complex optical potential was used to describe the electron–molecule interaction. Calculations in the independent-atom framework were also performed. Angular distributions of the elastically scattered electrons were measured in the 10 $$^{\circ }$$ –130 $$^{\circ }$$ and 30–800 eV ranges using a crossed beams geometry. Absolute values of differential cross sections were derived by the relative flow technique. In addition, integral and momentum transfer cross sections were obtained from the experimental and theoretical differential cross sections via a numerical integration procedure. Comparison between our results and previous experimental and theoretical calculations is made.

Conventional and Evolutionary Order Reduction Techniques for Complex Systems

Order reduction of the large-scale linear dynamic systems (LSLDSs) using stability equation technique mixed with the conventional and evolutionary techniques is presented in the paper. The reduced system (RS) is obtained by mixing the advantages of the two methods. For the conventional technique, the numerator of the RS is achieved by using the Pade approximations and improved Pade approximations, whereas the denominator is obtained by the stability equation technique (SET). For the evolutionary technique, numerator of the RS is obtained by minimizing the integral square error (ISE) between transient responses of the original and the RS using the genetic algorithm (GA), and the denominator is obtained by the stability equation method. The proposed RS retains almost all the essential properties of the original system (OS). The viability of the proposed technique is proved by comparing its time, frequency responses, time domain specifications, and ISE with the new and popular methods available in the literature.

Barrier Lyapunov function-based adaptive fuzzy attitude tracking control for rigid satellite with input delay and output constraint

This paper investigates the adaptive attitude tracking problem for the rigid satellite involving output constraint, input saturation, input time delay, and external disturbance by integrating barrier Lyapunov function (BLF) and prescribed performance control (PPC). In contrast to the existing approaches, the input delay is addressed by Pade approximation, and the actual control input concerning saturation is obtained by utilizing an auxiliary variable that simplifies the controller design with respect to mean value methods or Nussbaum function-based strategies. Due to the implementation of the BLF control, together with an interval notion-based PPC strategy, not only the system output but also the transformed error produced by PPC are constrained. An adaptive fuzzy controller is then constructed and the predesigned constraints for system output and the transformed error will not be violated. In addition, a smooth switch term is imported into the controller such that the finite time convergence for all error variables is guaranteed for a certain case while the singularity problem is avoided. Finally, simulations are provided to show the effectiveness and potential of the proposed new design techniques.

Parallel-connected battery module modeling based on physical characteristics in multiple domains and heterogeneous characteristic analysis

Cell inconsistencies inevitably occur inside a battery module. Particularly, the inconsistencies in current distribution and heat generation in a parallel-connected battery module may lead to battery degradation and potential safety issues. Consequently, it is imperative to evaluate and reduce cell inconsistencies in a battery module. In this paper, an extended single particle model of a battery cell is constructed using the Pade approximation and the first-order Taylor expansion to simplify the conventional electrochemical mechanism model. On this basis, a multidomain electrochemical mechanism simulation model of a parallel-connected battery module is attained. Then, the influence of cell inconsistencies on the battery module voltage, internal current distribution and heat generation under different aging situation is assessed by a parameter sensitivity analysis method. Moreover, based on the contribution of each battery model parameter to the inconsistency of the parallel-connected battery module, a battery cell sorting method is proposed. Finally, this proposed sorting method for secondary applications of batteries is validated based on 15 aged batteries. Results indicate that the average standard deviation of the cell current in the NCM parallel-connected module can be reduced from 0.209 A to 0.060 A. The proposed approach is helpful to the fault analysis of electric vehicle battery modules, module level grading or the secondary applications of retired batteries.

Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

A low-dispersive padé approximation method for wave propagation in isotropic and anisotropic poroelastic medium

ABSTRACT Seismic exploration of complex geological structure model has been paid much attention as the increasing complexity of geological structure in seismic exploration, and especially, the research on saturated porous medium saturated with fluid has become a hot topic. In our study, we propose a new Padé approximation (PAM) method of solving the elastic wave equation in the porous medium of low-frequency case. This method uses an implicit scheme derived from the rational function of time difference operator for the time discretization, which has the characters of low-dispersion and high-efficiency for time advancing. An explicit iteration for this implicit algorithm is obtained for avoiding solving a large linear system with a block tridiagonal coefficient matrix at each time step. Then, we employ the stereo-modelling method with the eighth-order accuracy for space discretization, which uses linear combination of wave field displacements and their gradients to discretize spatial derivatives and obtains a high-order approximation. Compared with the traditional method, this discretization operator has shorter operator radius and better compactness, which is beneficial for increasing the precision and imaging quality of seismic inversion and seismic migration. Theoretical analysis and numerical experiments verify that the PAM method is an accurate forwarding modelling tool. Waveforms obtained by the PAM method can well match the analytical solutions. Moreover, the seismic wave fields including fast P wave, S wave, and slow P wave for the saturated fluid porous medium can be observed clearly on coarser grids. This is in contrast with the FD method, which suffers from serious numerical dispersion.

Analytical solution of systems of Volterra integro-differential equations using modified differential transform method

In this paper, a hybrid technique based on the combining of differential transform approach, the Laplace transformation, and Pade approximant is successfully implemented for solving Volterra Integro differential (VID) equations subject to appropriate conditions. The proposed method known as modified differential transform method (MDTM) enlarges the domain of convergence with less computational time. One of the interesting features of this algorithm is the ability to produce an analytic approximate solution in convergent power series form with few numbers of computable terms. Preliminary results on some test problems show that the new algorithm is efficient and promising for Volterra integro-differential equations and can be applied as alternatives for the integral and integro-differential (IID) equation that have no analytical solution.

FastCTF: A robust solver for conduction transfer function coefficients and thermal response factors

Conduction transfer functions (CTF) are commonly used in the building services to quickly estimate hourly conduction heat loads through multilayered walls without resorting to expensive, time-consuming solutions of the heat equation. It is essential for any software developed for this purpose to be able to simulate walls of varying weight with a high degree of accuracy. A robust algorithm for computing CTF coefficients and thermal response factors based on power series expansions of solutions of the governing equations in the complex s-domain is presented and validated. These series expansions are used to construct Pade approximants of the system’s transfer functions, which greatly simplifies the inversion of the solution from the complex domain to the time domain, and allows for an easy recovery of a time series representation via the Z-transform. The algorithm is also implemented in an open-source C++ code. Its performance is validated with respect to exact theoretical frequency characteristics and its results are compared with data generated by previously established methods for computing CTF coefficients/response factors.

A simplified adaptive tracking control for nonlinear pure‐feedback systems with input delay and full‐state constraints

SummaryThis article investigated the adaptive backstepping tracking control for a class of pure‐feedback systems with input delay and full‐state constraints. With the help of mean value theorem, the system is transformed into strict‐feedback one. By introducing the Pade approximation method, the effect of input delay was compensated. Radial basis function neural networks are utilized to approximate the unknown nonlinear functions. Furthermore, in order to reduce the computational burden by introducing backstepping design technique, dynamic surface control technique was employed. In addition, the number of the adaptive parameters that should be updated online was also reduced. By utilizing the barrier Lyapunov function, the closed‐loop nonlinear system is guaranteed to be semi‐globally ultimately uniformly bounded. Finally, a numerical simulation example is given to show the effectiveness of the proposed control strategy.

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

Error estimation for Differential Transform Method (DTM)solution of non-linear SIQRM biological model

Solving a non-linear differential equation most times is difficult and requires some technicalities. Many semi-analytical methods were derived in literature to provide series solution to non-linear problem, with each method giving some level of accuracy when compared with their equivalent exact solution (or numerical solution incase exact does not exist). Thus, system of ordinary differential equations (ODEs) arising from a formulated Susceptible- Infected-Quarantine-Recovered-Immunity (SIQRM) mathematical model of a disease dynamics were solved using DTM and Pade approximation; and their results numerically compared with Runge-Kutta order 4 (RK4). The table of result shows that DTM is reliable to tackle non-linear DE while Pade approximant improves its (DTM) accuracy.