Pade Approximants Research Articles

Effect of Initial Geometric Imperfections on Nonlinear Vibration of Thin Plate by an Asymptotic Numerical Method

In this work, the influence of initial geometric imperfections on geometrically nonlinear free vibrations of thin elastic plates has been investigated by an asymptotic numerical method. The nonlinear strain displacement relationship of von Karman theory is adopted to calculate the elastic strain energy. The harmonic balance approach and Hamilton’s principle are used to convert the equation of motion into an operational formulation. The nonlinear problem is transformed into a sequence of linear ones having the same stiffness matrix, which can be solved by a classical finite-element method. To improve the validity range of the power series, Pade approximants are incorporated and a continuation technique is also used to get the whole solution. Numerical results are discussed and compared to those available in the literature and convergence of the solution is shown for various amplitudes of imperfection.

Numerov-Padé Scheme for the One-Way Helmholtz Equation in Tropospheric Radio-Wave Propagation

This letter is devoted to applying the higher order finite-difference approximations to the tropospheric radio-wave propagation problem. It is proposed to use the fourth-order approximation (Numerov discretization) of the transverse differential operator, and rational Pade approximations of the propagation operator. A discrete transparent nonlocal boundary condition that can take into account the modified refractive index of the troposphere has been constructed. An error analysis of the considered scheme with emphasis on tropospheric propagation has been carried out. A comparison with the existing finite-difference schemes and the split-step Fourier method has been performed. The effectiveness of the proposed approach has been shown.

Adaptive tracking control for a class of stochastic non‐linear systems with input delay: a novel approach based on multi‐dimensional Taylor network

In this study, by means of the techniques of adaptive multi-dimensional Taylor network (MTN) control, the tracking problem of a class of stochastic non-linear systems with input delay is studied. Firstly, based on the Pade approximation method, a new variable is employed to overcome the problem of input delay, and MTNs are applied to estimate the non-linear functions in the design of the controller. Secondly, an adaptive MTN controller design scheme is developed in the framework of backstepping. Thirdly, the stability of the closed-loop system is analysed using Lyapunov's stability theory. Finally, the effectiveness of the proposed design approach is demonstrated by two examples.

Structural and magnetic properties of Cd-Ni spinel ferrites: density functional theory calculations and high-temperature series expansions

ABSTRACT First-principles calculations of density functional theory (DFT) using the generalized gradient approximation (GGA) and generalized gradient approximation with the Hubbard U (GGA + U) for the exchange-correlation energy functional are used to investigate the structural, electronic, and magnetic properties of the spinel system in the range . High-temperature series expansions (HTSE) combined with the Padé approximants (PA) method and mean field theory (MFT) are applied to the systems. The lattice parameter and magnetic moment of Fe and Ni ions increase when Cd content x increases. Density of States (DOS) calculations show that the has a semiconductor behaviour. The obtained theoretical results show that the Curie temperature decreases with x. These obtained results are in agreement with experimental ones.

New solutions of hyperbolic telegraph equation

<p style='text-indent:20px;'>We present a new method based on unification of fictitious time integration (FTI) and group preserving (GP) methods. The GP method is applied in numerically discretized ordinary differential equations obtained from application of FTI method to a given partial differential equation (PDE). The algorithm is applied to hyperbolic telegraph equation and utilizes the Cayley transformation and the Pade approximations in the Minkowski space. It avoids unauthentic solutions and ghost fixed points which is one of the advantages of the present method over other related numerical methods in the literature. The technique is tested on three specific examples for various parameter values appearing in the telegraph equation and discretization steps. Such solutions of the telegraph equation are obtained first time in this paper. Illustrative figures are provided. Efficiency of the method is determined by an error analysis which is achieved by comparing numerical solutions with exact solutions.

On the Convergence of Multi-Level Hermite–Padé Approximants for a Class of Meromorphic Functions

The present paper deals with the convergence properties of multi-level Hermite–Pade approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.

Double Exponent Fractional-Order Filters: Approximation Methods and Realization

The main goal of this work is to exploit different tools in order to approximate a general double exponent fractional-order transfer function. Through the appropriate selection of the two fractional orders of this function, different types of filters can be derived. The investigated approximation tools are either curve fitting based tools or the Pade approximation tool, and the derived approximated transfer functions in all cases have the form of rational integer-order polynomials, which can be easily realized electronically.

Propagation Operator Based Boundary Condition for Finite Element Analysis

A new efficient boundary condition of finite element method (FEM) by using propagation operator is proposed. In this method, input, and output ports are terminated on their own boundaries instead of using perfectly matched layer (PML) which requires expanding the computational window. Moreover, this boundary condition can consider all modes including radiation modes without mode expansion. The propagation operator is efficiently calculated by Denman-Beavers iteration (DBI). The electromagnetic field on the POM boundary can be accurately propagated outside the boundary by using the propagation operator. In addition, we present a technique based on scattering operator method which can reduce the computational complexity of FEM. Three numerical results show that the present scheme is more accurate, and stable than conventional approximate boundary conditions such as using Pade approximation in both TE, and TM modes.

Solutions of classes of differential equations using modified differential transform method

This paper proposed a numerical approximation scheme for solving singular two-point boundary value problems. This approach is based on a truncated series obtained by differential transform (DT) method, the Laplace transformation (LT) approach and process of Pade approximants. The modified differential transform method ´ (MDTM) aims to overcome the difficulty of solving these types of problems and gives a good approximation for the true solution in a large region. Some illustrative examples are presented to demonstrate the effectiveness and applicability of the new method and the obtained results are compared with the exact solution.

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.

Bandwidth optimization of information application system under fine integral method of fuzzy fractional order ordinary differential equations

In order to solve the problems that the network bandwidth of previous information application systems can’t guarantee the quality of big data transmission, resulting in low transmission efficiency and slow data processing, etc., an improved information application system is proposed by optimizing the fine integration method of fuzzy fractional ordinary differential equations and combining with software-defined networking (SDN). Firstly, the fractional-order fuzzy differential equation fine integration method of Pade approximation is derived. Then, based on the error analysis theory, the optimization formula for the adaptive selection of weighted parameter N and expanded item number q is obtained. Combined with SDN, an improved information application system is designed, and the improved algorithm and system are detected by example simulation and performance test. The results show that the numerical accuracy and the computational efficiency of the improved algorithm are higher than that of the improved algorithm. The improved port data merge rate and task completion efficiency under different bandwidth are also significantly higher than the original system. It shows that the information application system proposed in this research can better solve the problems of insufficient bandwidth and low communication speed of traditional systems, and provide a new perspective for bandwidth optimization of big data.

A Modification of Differential Transform Method for Solving Systems of Second Order Ordinary Differential Equations

The method of differential transform (DTM) is among the famous mathematical approaches for obtaining the differential equations solutions. This is due to its simplicity and efficient numerical performance. However, the major drawback of the DTM is obtaining a truncated series solution which is often a good approximation to the true solution of the equation in a specified region. In this study, a modification of DMT scheme known as MDTM is proposed for obtaining an accurate approximation of ordinary differential equations of second order. The scheme whose procedure is designed via DTM, the Laplace transforms and finally Pade approximation, gives a good approximate for the true solution of the equations in a large region. The proposed approach would be able to overcome the difficulty encountered using the classical DTM, and thus, can serve as an alternative approach for obtaining the solutions of these problems. Preliminary results are presented based on some examples which illustrate the strength and application of the defined scheme. Also, all the obtained results corresponded to exact solutions.

Determination of Individual Parameters for the Model Mapping of a Boundary Curve in the Phase Diagram of a Stratifying Liquid Mixture

The properties of a model based on the analogy between binary stratifying liquid mixtures and coupled self-oscillating systems is studied. In this paper, an expression representing a curve mapping in the phase diagram of a binary system is derived. The mapping of a boundary curve is presented in the form of the dependence between the ratio of characteristic frequencies assigned to pure components and the mixture concentration. In the framework of this model, the temperature dependence of the squared ratio of frequencies $$({\nu}_{a}/{\nu}_{b})^{2}$$ is also determined. One important aspect of this model is that the temperature-dependent characteristic frequencies $$\nu_{a}$$ and $$\nu_{b}$$ reflect the properties of pure components (instead of solutions). The problem is posed of investigating the physical aspects of the model proposed in this paper in more detail. In the present study, it has been shown that the use of Pade approximants makes it possible to determine the character of the temperature dependence of each characteristic frequency separately on the basis of the temperature dependence derived by linking to experimental data for the ratio $$({\nu}_{a}/{\nu}_{b})^{2}$$ .

Structural and Magnetic Properties of CrN: Investigated by First-Principles Calculations, Monte Carlo Simulation, and High-Temperature Series Expansions

We have employed first-principles calculations of density functional theory (DFT) with GGA and GGA+U exchange correlation to study structural, magnetic, and electronic properties of chromium nitride (CrN) in the rock salt (NaCl) structures. The obtained data from DFT calculations is used as an input in high-temperature series expansions (HTSEs) combined with the Pade approximants (PA) method and Monte Carlo simulation (MC) for the Ising model to investigate the magnetic properties of CrN. The obtained results show that CrN is more stable in the antiferromagnetic order, and undergoes structural and magnetic transitions from an antiferromagnetic NaCl structure to a nonmagnetic Pnma phase at 201 GPa. The density of states (DOS) with GGA approach shows that CrN has a metallic behavior in both FM and AFM configurations whereas with GGA+U CrN is semiconductor in AFM and half-metallic in FM order. The Neel temperature (TN) obtained by HTSE and MC is in agreement with available experimental data.

Two-Way Parabolic Equation Method for Radio Propagation Over Rough Sea Surface

This article presents a two-way parabolic equation (2WPE) method based on the Pade(2,2) approximation form, which can model bidirectional electromagnetic wave (EW) propagation in a sea environment. Compared with other common approximation forms, the Pade(2,2) approximation has better convergence and accuracy. To accurately simulate the reflection and refraction effects of EW propagation near the sea surface, an improved fractal sea surface model is used to construct the geometric rough sea surface. The results of the Pade(2,2) approximation PE model are compared with those of the Miller–Brown model, and good agreement is observed. Numerous experiments prove that the 2WPE model is more accurate than the one-way parabolic equation model for predicting large-scale radio wave propagation in marine environments. Furthermore, we investigate the relationship between the backward-scattered EWs and the wind speed. The polynomial fitting method is adopted to process a large amount of sampled data for backward-scattered EWs with different wind speeds and frequencies. Through accurate judgment of the points of the polynomial fitting curve in a certain frequency range, the wind speed over the sea surface can be precisely determined. Thus, this article provides a novel theoretical method for marine remote sensing.

Adaptive fuzzy control for non-strict feedback nonlinear systems with input delay and full state constraints

In this paper, the problem of input delay and full state constraints for non-strict feedback nonlinear systems is studied. The influence of input delay on the control effect is eliminated by the Pade approximation method, and the barrier Lyapunov function (BLF) is adopted to constrain the system states. In the design process, using fuzzy logic systems (FLSs) to estimate the unknown nonlinear functions in the systems. The backstepping method is applied to the design of adaptive fuzzy controller. On the basis of Lyapunov stability theory, the stability of the closed-loop systems is proved, and all variables are semi-globally uniformly ultimately bounded (SGUUB). Two simulation examples verify the performance of the adaptive controller.

Short-Term and Long-Term Solutions of Two-Dimensional Shallow Water Equations Using the Modified Decomposition Method

In this paper, two modifications of the decomposition method (DM) including its multistage form (say here MSDM) and its combination with the Pade approximants (say here DMPA) are proposed and applied for the short-term and long-term solutions of two-dimensional shallow water equations (SWEs) including friction effects. Due to nonlinear behavior of the model, the DM is used to facilitate the computations on the nonlinear terms. However, the MS technique is used to obtain a solution for longer periods of time and to prevent accuracy issues associated with the standard DM at large times. The results of the proposed method are compared with those obtained using the PAs as an after treatment technique which also yields an accurate solution in our work. It has been shown that the MS technique can efficiently be used in the solution of the nonlinear coupled SWEs and would extend the domain of validity of the solution obtained via the standard DM. However, the necessity of predicting the long-term behavior of flow in some applications makes the PAs superior to the MS technique with respect to accuracy and CPU time. Numerical and graphical results of the flow characteristics for a given application of two-dimensional flood wave over two types of mild and steep slopes are presented to elucidate the efficiency of the proposed modifications in the study of the flow behavior. Moreover, a convergence analysis and the time evolution of the residuals as an index of the solution error are also presented in tabular and graphical forms.

Physics-based fractional-order model with simplified solid phase diffusion of lithium-ion battery

Abstract Solid phase diffusion plays an important role in the long-term performances of lithium-ion batteries. Current mechanistic model for describing the solid phase diffusion has low computational efficiency and obscure physical meanings. A simple but effective physics-based solid phase diffusion model is of great significance for the impedance characterization and aging diagnosis of lithium-ion batteries. In this paper, a physics-based fractional-order model is established by simplifying the solid phase diffusion process through Pade approximation. A full-cell fractional-order model is constructed by combining the medium-high frequency dynamic model. The proposed model has a clear physical meaning and can be used to characterize full-cell performances. Furthermore, this paper provides solutions to identifying the parameters of the full-cell fractional-order model by decoupling the dynamics in frequency and spatial domain. The established full-cell fractional-order model is validated under various working loads. The results show that the proposed physics-based fractional-order model with simplified solid phase diffusion improves the computational efficiency with little loss of accuracy, indicating that the proposed model is an appropriate candidate for online applications.

Linear independence of values of G-functions, II: outside the disk of convergence

Given any non-polynomial G-function $$F(z)=\sum _{k=0}^\infty A_kz^k$$ of radius of convergence R and in the kernel a G-operator $$L_F$$ , we consider the G-functions $$F_n^{[s]}(z)=\sum _{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$$ for every integers $$s\ge 0$$ and $$n\ge 1$$ . These functions can be analytically continued to a domain $${\mathcal {D}}_F$$ star-shaped at 0 and containing the disk $$\{\vert z\vert <R\}$$ . Fix any $$\alpha \in {\mathcal {D}}_F \cap \overline{{\mathbb {Q}}}^*$$ , not a singularity of $$L_F$$ , and any number field $${\mathbb {K}}$$ containing $$\alpha $$ and the $$A_k$$ ’s. Let $$\Phi _{\alpha , S}$$ be the $${\mathbb {K}}$$ -vector space spanned by the values $$F_n^{[s]}(\alpha )$$ , $$n\ge 1$$ and $$0\le s\le S$$ . We prove that $$u_{{\mathbb {K}},F}\log (S)\le \dim _{\mathbb {K}}(\Phi _{\alpha , S })\le v_FS$$ for any S, for some constants $$u_{{\mathbb {K}},F}>0$$ and $$v_F>0$$ . This appears to be the first general Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [Linear independence of values of G-functions, J. Europ. Math. Soc. 22(5), 1531–1576 (2020)], where $$\alpha \in \overline{{\mathbb {Q}}}^*$$ was assumed to be such that $$\vert \alpha \vert <R$$ . Its proof relies on an explicit construction of a Pade approximation problem adapted to certain non-holomorphic functions associated to F, and it is quite different of that in the above mentioned paper. It makes use of results of Andre, Chudnovsky and Katz on G-operators, of a linear independence criterion a la Siegel over number fields, and of a far reaching generalization of Shidlovsky’s lemma built upon the approach of Bertrand–Beukers and Bertrand.

Closed-Form Approximation for Horizontal Grounding Electrodes Transient Analysis

The method of moments (MoM) is widely used in several areas of knowledge to solve integral equations. In the specific case of solving electrical grounding transient behavior problems, the computational time is usually high. This paper presents new alternative solutions for solving the double integrals in the well-known hybrid electromagnetic model (HEM) applied to horizontal grounding electrodes. These new alternative solutions correspond to using an approximation to the exponential term appearing in the finite integral needed to obtain the parameters. Two approaches are considered either using Maclaurin series or Pade approximation. The corresponding results are compared with those obtained by using HEM with MoM. It is shown that the proposed solutions allow reducing the computational time, without jeopardizing accuracy.