Series Expansion Method Research Articles

State of charge estimation of lithium-ion batteries with non-negligible outlier observations based on Student’s-T filter

In the state of charge (SOC) estimation problem, when some outliers appear in the data measured by the sensor, the estimation effect will become worse, and the heavy-tailed distribution is more suitable to describe this complex interference noise with outliers than the Gaussian distribution. To solve the SOC estimation problem in this case, the SOC estimation scheme based on Student’s-T distribution is proposed, and the Taylor series expansion method is utilized to make the Student’s-T filter suitable for the nonlinear system of battery. Then, the SOC estimation experiment of batteries affected by different degrees of outlier noise in high and low temperature conditions are carried out by using the extended Student’s-T filter (ESTF), robust extended Student’s-T filter (RESTF), and Kullback–Leibler divergence extended Student’s-T filter (KLDESTF). The experiment results demonstrate that the proposed scheme can effectively improve the estimation effect of SOC under various working conditions when the system is polluted by different degrees of outliers, which shows that the proposed scheme has good robustness and is more suitable for SOC estimation under actual complex interference conditions.

Blocking Mechanism and Evaluation Model of Insensitive Effect Under Multi-Frequency Electromagnetic Radiation

Due to the insensitive blocking effect on frequency equipment, the current effect evaluation model based on the third-order power series expansion is disabled to the insensitive blocking effect evaluation under multi-frequency electromagnetic radiation. We adopt the fifth-order power series expansion to analyze the insensitive blocking interference effect. In addition to revealing the mechanism of the insensitive blocking effect under multi-frequency electromagnetic radiation, we also build an improved effect evaluation model in this research. It defines the effect evaluation index. Subsequently, a ranging radar and a communication radio station are employed to verify the effectiveness of the proposed effect evaluation model separately. The results show that the proposed model can be applied to the electromagnetic radiation effect evaluation, regardless of whether the multi-frequency insensitive jamming effect exists on tested equipment or not. All effect evaluation indexes of the new model are smaller than that of the previous model based on the third-order power series expansion method. The absolute value of relative errors could be less than 1 dB. Not only is the theoretical analysis result of the multi-frequency insensitive blocking effect mechanism verified, but also the problem of poor universality of the current electromagnetic radiation effect evaluation model is figured out.

Dynamic analysis of piecewise nonlinear systems with fractional differential delay feedback control

This paper mainly analyzes a class of piecewise nonlinear systems with fractional differential delay feedback control. The average method is used to solve the nonlinear system to obtain the amplitude-frequency relationship of the system. The analytical solution of the system can quantitatively analyze the system. At the same time, the correctness of the analytical solution of the system is verified by numerical solution. The effects of linear damping, linear stiffness and nonlinear stiffness of the system and the proportion, fractional order coefficient and fractional order number of the feedback controller on the amplitude-frequency characteristics of the system are analyzed. Finally, the dynamic analysis of the system is carried out. Taking the fractional order coefficient as the bifurcation parameter, the global characteristics of the bifurcation diagram, time history diagram and phase diagram system under different parameter perturbations are studied by power series expansion method. It is found that the system exhibits periodic motion, period doubling motion and chaos with the change of perturbation parameters.

Analytical Solution of Black-Scholes Model for Pricing Barrier Option using Method of Partial Taylor Series Expansion

In this work, Black-Scholes differential equation for barrier/traditional option is solved using partial Taylor series expansion method. The developed solutions are in very good agreement with the closed-form solutions of the Black Scholes equation for the powered ML-payoff functions. Also, the analytical solutions of the new method in this present study give the same expressions as the solutions of projected differential equations and homotopy perturbation method as presented in the literature. Moreover, the reliability, speed, accuracy, and ease of application of the proposed method show its potential for wide areas of applications in science, financial mathematics, and engineering.

A Series Solution for SH Wave Scattering by Irregularly Shaped Surface Topographies

ABSTRACT This paper proposes a series solution for studying the ground motion of twin-peaks hill, double triangular hills, and hill-canyon composite topography. The propagation medium of SH wave is linearly elastic, isotropic, and homogeneous. A flexible multi-region-matching technique (MRMT) and complex function method are adopted to construct the wave field expression in each sub-region. To solve the unknown coefficients in the wave field expressions, Fourier series expansion method in the complex domain is adopted. Finally, some typical numerical examples are calculated to analyse the influence of the topography shape parameters, wave number, and the incident angle on ground motion.

Valuation of a DB underpin hybrid pension under a regime-switching Lévy model

This paper studies the valuation problem of the defined benefit (DB) underpin guarantee. We consider that the salary process follows a geometric Brownian motion, and the stochastic price index process of the funds in the defined contribution (DC) account is modeled by a regime-switching Lévy process. Under this framework, the explicit valuation formula of the DB underpin option is derived by the Fourier cosine series expansion (COS) method, and the corresponding error analysis is provided. Numerous simulation experiments are performed to illustrate the accuracy and efficiency of the proposed method. In addition, the convergence of this method and its sensitivity with respect to various model paraments are analyzed.

Robust adaptive neural networks control for a class of time-delay parabolic systems with nonlinear periodic time-varying parameter

This paper mainly focuses on the stabilization of a class of parabolic systems with time-varying delay, external disturbance and nonlinear periodic time-varying parameter (NPTVP). Firstly, combined with Fourier series expansion (FSE) method and neural networks (NNs) approximation technology, a NNs approximator is designed to approximately describe the uncertain dynamic term with NPTVP. Then, based on adaptive control theory, NNs approximation technology and reparameterization method, two robust adaptive neural network control (RANNC) algorithms are designed to make the parabolic systems with time-varying delay, external disturbance and NPTVP achieve asymptotic stability and meet a prescribed adaptive H∞ performance of disturbance attenuation. Sufficient conditions of the asymptotical stability while satisfying the prescribed adaptive H∞ performance index for the resulting closed-loop systems are also derived. Finally, several simulations are carried out to verify the effectiveness of the designed RANNC algorithms.

Data-driven bond-based peridynamics with nonlocal influence function for crack propagation

In this paper, a data-driven method is proposed to define nonlocal influence function of bond-based peridynamics. In the proposed method, the data-driven algorithm coupling with Taylor series expansion and Adam optimization method shows strong stability and high accuracy in complex nonlocal influence function regression process. Moreover, two dataset acquisition methods including experimental dataset acquisition and the manufactured dataset are given out. Meanwhile, this algorithm is tested by the manufactured dataset, and the numerical results are in good agreement with the manufactured nonlocal influence function, which verifies the feasibility of the algorithm. Finally, two numerical examples are used to qualitatively and quantitatively illustrate its ability to regress the nonlocal influence function of bond-based peridynamics.

Analytical solution for the consolidation of unsaturated vertical drain foundation under depth‐dependent initial conditions

AbstractInsufficient bearing capacity of foundations is hazardous for construction engineering, and improper construction scheme can also lead to unexpected problems. Vertical drains are always installed in the soil foundations to accelerate the consolidation process and thus improve the bearing capacity. According to the consolidation theory proposed by Fredlund, the consolidation of unsaturated vertical drain foundation is investigated with analytical solutions, in which smear effect, drain resistance, and depth‐dependent initial excess pore pressures (i.e., excess pore‐air and pore‐water pressures) are considered. In the derivative process, the decoupling method and Fourier series expansion method are applied. The reliability of the current solutions is verified by comparing the degenerate ones with existing solutions. Sensitivity analysis is conducted to assess the consolidation characteristics of the proposed model against modeling sizes, smear coefficients, drain resistance factors, and distribution coefficients. The analytical model has the advantages of simplicity and easy to implement in practical engineering.

Performance Analysis of MIMO System Using Fish Swarm Optimization Algorithm

During the signal identification process, massive multiple-input multiple-output (MIMO) systems must manage a high quantity of matrix inversion operations. To prevent exact matrix inversion in huge MIMO systems, several strategies have been presented, which can be loosely classified into similarity measures and evolutionary computation. In the existing Neumann series expansion and Newton methods, the initial value will be taken as zero as a result wherein the closure speed will be slowed and the prediction of the channel state information is not done properly. In this paper, fish swarm optimization algorithm is proposed in which initial values are chosen optimally for ensuring the faster and accurate signal detection with reduced complexity. The optimal values are chosen between 0 to 1 value and the initial arbitrary values are chosen based on number of input signals. In the proposed work, Realistic condition based channel state information prediction is done by using machine learning algorithm. Simulation results demonstrate that the suggested receiver's bit error rate performance characteristics employing the Quadrature Amplitude Modulation (QAM) methodology outperform the existing Neumann series expansion and Newton methods.

Efficient generation of geodesic random fields in finite elements with application to shell buckling

Structures contain inherent deviations from idealized geometry and material properties. Quantifying the effects of such random variations is of interest when determining the reliability and robustness of a structure. Generating fields that follow complex shapes is not trivial. Generating random fields on simple shapes such as a cylinder can be done using series-expansion methods or analytically computed distances as input for a decomposition approach. Generating geodesic random fields on a mesh representing complex geometric shapes using these approaches is very complex or not possible. This paper presents a generalized approach to generating geodesic random fields representing variations in a finite element setting. Geodesic distances represent the shortest path between points within a volume or surface. Computing geodesic distances of structural points is achieved by solving the heat equation using normalized heat gradients originating from every node within the structure. Any element (bar, beam, shell, or solid) can be used as long as it can solve potential flow problems in the finite element program. Variations of the approach are discussed to generate fields with defined similarities or fields that show asymmetric behavior. A numerical example of a gyroid structure demonstrates the effect of using geodesic distances in field generation compared to Euclidean distances. An anisotropic cylinder with varying Young’s modulus and thickness is taken from literature to verify the implementation. Variations of the approach are analyzed using a composite cylinder in which fiber angles are varied. Although the focus of this paper is thin-walled structures, the approach works for all types of finite element structures and elements.

Two-Stage Chance-Constrained Coordinated Operation of an Integrated Gas–Electric System

Under the background that the high penetration of renewable energy generation, which mainly consists of wind power, will have a significant impact on electric power systems due to the volatility and uncertainty of renewable energy, energy systems with gas–electric coupling and interconnections have been widely studied to accommodate renewable energy generation. This paper proposes a two-stage chance-constrained coordinated operation model of an integrated gas–electric system and fully considers the uncertainty and high penetration of wind power. The Taylor series expansion method is used to linearize the Weymouth gas flow equation of a natural gas system and finally obtains a mixed integer linear programming model. Case studies show the effectiveness of the integrated energy system for peak shaving, valley filling, and promoting wind power accommodation. The proposed model ensures the consumption of wind power generation and also reduces the operation cost by about 0.7%.

Uncertain Quasi-static and Nonlinear Dynamic Analysis of Viscoelastic Dielectric Elastomer with Interval Parameters

Dielectric elastomers as a soft active material have been widely used in the field of artificial muscle actuator, acoustic actuator, loudspeaker, active control of vibration, soft robots and membrane resonators. Compared with traditional materials, there are many unknown uncertainties in the properties of the DE actuators. In this work, a viscoelastic dynamic model of dielectric elastomer is proposed with considering the uncertainties in material parameters, external mechanical load and voltage. By introducing the interval perturbation method and first-order Taylor series expansion method, the creep analysis, relaxation analysis and dynamic analysis of the dielectric elastomer with interval uncertain parameters are implemented. The effectivity of the proposed interval method is verified by the Monte Carlo simulation. This uncertain prediction method could be used in the design of active control systems with dielectric elastomers as actuators or sensors in the future.

Free Convective MHD Radiative Flow Past a Porous Vertical Plate in a Porous Medium with Chemical Reaction

An analytical solution for two-dimensional steady, viscous, incompressible hydromagnetic free convective flow past a uniformly moving vertical porous plate embedded in a porous medium in the presence of the Soret effect and chemical reaction is investigated in this paper. A uniform magnetic field is applied perpendicular to the plate and directed into the fluid region. The asymptotic series expansion method is used to solve the non-dimensional governing equations. The effects of thermal diffusion, magnetic field, thermal radiation, Grashof number, and chemical reaction are mainly focused on discussing the present problem. One of the significant findings of this study includes that an upsurge in the chemical reaction causes a downfall in the fluid temperature. It is seen that the Soret number enhances the velocity and temperature. Further, the fluid's velocity, concentration, and temperature decrease with an increase in the magnetic field parameter. In most works dealing with convective MHD flow, ohmic dissipation and viscous dissipation are neglected. In this work, these two effects are considered. Further, the Soret effect and the effect of first-order chemical reactions are also considered.

Gerber-Shiu analysis in the compound Poisson model with constant inter-observation times

In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

Solutions of the Schrödinger equation with Hulthén-screened Kratzer potential: Application to Diatomic Molecules

In this study, the Schrödinger equation with the Hulthén plus screened Kratzer potentials (HSKP) are solved via the Nikiforov-Uvarov (NU) and the series expansion methods. We obtained the energy equation and the wave function in closed form with Greene-Aldrich approximation via the NU method. The series expansion method was also used to obtain the energy equation of HSKP. Three distinct cases were obtained from the combined potentials. The energy eigenvalues of HSKP for HCl, LiH, H2, and NO diatomic molecules were computed for various quantum states. To test the accuracy of our results, we computed the bound states energy of HCl and LiH, for a special case of Kratzer and screened Kratzer potentials, which are in excellent agreement with the report of other researchers.

Lamb Waves Propagation Characteristics in Functionally Graded Sandwich Plates.

Functionally graded materials (FGM) have received extensive attention in recent years due to their excellent mechanical properties. In this research, the theoretical process of calculating the propagation characteristics of Lamb waves in FGM sandwich plates is deduced by combining the FGM volume fraction curve and Legendre polynomial series expansion method. In this proposed method, the FGM plate does not have to be sliced into multiple layers. Numerical results are given in detail, and the Lamb wave dispersion curves are extracted. For comparison, the Lamb wave dispersion curve of the sliced layer model for the FGM sandwich plate is obtained by the global matrix method. Meanwhile, the FGM sandwich plate was subjected to finite element simulation, also based on the layered-plate model. The acoustic characteristics detection experiment was performed by simulation through a defocusing measurement. Thus, the Lamb wave dispersion curves were obtained by V(f, z) analysis. Finally, the influence of the change in the gradient function on the Lamb wave dispersion curves will be discussed.

A Discrete Dynamics Approach to a Tumor System

In this paper, we present a cancer system in a continuous state as well as some numerical results. We present discretization methods, e.g., the Euler method, the Taylor series expansion method, and the Runge–Kutta method, and apply them to the cancer system. We studied the stability of the fixed points in the discrete cancer system using the new version of Marotto’s theorem at a fixed point; we prove that the discrete cancer system is chaotic. Finally, we present numerical simulations, e.g., Lyapunov exponents and bifurcations diagrams.

Numerical solution of the two-dimensional regularized long-wave equation with a conservative linearized high order finite difference scheme

In this article, a finite difference scheme is investigated to solve a two-dimensional regularized long-wave equation. The original equation is deformed into another formulation. The fourth order compact difference scheme and the fourth order Padé scheme are used for the treatment of the second and first derivatives of the deformation equation in space direction, and the fourth order backward difference formula is used for the discretization of the time derivative. The nonlinear term in the proposed scheme is linearized by Taylor series expansion method. The new scheme is shown to be of the fourth order accuracy in both temporal and spatial dimensions. The existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the new scheme is shown to be conservative and unconditionally stable. Finally, numerical experiments are provided to verify our theoretical analysis results of the new scheme.

Acoustic radiation torque of a cylindrical quasi-Gauss beam on a viscoelastic cylindrical shell near an impedance boundary

This work presents a comprehensive analytical formalism to calculate the acoustic radiation torque acting on a viscoelastic cylindrical shell near an impedance boundary, which is exerted by a cylindrical quasi-Gauss beam with an arbitrary incident angle propagating in an ideal fluid. An analytical expression in finite series form for the acoustic radiation torque is derived by applying the partial-wave series expansion method, the image theory and the translational addition theorem. Numerical computations for a viscoelastic cylindrical polyethylene shell immersed in water are provided, with particular emphasis on the effects of the reflecting coefficient of the interface, the shell-boundary distance, the position of the shell, the incident angle and the beam waist. Changing the relative thickness and the interior fluid medium are also considered. It is shown that the cylindrical shell can be rotated counter-clockwise or clockwise, depending on the beam frequency, the shell size, the reflecting coefficient and the incident angle. The periodical oscillation of acoustic radiation torque is also observed as the shell moves away from the boundary. The results of this study can improve our understanding of the acoustic radiation torque behaviors for viscoelastic particles near the boundary, which is commonly encountered in biomedical ultrasound and fluid dynamics.