Series Expansion Technique Research Articles

Stochastic Lighthill-Whitham-Richards traffic flow model for nonlinear speed-density relationships

Stochasticity is becoming increasingly essential in traffic flow research, given its notable influence in several applications, such as real-time traffic management. To consider stochasticity in macroscopic traffic flow modeling, this paper introduces a stochastic Lighthill-Whitham-Richards (SLWR) model, which not only captures equilibrium values in steady-state conditions but also describes stochastic variabilities. The SLWR model follows a conservation law, in which the free-flow speed is randomized to represent heterogeneities of drivers. To more accurately reflect real-life traffic patterns, a nonlinear speed-density relationship is considered. For addressing this highly nonlinear problem, a dynamically bi-orthogonal (DyBO) method is coupled with the Taylor series expansion technique. The results of simulation experiments show that the SLWR model can effectively describe the evolution of stochastic dynamic traffic with temporal or geometric bottlenecks. Moreover, the DyBO solutions exhibit reasonable accuracy while significantly reducing computation costs compared with the Monte Carlo method.

Hierarchical Distributed Adaptive Fault-Tolerant Control of Nonlinear Fractional-Order Multiagent Systems With Faults and Periodic Disturbances Using Event-Triggered Communication.

This article presents a distributed fault-tolerant control (FTC) scheme for nonlinear fractional-order (FO) multiagent systems (MASs) with the order lying in (0, 1], such that the proposed control architecture can be directly applied to both FO and integer-order (IO) systems without any modifications. To handle the unexpected actuator faults encountered by the FO MASs, a hierarchical FTC mechanism is developed for each system by constructing an event-triggered distributed FO estimator at the upper layer to estimate the leader system's output via conditionally triggered neighboring information, and an FTC unit at the lower layer to counteract the loss-of-effectiveness faults via Nussbaum function with FO criteria. To further address the unknown nonlinear functions involving bias faults and periodic disturbances, the Fourier series expansion technique is used to construct the input variables of fuzzy neural networks (FNNs), such that the FNNs with dynamically adjusted weight matrices, centers, and widths can be developed for each FO system to act as the learning module. It is shown by FO Lyapunov stability analysis that all follower systems can track the leader system against faults and periodic disturbances. Simulation results on FO systems and hardware-in-the-loop experiment results on IO fixed-wing unmanned aerial vehicles show the extensive feasibility of the developed scheme.

Closed-Form Method for Unified Far-Field and Near-Field Localization Based on TDOA and FDOA Measurements

When the near-field and far-field information of a target is uncertain, it is necessary to choose a suitable localization method. The modified polar representation (MPR) method integrates the two scenarios and achieves a unified localization with direction of arrival (DOA) estimation in the far field and position estimation in the near field. Previous studies have only proposed solutions for stationary environments and have not considered the motion factor. Therefore, this paper proposes a new unified positioning algorithm using multi-sensor time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements without prior target source information. The method represents the position of the target source using MPR and describes the localization problem as a weighted least squares (WLS) problem with two constraints. We first obtain the initial estimates by WLS without considering the constraints and then investigate a two-step error correction method based on the constraints. The first step corrects the initial estimate using the Taylor series expansion technique, and the second step corrects the DOA estimate in the previous step using the direct error compensation technique based on the properties of the second constraint. Simulation experiments show that the method is effective for the unified positioning of moving targets and can achieve the Cramer–Rao lower bound (CRLB).

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

Performance analysis of a collision channel with abandonments

We consider a Markovian retrial queueing system with customer collisions and abandonment in the context of carrier-sense multiple access systems. Using z-transform techniques, we find a set of first-order differential equations for the probability generating functions of the orbit size when the server is empty, busy, or in the collision phase. We then rely on series expansion techniques to extract approximations for relevant performance measures from this set of differential equations. More precisely, we construct a numerical algorithm to calculate the terms in the series expansions of various factorial moments of the orbit size. To improve the accuracy of our series expansion approach, we apply Wynn’s epsilon algorithm which not only speeds up convergence, but also extends the region of convergence. We illustrate the accuracy of our approach by means of some numerical examples, and find that the method is both fast and accurate for a wide range of the parameter values.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Optimization of Covert Communication in Multi-Sensor Asymmetric Noise Systems

This work investigates wireless covert communication in a multi-sensor asymmetric noise scenario. We adopt KL (Kullback–Leibler) divergence as the covertness constraint metric and mutual information as the transmission rate metric. To accurately approximate KL divergence and mutual information in covert communication, we employ the Taylor series expansion technique. Analytical expressions for KL divergence and mutual information in covert communication are derived, and we optimize the amplitude gain and phase angles based on these analytical expressions. Our findings underscore the importance of phase angle selection in covert communication within asymmetric noise systems. We propose an effective method for optimizing the transmission amplitude gain and phase angles in scenarios with asymmetric noise. Numerical results validate the effectiveness and superiority of our proposed method.

Dynamic behaviour of carbon-nanotube reinforced functionally graded double-arch systems

The goal of the current investigation is to determine the dynamic behaviour of double-arch systems: the system is made of two arches reinforced by three different functionally graded patterns of carbon nanotubes and connected with an elastic layer of spring bed. The carbon-nanotube functionally graded patterns considered are uniformly distributed, FG-X and FG-O. Two different boundary conditions of movable and immovable simply supported (named SS1 and SS2) are studied for the reinforced double-arches, where the displacements along the curve-line are constrained for the ends of SS2 arches. Both the Hamilton principle and force-moment technique are utilised to formulate the coupled equations of motion. A series expansion technique is then used to solve the equations. A validation is performed and consistent agreement between the proposed methodology and simplified version of the double-arch system is achieved. One prominent observation arising from this study is that an increase in the opening angle of the double-arch system results in a decline in both the series for the transverse natural frequencies for SS1. Conversely, the systems with SS2 boundary conditions exhibit an initial rise in both the series of natural frequencies as the opening angle increases, followed by a gradual decrease. A thicker carbon-nanotube reinforced functionally graded double-arch system demonstrates an increased natural-frequency sensitivity to variation in opening angles. Lastly, increasing the elastic layer coefficient of stiffness causes an increment in the second series natural frequency of the system.

A study of the 3-phase lag model to a two-dimensional isotropic micro-polar thermoelastic medium with memory-dependent properties

This article enlightens the two-dimensional (2D) isotropic micro-polar thermoelastic problem of the three-phase-lag (3PHL) model and heat conduction equation is formulated in the context of memory-dependent derivative (MDD). The equations are converted into the domain of the Laplace transform vector matrix differential equation form and solved by using the eigenvalue technique. To obtain an analytical solution of displacement, temperature, and stress components Silicon material properties are used. Inversion of the Laplace transform with Fourier series expansion technique is used to obtain the numerical solution. For obtaining graphical results Mathematica software is used. For the purpose of exhibiting the beauty of MDD in the present model comparisons are made between the time delay parameters and kernel functions (constant, linear, and non-linear kernels), respectively, over the micro-polar panel. The results obtained have a valuable impact on structural analysis, especially in the design of rotating machinery structures using accurate material properties.

Reduced order isogeometric boundary element methods for CAD-integrated shape optimization in electromagnetic scattering

This paper formulates a model order reduction method for electromagnetic boundary element analysis and extends it to computer-aided design integrated shape optimization of multi-frequency electromagnetic scattering problems. Firstly, a series expansion technique is adopted to decouple frequency-dependent terms from the integrands in boundary element formulation, and the second-order Arnoldi procedure is utilized to reduce the order of original systems. Secondly, to ensure geometric exactness and avoid re-meshing during shape optimization, the isogeometric boundary element method is employed, which uses the Non-Uniform Rational B-splines to represent the geometry and to discretize boundary integral equations. Lastly, we employ the Grey Wolf Optimization-Artificial Neural Network as a surrogate model for shape optimization in multi-frequency electromagnetic scattering problems, with the radar cross section as the objective function. Several examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Restoration of the orientation distribution function for materials with low lattice and sample symmetry using the harmonic method

A lot of the properties polycrystalline materials depend on their crystallographic texture. The most complete information about the texture can be obtained from the orientation distribution function (ODF). We present the results of recovering ODF using series expansion technique for materials with low crystal and sample symmetry. The technique of ODF restoration is based on its Fourier series expansion with symmetrical spherical harmonic functions. Real spherical harmonics which are linear combinations of general spherical harmonics were used. The model single-component texture as well as the real texture of magnesium alloy sample subjected to equal-channel angular pressing have been studied. Textures are characterized by hexagonal crystal symmetry and triclinic sample symmetry. In both cases RP-factors and ODF calculation errors that were used as reliability criteria of ODF reconstruction showed good agreement between the calculated and experimental data. It was also revealed that the ODF of a magnesium alloy sample subjected to equal-channel angular pressing contains two texture components (1216)[1211] and (1216)[1211] with maximum intensity values of 13.81 and 2.23, respectively. The results obtained can be used for texture studies of ceramics, rocks and other non-metallic materials characterized by a lower symmetry.

Analytical solutions for steady vertical flux through unsaturated soils with generalized hydraulic properties

In this paper, analytical solutions for one-dimensional nonlinear steady-state vertical flux through unsaturated soils are presented. The analytical solutions are applicable for homogeneous soils for which the hydraulic conductivity function is a generalized form of the well-known Brooks-Corey and rational models. The adopted generalized soil–water retention model involves two additional parameters which reflect the curvature of hydraulic functions near saturation. The main advantage of the used generalized hydraulic conductivity function is that it fits better experimental data. The soil domain is a finite-depth medium overlying a fixed groundwater level. The derived analytical solutions are obtained using the Maclaurin series expansion technique. Analytical solutions for both infiltration and evaporation are developed. For evaporation from a fixed groundwater level, the derived solutions can be used to predict the existence of a vaporization plane below the soil surface. For this case, analytical expression of the position of drying front separating liquid and gas regions is derived. The analytical results are compared to numerical ones for various infiltration and evaporation examples and excellent agreements are obtained. The generalized model is also used to fit experimental data obtained for two soils. The results show that the use of the standard Brooks-Corey model may lead to an overestimation of the position of the drying front.

Vibrations of porous functionally graded CNT reinforced viscoelastic beams connected via a viscoelastic layer

This work analyses the vibrational response of porous viscoelastic functionally graded (FG) carbon nanotube (CNT) reinforced double beams. The beams are simply supported, connected through a viscoelastic layer, where four functionally graded patterns of CNT in the thickness directions (uniformly distributed (UD), FG-V, FG-O, and FG-X) are considered. The porosity models considered are UD, PO-V and PO-X. After formulating the coupled equations of motion via an energy-based method, and incorporating the effects of the viscoelastic beams and viscoelastic layers, the equations are solved using an assumed series expansion technique. The equations and solution methodology are verified utilising simplified models from existing literature and also through development of a finite element method (FEM) based simplified model; very good agreement has been achieved. Effects of the stiffness and viscous coefficients of the viscoelastic layer, the structural viscosity parameter of the porous CNT reinforced double beams, the porosity coefficient, and porosity models are examined. Overall, the results indicate that the stiffness and viscous coefficients of the viscoelastic layer have increasing effects on the real and imaginary components of second series fundamental transverse natural frequency of the double beam system, respectively. An increase in the structural viscosity parameter of the porous CNT reinforced double beams leads to an increase in the imaginary components of the fundamental frequencies of both the series. Moreover, a range of different effects on the real and imaginary components of the natural frequencies were noted while varying the porosity coefficient and models. The FG-X CNT porous reinforced viscoelastic double beams with a viscoelastic layer have the highest natural frequencies of all the cases. The study demonstrates the importance of the porosity, viscoelastic layer and viscoelastic material properties on the dynamics of the system.

New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

We present in this paper two new families of bi-parametric multipoint higher order iterative methods of optimal order for determining simple roots of the nonlinear equation Ω(s)=0. The proposed families of methods are derivative-free with the optimal three-point fourth order methods requiring only three function evaluations per iteration and the optimal four-point eighth order methods consuming four function evaluations per iteration. Taylor’s series expansion and divided difference techniques are employed for the formulation of the methods. Their theoretical convergence properties are thoroughly analysed through the main theorems. Numerical experiments on nonlinear functions with some engineering applications are presented and are compared with some existing methods to demonstrate the effectiveness, applicability and validity of the proposed families of methods. Finally, graphical comparison is made through the basins of attraction which further provide useful information about their dynamical behaviour in the complex plane.

Employment of Fracture Mechanics Criteria for Accurate Assessment of the Full Set of Elastic Constants of Orthorhombic/Tetragonal Mono-Crystalline YBCO

The effect of elastic constants, cij, on the nature (easy or difficult) of a cleavage system in mono-crystalline YBa2Cu3O7−δ is investigated by employing a novel three-dimensional eigenfunction expansion technique, based in part on the separation of the thickness variable and partly on a modified Frobenius-type series expansion technique in conjunction with Eshelby–Stroh formalism. Out of the three available, complete sets of elastic constants, only the experimental measurements using resonant ultrasound spectroscopy merit serious attention, despite reported values of c12 and, to a lesser extent, c66 being excessively high. The present investigation considers six through-thickness crack systems weakening orthorhombic mono-crystalline Yttrium barium copper oxide (YBCO) plates. More importantly, the present investigation establishes sufficient conditions for crack path stability/instability, which entail a cleavage system being easy or difficult, i.e., whether a crack would propagate in its original plane/direction or deflect to a different one. This criterion of fracture mechanics is then employed for accurate determination of the full set of elastic constants of superconducting mono-crystalline YBCO. Finally, heretofore unavailable results pertaining to the through-thickness variations of stress intensity factors and energy release rates for a crack corresponding to symmetric and skew-symmetric hyperbolic cosine loads, which also satisfy the boundary conditions on the plate surfaces, bridge a longstanding gap.

Random-bond Ising model and its dual in hyperbolic spaces.

We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature T and the fraction of antiferromagnetic bonds p. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.

Transient solute dispersion in electro‐osmotic viscoplastic flow in a microchannel

AbstractThe transport of a neutral solute in incompressible electro‐osmotic flow of Bingham plastic non‐Newtonian liquid flowing through a microchannel is studied theoretically. The flow is driven by a constant axially applied electric field. The non‐dimensional conservation equations with associated boundary conditions are solved with Gill's series expansion technique. The whole transport process is analyzed using three transport coefficients, namely, advection coefficient, dispersion coefficient and the apparent asymmetry coefficient, respectively. The mean concentration distribution of the solute is calculated via a third‐order approximation of the series expansion. The study investigates the collective effects of yield stress and electric double layer (EDL) thickness (inverse Debye length) on the transport of solute. The long‐term behaviour of mean concentration distribution is shown to be accurately predicted by second‐order approximations; however, utilizing third or higher order approximations in Gill's series expansion enables a more refined analysis of the small and moderate time behavior of transport coefficients. The present analysis is relevant to emerging applications in bio‐microfluidics exploiting electroviscous fluent media.

Application of Nonlinear Differential Equation in Electric Automation Control System

Abstract This article uses fifth-order nonlinear differential equations to describe the dynamic process of electrical automation control systems. This method first derives the equivalent system of the nonlinear fuzzy global system and then uses the orthogonal polynomial series expansion technique and its integral operation matrix. The local manifold at the dominant unstable equilibrium point of a single-machine infinite-bus system after a failure described by a two-dimensional quadratic nonlinear differential equation is calculated, and the stability boundary of the power system is obtained. The research results show that the output frequency fluctuation of the electrical automation control system is small after the algorithm is adopted, and the intelligent control system can accurately diagnose and warn the electrical faults. The system can meet the requirements of online voltage coordinated control.

ПОЛЯРИЗАЦІЙНО ТА ЧАСТОТНО СЕЛЕКТИВНІ ХАРАКТЕРИСТИКИ ХІРАЛЬНОЇ ПОВЕРХНІ, ЩО СКЛАДАЄТЬСЯ З ПЕРІОДИЧНО РОЗТАШОВАНИХ ДІЕЛЕКТРИЧНИХ КВАДРАТНИХ СПІРАЛЕЙ

Subject and Purpose. The present study is concerned with the linearly polarized electromagnetic wave transmission through a chiral metasurface composed of periodically assembled square dielectric helices. We expect that the metasurface of the kind has a wider range of functional capabilities to transform a polarized wave into a cross-polarized one when compared to a similar metasurface composed of metal helices. Methods and Methodology. To find the scattering coefficients of the considered structure, the well-established method of integral functionals is followed. A set of volume integral equations in the vectorial form is solved for the equivalent electric and magnetic polarization currents of the analyzed periodic layer. A distinctive feature of the method is that the internal electromagnetic fields of the structure are initially found, whence the fields scattered by this structure are sought. The equations are discretized in terms of integral functionals related to the polarization currents and through the use of the double Floquet–Fourier series expansion technique. Results. It has been found that the metasurface transmission coefficients depend critically on the number of bars making the square helical particle. In the case of an even bar number, the chiral metasurface exhibits the same transmission coefficients for co- polarized field components in the event that linearly x- and y-polarized waves are incident. For cross-polarized field components, the transmission coefficients differ and can reach peak values at different frequencies. Finally, transmission coefficients of these polarizers have been investigated versus dimensions of helice-making bars. Conclusion. A wide variety of transmission properties observed in the metasurfaces make them particularly attractive for use in polarization converting and separating devices. The metasurface can feature dichroic asymmetric transmission and be used as a dichroic filter with polarization transformation. It can be put to use in differential phase sections and, also, as an effective dichroic cross-polarization converter (twist polarizer)

Jacobi stochastic volatility factor for the LIBOR market model

We propose a new method to efficiently price swap rate derivatives under the LIBOR market model with stochastic volatility and displaced diffusion. This method applies series expansion techniques built around Gaussian (Gram–Charlier) or Gaussian mixture densities to polynomial processes. The standard pricing method for the considered model relies on dynamics freezing to recover a Heston-type model for which analytical formulas are available. This approach is time-consuming, and efficient approximations based on Gram–Charlier expansions have been proposed recently. In this article, we first discuss the fact that for a class of stochastic volatility model, including the Heston one, the classical sufficient condition ensuring the convergence of Gram–Charlier series is not satisfied. Then we propose an approximating model based on a Jacobi process for which we can prove the stability of Gram–Charlier-type expansions. For this approximation, we have been able to prove a strong convergence towards the original model; moreover, we give an estimate of the convergence rate. We also prove a new result on the convergence of the Gram–Charlier series when the volatility factor is not bounded from below. We finally illustrate our convergence results with numerical examples.