Difference Approximation Method Research Articles

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System.

Magnetic anomaly detection (MAD) technology based on the magnetic gradient tensor (MGT) has broad application prospects in fields such as unexploded ordnance detection and mineral exploration. The difference approximation method currently employed in the MGT measurement system introduces measurement errors. Designing reasonable geometric structures and configuring optimal structural parameters can effectively reduce measurement errors. Based on research into differential MGT measurement, this paper proposes three simplified planar MGT measurement structures and provides the differential measurement matrix. The factors that affect the design of the baseline distance of the MGT measurement system are also theoretically analyzed. Then, using the magnetic dipole model, the error analysis of the MGT measurement structures is carried out. The results demonstrate that the planar cross-shaped structure is optimal, with the smallest measurement error, only 3.15 × 10-10 T/m. Furthermore, employing the control variable method, the impact of sensor resolution constraints, noise level, target magnetic moment, and detection distance on the design of the optimal baseline distance of the MGT measurement system is simulated and verified. The results indicate that the smaller the target magnetic moment, the farther the detection distance, the lower the magnetometer resolution, the greater the noise, and the greater the baseline distance required. These conclusions provide reference and guidance for the construction of the MGT measurement system based on triaxial magnetometers.

Theoretical studies for the detailed electronic structure of organotin(IV) derivatives of 4-fluoroanthranilic acid, X-ray structure of n-dibutyltin bis(4-fluoroanthranilate)

The present study describes the detailed DFT-based electronic structure calculations without any symmetry constraints being performed on di- and triorganotin derivatives of 4-fluoroanthranilic acid (AFA), viz. Me2SnL2 (1), n-Bu2SnL2 (2), Me3SnL (3), and Ph3SnL (4) (where L= monoanion of AFA). The structural and atomic charge analysis have confirmed the previously reported our experimental results, i.e. bicapped tetrahedral and distorted tetrahedral geometry for di- and triorganotin derivatives, respectively, and all the complexes 1–4 contain a positively charged central tin atom surrounded by a negatively charged system. The single crystal X-ray structure of complex 2 (n-Bu2SnL2) also suggests that AFA acts as a monoanion and bidentate ligand resulting a bicapped tetrahedral environment around tin. Various population analysis such as Mulliken (MPA), Hirshfeld (HPA), and natural population analysis (NPA) have been employed to calculate the charges at all the atoms. A finite difference approximation method has been used to explain the charge distribution within the studied complexes 1–4 based on the conceptual global and local reactivity DFT-based descriptions, frontier molecular-orbital analysis (FMOA), and molecular electrostatic potential maps (MEP). Further, the conceptual DFT-based global reactivity descriptors and frontier molecular orbital analysis show that the interactions between CT-DNA and complexes 1–4 may be groove binding or electrostatic. The integral equation formalism-polarizable continuum model (IEF-PCM) has been employed to calculate the UV–visible spectra of 1–4 in the solvent field, whereas the same in the gas phase has been obtained with the help of time-dependent DFT (TD-DFT) at the same level of theory. The intramolecular charge distribution in 1–4 has been investigated with the help of natural bond orbital (NBO) analysis. The topological and energetic parameters at the selected bond critical points in the coordination sphere of complexes 1–4 have been calculated using atoms-in-molecules (AIM) theory. The analysis of different kinds of non-covalent interactions (NCI) in complexes 1–4 has also been investigated.

Direct detection Φ-OTDR based on UWFBG array using linear-phase-modulated double-pulse

We propose what we believe to be a novel direct detection phase-sensitive optical time-domain reflectometry (Φ-OTDR) based on ultra-weak fiber Bragg grating (UWFBG) array to achieve distributed vibration measurements with exceptional sensitivity and remarkable stability. Our system employs a pulse modulator to generate a double pulse and achieves linear phase modulation of one pulse by one cycle through a phase modulator. The phase change can be quantitatively demodulated using our proposed N-step phase-shifted demodulation algorithm. This method effectively mitigates the influence of phase noise of the laser and the pulse modulator, while also eliminating fluctuations in the half-voltage of the phase modulator. Compared with the existing phase modulation methods, our method avoids stringent requirements for the stability and precision of phase modulation. Moreover, we propose a phase-shifted approximation method, breaking the limitation of sensing length on the traditional differential approximation method and improving the accuracy significantly. The technique's effectiveness is experimentally demonstrated on a 1 km UWFBG array with a reflectivity of −40 dB to −45 dB and a spatial resolution of 10 m. Vibrations with different amplitudes are measured quantitatively with good linearity. The low-frequency self-noise is greatly suppressed and the overall self-noise is −54.3 dB rad2/Hz.

HNS: An efficient hermite neural solver for solving time-fractional partial differential equations

Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for addressing time-fractional derivatives within neural network solvers. However, we have discovered that neural network solvers based on L1 interpolation approximation are unable to fully exploit the benefits of neural networks, and the accuracy of these models is constrained to interpolation errors. In this paper, we present the high-precision Hermite Neural Solver (HNS) for solving time-fractional partial differential equations. Specifically, we first construct a high-order explicit approximation scheme for fractional derivatives using Hermite interpolation techniques, and rigorously analyze its approximation accuracy. Afterward, taking into account the infinitely differentiable properties of deep neural networks, we integrate the high-order Hermite interpolation explicit approximation scheme with deep neural networks to propose the HNS. The experimental results show that HNS achieves higher accuracy than methods based on the L1 scheme for both forward and inverse problems, as well as in high-dimensional scenarios. This indicates that HNS has significantly improved accuracy and flexibility compared to existing L1-based methods, and has overcome the limitations of explicit finite difference approximation methods that are often constrained to function value interpolation. As a result, the HNS is not a simple combination of numerical computing methods and neural networks, but rather achieves a complementary and mutually reinforcing advantages of both approaches. The data and code can be found at https://github.com/hsbhc/HNS.

Construction of Piano Performing Arts Talent Cultivation and Creative Thinking Based on the Constant Differential Error Approximation Method

Abstract Piano talent cultivation mostly focuses on repetitive training only, and lacks a clear cultivation program for theoretical knowledge and skill points. In this paper, we adopt the curriculum knowledge graph to extract the knowledge point subgraph of piano learning and provide an initialized knowledge graph structure for the learning path under the cultivation scheme. A multi-objective optimization model of the path containing constraints such as the difficulty and mastery of knowledge points is set, and the relevant parameters of the model are defined and calculated to establish a multi-optimization objective function. A solution method that uses constant differential error approximation is proposed as a means of approaching the Pareto optimal solution. To develop the LSPIA algorithm with reciprocal weights, the adjustment vectors are modified using a least squares asymptotic iterative approximation with minimum squares values. Following the discovery of the most suitable route, a teaching experiment was carried out. The results show that the posttest P-value of the total creative thinking score of the experimental and control classes is 0.022 less than 0.05, which indicates that there is a significant difference between the creative thinking of the students in the two classes in terms of affective characteristics. The experimental group achieved a mean score of 122.69 on the post-test, which was a significant improvement from the mean score on the pre-test. An innovative approach to cultivating piano playing talents involves optimizing learning paths.

MHD Fluid Flow Bounded by Two Parallel Vertical Plates in A Porous Media with Heat Transfer

We are investigating Magneto hydrodynamic fluid flow between two parallel vertical plates in a porous media with heat transfer. We will nondimensionalize the governing equations and then transform the resulting equations using the central difference approximation method. We will simulate the resulting equations using MATLAB to establish the effect of non dimensionalized parameters on the fluid velocity and temperature. We will graphically present the effect of the Darcy, Hartmann, Prandtl, and Reynolds numbers on the velocity and temperature profile. We have made the following conclusions from this study. The fluid velocity is inversely proportional to Darcy’s and Hartmann’s numbers. Fluid velocity increases as the Reynolds number is increased. Fluid temperature declines with a rise in the Prandtl number when the distance between the plates is greater than one and increases the fluid temperature when the distance between plates is less than one. Fluid temperature increases as Hartmann and Darcy numbers are increased. No velocity and temperature profile effect exists on varying Prandtl and Reynolds numbers. The findings have practical applications in industrial processes such as cooling systems.

Analytical solution of the fractional neutron point kinetic equations using the Mittag-Leffler function

A new analytical solution of the fractional point kinetic equations is developed in the present work, which considers one group of delayed neutrons as well as a constant reactivity. One of the main novelties and theoretical contributions of the developed solution is that it does not require approximating the Inverse Laplace transform by numerical means, as other solutions do, and instead it is obtained analytically using the Green and the Mittag-Leffler functions. A set of algorithms and MATLAB codes are developed with the purpose to implement and compute the developed solution in an adequate way, which represents the major computational contribution. Numerical experiments show that the developed solution is very accurate solving short-time dynamics phenomena, being able to reproduce, in a precise way, data that is reported in literature. The development of the proposed analytic solution represents an important contribution to validate numerical methods that are currently used to solve the neutron fractional point kinetic equations. Program SummaryProgram Title: Mittag-Leffler FNPKECPC Library link to program files:https://doi.org/10.17632/v4kfhzw7sf.1Developer's repository link:https://github.com/Cruz-Lopez-Carlos-Antonio/Mittag-Leffler-FNPKE/tree/mainLicensing provisions: Creative Commons by 4.0Programming language: MATLAB (2021a) / Python 3Nature of problem:Fractional Neutron Point Kinetic Equations describe the neutron population in a nuclear reactor, considering the changes on transport and thermohydraulic properties of the system. For such task, a mass balance approach based on fractional and integral derivatives is used, which improves and generalizes the integer case that uses the classical definitions of these mathematical operators. From a physical point of view, this new approach takes into account memory and non-local effects that are omitted in the standard neutron diffusion theory and which are relevant in scenarios where the current neutron density undergoes strongly variations on time. From a mathematical point of view, using fractional derivatives leads to a new and more general type of differential equations of arbitrary order that requires different methods of solution and approximation, and which must include the integer approach as particular case. In such context, the development of analytical solutions is a fundamental problem because it provides a framework to carry out verifications of numerical methods, as well as an insight to perform theoretical comparisons with the classical model. The developed Mittag-Leffler FNPK codes analytically solve the Fractional Neutron Point Kinetic Equations defined in terms of the Caputo's derivatives and subject to classical initial conditions.Solution method:The developed codes are written in MATLAB and Python programming languages, and solve the mentioned fractional differential equation system using a power series expansion as well as the Laplace transform. Such solution considers a single group of precursors of delayed neutrons and it is expressed in terms of the 2-parameter Mittag-Leffler function, being accurately computed considering a small number of terms of the infinite sum. Unlike other approaches, in the present case it is not necessary to use a numerical procedure to find the inverse Laplace transform, because it is analytically found using the Green function. Even when the developed solution is found for a constant reactivity case, the Mittag-Leffler FNPKE codes can extended it to more general scenarios where the reactivity is a function of time or where it depends on the neutron density itself. In these cases, it is necessary to apply a subinterval methodology where the time is divided in small steps, assuming a constant reactivity in each of them. The smaller of the intervals, the more accurate the approximation. With the purpose to overcome the numerical issues related to the computation of the 2-parameter Mittag-Leffler function, the developed codes integrate the most advanced algorithm reported in literature, which is useful also to estimate the high-order derivatives of such function.Additional comments including restrictions and unusual features:The developed codes do not face issues related to arithmetical precision digits, as other solutions do, because it is not expressed in terms of high order polynomials and instead it is written as a linear combination of the Green function. Additionally, several steps can be precomputed without regard the neutron parameters and initial conditions, which represents an important advantage in terms of the computational time.

Spatio-Temporal Validation of GNSS-Derived Global Ionosphere Maps Using 16 Years of Jason Satellites Observations

Existing ionospheric models perform very well in mapping the calm state of the ionosphere. However, the problem is accurately determining the total electron content (TEC) for disturbed days. Knowledge of the exact electron density is essential for single−frequency receivers, which cannot eliminate the ionospheric delay. This study aims to investigate temporal and spatial variability in the distribution of TEC based on differences between maps of individual Ionospheric Associated Analysis Centers (IAACs) of the International GNSS Service (IGS) and aligned altimetry−TEC from 2005–2021. Based on the temporal distribution, we have observed a significant effect of solar activity on the mean and standard deviation behavior of the differences between global ionospheric maps (GIMs) and Jason−derived TEC. We determined the biases for the entire calculation period, through which it can be concluded that the upcg-Jason and igsg-Jason differences have the lowest standard deviation (±1.81 TECU). In addition, the temporal analysis made it possible to detect annual, semi−annual, and 117-day oscillations occurring in the Jason−TEC data, as well as 121-day oscillations in the GIMs. It also allowed us to analyze the potential sources of these cyclicities, solar and geomagnetic activity, in the case of the annual and semi−annual periodicities. When considering spatial variations, we have observed that the most significant average differences are in the intertropical areas. In contrast, the smallest differences were recorded in the southern hemisphere, below the Tropic of Capricorn (23.5°S). However, the slightest variations were noted for the northern hemisphere above the Tropic of Cancer (23.5°N). Our research presented in this paper allows a better understanding of how different methods of GNSS TEC approximation affect the model’s accuracy.

Meshfree algorithms for analysis and computational modeling of multidimensional hyperbolic wave models

PurposeThis article offered two meshfree algorithms, namely the local radial basis functions-finite difference (LRBF-FD) approximation and local radial basis functions-differential quadrature method (LRBF-DQM) to simulate the multidimensional hyperbolic wave models and work is an extension of Jiwari (2015).Design/methodology/approachIn the evolvement of the first algorithm, the time derivative is discretized by the forward FD scheme and the Crank-Nicolson scheme is used for the rest of the terms. After that, the LRBF-FD approximation is used for spatial discretization and quasi-linearization process for linearization of the problem. Finally, the obtained linear system is solved by the LU decomposition method. In the development of the second algorithm, semi-discretization in space is done via LRBF-DQM and then an explicit RK4 is used for fully discretization in time.FindingsFor simulation purposes, some 1D and 2D wave models are pondered to instigate the chastity and competence of the developed algorithms.Originality/valueThe developed algorithms are novel for the multidimensional hyperbolic wave models. Also, the stability analysis of the second algorithm is a new work for these types of model.

Improving the quality of preschool education management based on the ordinary differential error approximation method

Abstract Improving the quality of preschool education management is the basic guarantee of national talent training. In this paper, the data envelopment analysis of resource allocation efficiency of preschool education is conducted by analyzing the characteristics of preschool education and the allocation process of educational resources. Based on the original input and original output, the productivity of preschool education is analyzed by combining different environmental variables in the CCR model. The asymptotic approximation method was used to improve the fit of the initial data and to speed up the convergence of the fit by weighting. Based on the study of preschool education in County M, kindergartens with resource allocation at a high level ( 0.8≤ θ <1.0) or a medium level ( 0.6≤ θ < 0.8) remained largely unchanged. The mean values of the combined technical efficiency of each kindergarten were 0.657, 0.702, and 0.759, respectively, and more than half of them were at a higher level. The data analysis based on the error approximation method can effectively avoid the problem of data overfitting and thus extract the key factors of preschool education management.

Damage identification in steel plate structures considering bending and torsion moments using estimated seismic responses

This paper proposes a model-free damage identification method and suggests an effective index for damage diagnosis in steel plate-like structures considering both the torsion and bending moments using only acceleration data, with no need for geometric characteristics. The basis for damage detection is the difference between healthy and damaged structural responses to different ground accelerations. In doing so, two Peak Ground Acceleration (PGA) scales, 0.05 (g), and 0.5 (g), labeled for the Loma Prieta ground motion, are considered. Four-node meshes are applied to reflect the torsion moments in the steel plate behavior. A multi-objective function, Fisher Information Matrix, is utilized, and four sensor locations are optimized to identify the locations most sensitive to damage. AutoRegressive Moving Average with exogenous input (ARMAX) system identification method is employed to obtain the structural displacement from the acceleration data and, subsequently, the displacement state equations of the nodes. With the aid of the structural displacement and its derivatives obtained through the central finite difference approximation method, damage to the plate structure is detected at PGA = 0.05 (g) and PGA = 0.50 (g) considering thickness changes based on the peak values manifested in the proposed damage detection index. Furthermore, the recorded structural acceleration responses at the desired scales of PGAs are fed into the 2D wavelet transform to detect damage imposed on the plate structure using the diagonal component of the wavelet transform. The results confirm that the proposed nonmodel-based approach successfully detects the single and multiple damage scenarios and can be used for a quick and judicious assessment of the steel plate behavior immediately after damage.

Analysis of the framework for realizing core literacy in teaching physical education and health curriculum based on the ordinary differential error approximation method

Abstract The key to implementing the fundamental task of establishing moral education is how the abstract and broad physical education and health curriculum core literacy is translated into concrete and visible physical education and health subject curriculum and teaching practice. This paper firstly explains the framework for the realization of core literacy in PE and health curriculum in teaching and learning, specifically from three perspectives: context, health behavior, and physical virtue. Secondly, the abstract core literacy of physical education and health curriculum is transformed into a side problem of ordinary differential equation with unknown parameter initial values by using the method of ordinary differential error approximation, which can obtain the parametric expressions of the approximation of function values at the nodes and realize the concretization. The experimental results show that there is a significant difference between the core literacy of physical education and health curriculum based on the ordinary differential error approximation method before and after the application of the core literacy in teaching, in which the comprehensive improvement of motor ability is 23.7%, the comprehensive improvement of health behavior is 19.2%, and the comprehensive improvement of physical morality is 26.6%. This paper successfully visualized the core literacy issues of physical education and health curriculum based on the ordinary differential error approximation method, which provided a reference for physical education teaching, improved the professional competence and quality of school physical education teachers, and formed the characteristics of school physical education and health curriculum.

Data Offloading Enabled by Heterogeneous UAVs for IoT Applications Under Uncertain Environments

With the continuous expansion of the Internet of Things (IoT) application scope, there are growing IoT scenarios that lack the coverage of wireless communication networks have the demand for data offloading. Efficient data transmission has been the main concern of these applications. Thus, data offloading within the limited communication environment has become a hotspot in both industry and academia. Since unmanned aerial vehicles (UAVs) can move across regions to make up for the communication gap caused by the loss of wireless communication networks, a lot of in-depth studies on UAV-enabled data offloading have been conducted. Nevertheless, few studies to date consider the uncertain user status and the heterogeneous UAV capabilities, which is however more practical and needs more attention. In this article, we propose an innovative framework to dynamically estimate user status information and determine the UAV scheduling strategy. On this basis, the heterogeneous UAV-enabled data offloading is modeled as a constrained multiobjective optimization problem, whose purpose is to lower the user data queue length while extending the working time of the UAV. Moreover, a differential evolution-based dynamic objective approximation method—RUDDER is proposed to solve the constrained multiobjective optimization problem. Through rigorous mathematical proof, we prove that RUDDER can consistently guide the population to approach the optimization solution with polynomial-level time complexity. To verify the effectiveness of the proposed RUDDER, extensive experiments are conducted to compare it with five comparison algorithms. The experimental results demonstrate the superiority of the RUDDER in terms of energy saving, time efficiency, and adaptability.

RESULTS OF INFLAMMATORY AND IMMUNOLOGICAL PARAMETERS OF THE ORAL CAVITY AFTER CYSTECTOMY WITH DIFFERENT METHODS OF CONNECTING THE EDGES OF THE OPERATED AREA.

The aim: To analyze the results of inflammatory and immunological parameters of the oral cavity after cystectomy with different methods of connecting the edges of the operated area. Materials and methods: The research was conducted in 87 patients who sought surgical treatment of odontogenic cysts of the jaws. Patients were divided into groups depending on the method of wound closure after surgery. We analyzed the results of laboratory tests (leukocytes, ESR, IL-1β, IL-6, TNF-α, IL-8, NO synthase, MMP-9). Results: The analysis of the effectiveness of using different methods of approximation of the oral mucosa wound on inflammatory and immunological pa¬rameters showed that when using welding of the edges of the operated area using the EKVZ-300 "Patonmed" apparatus, we get positive results of laboratory indicators of inflammatory markers faster (leukocyte counts are normal on day 30, ESR, IL-1β - on the 14th day, TNF-α - on the 7th day, IL-6, IL-8, NO synthase, MMP-9 - on the 30th day), that is, healing in such patients is faster than in patients with wound closure with classic suture material or laser. Conclusions: When comparing the methods of approximation of postoperative oral mucosa wounds by different methods, based on the results of these inflammatory and immunological parameters, the best results were obtained when using electric welding of tissues. Further research and use of the proposed method will facilitate and shorten the rehabilitation period of patients after surgery.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

Three-dimensional contact formulation for assessment of dynamic interaction of pantograph and overhead conductor rail system

Overhead conductor rail (OCR) is an important power supply device in tunnel railway lines. The penalty method is widely adopted to model the dynamic interaction behaviour of the pantograph-OCR system (POCRs), while it is difficult to determine a proper contact stiffness value. This paper proposes a three-dimensional contact formulation for the POCRs based on the Lagrange multiplier method. Accurate contact forces can be obtained easily by this contact formulation without the trouble of determining the contact stiffness. Contact force is defined in the global coordinate system as a three-dimensional vector. Coulomb law of friction is adopted in this paper to describe the tangential contact force, and the train speed is included in the contact formulation. The central finite difference approximation method is utilised, and an iteration procedure is elaborated to calculate the accurate contact force vector. The proposed contact formulation is verified by comparing simulation results with in-line measured data. The maximum statistical relative error of the contact force is only 2.63%. The advantages of the proposed contact formulation are also demonstrated by comparing it with the penalty method.

A computational point of view on teaching derivatives

The strong connection between mathematics and informatics highlights the important role of scientific computing in many applications. Unfortunately, mathematics is traditionally taught without investigating possible connections between abstract problem solving and the use of algorithms capable of being implemented on a computer. Since mathematical theory and computing practice are taught separately, many students fail to appreciate the utility of mathematics. In this paper, we briefly explain how a typical lecture on obtaining derivatives in differential calculus can benefit from examples implemented in high-level languages like MATLAB, Python or R. Such examples can help to guide the students to a better understanding of the theoretical concepts and limits of finite precision floating-point arithmetic. We argue that, while the historical findings of Leibniz and Newton are good starting points for introducing finite difference approximation methods, this does not preclude new approaches to numerically computing derivatives using Infinite Computer Arithmetic.

A second-order L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients

Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α∈(1,2) is proposed. This method is different from the known second-order methods, which is called the L2-1σ Crank-Nicolson method in this paper. Using the L2-1σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L2-norm and H1-norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α∈(1,2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L2-1σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes.

Study of cylindrical and spherical dust acoustic solitons and quasiperiodic structures in a quantum dusty plasma

The solitonic and quasiperiodic structures of dust acoustic (DA) waves are investigated in a three components quantum dusty plasma composed of mobile negative dust grains, ions, and inertialess electrons. The reductive perturbation method is employed to derive A deformed Korteweg–de Vries (dKdV) equation in planar and nonplanar geometries, and its numerical solutions are obtained using the two level finite difference approximation method. The influence of geometries on DA solitons is discussed. It is observed that in nonplanar geometries, DA solitons travel at different speeds in comparison to one-dimensional planar ones. Furthermore, in the planar geometry, the bifurcation of DA traveling waves has been analyzed on the framework of the dKdV equation. By adding an external periodic force to the derived dKdV equation, the quasiperiodic behaviors of DA waves are presented.

Closed-Form Evaluation of Mixed Potential Shielded Layered Media Green’s Functions With Spectral Differential Equation Approximation Method

A novel, robust, and computationally efficient approach is proposed for the evaluation of the Michalski–Zheng’s mixed-potential Green’s functions of general shielded layered media. A high-order variant of the spectral differential equation approximation method (SDEAM) is used to cast the spectra of Green’s functions’ components into pole-residue forms. The latter allows for closed-form evaluation of the Sommerfeld integrals (SIs) defining spatial components of the mixed-potential Green’s functions required by Method-of-Moment (MoM) discretization of different types of integral equations. The method produces Green’s functions for all elevations of interest at a fixed cost, making it substantially faster than popular alternative approaches. Numerical results validate the developed high-order SDEAM scheme and demonstrate its extended range of validity compared to the discrete complex image method (DCIM).