Fourth Approximation Research Articles

Ion mobility calculations of flexible all-atom systems at arbitrary fields using two-temperature theory.

Ion mobility spectrometry (IMS) separates and analyzes ions based on their mobility in a gas under an electric field. When the field is increased, the mobility varies in a complex way that depends on the relative velocity between gas and ion, their electrostatic potential interactions, and the effects from direct impingement. Recently, the two-temperature theory, primarily developed for monoatomic ions in monoatomic gases, has been extended to study mobilities at arbitrary fields using polyatomic ions in polyatomic gases, with some success. However, this extension poses challenges, such as inelastic collisions between gas and ion and structural modifications of ions as they heat up. These challenges become significant when working with diatomic gases and flexible molecules. In a previous study, experimental mobilities of tetraalkylammonium salts were obtained using a FAIMS instrument, showing satisfactory agreement with numerical two-temperature theory predictions. However, deviations occurred at fields greater than 100 Td. To address this issue, this paper introduces a modified high-field calculation method that accounts for the structural changes in ions due to field heating. The study focuses on tetraheptylammonium (THA+), tetradecylammonium (TDA+), and tetradodecylammonium (TDDA+) salts. Molecular structures were generated at various temperatures using MM2 forcefield. The mobility was calculated using IMoS 1.13 with two-temperature trajectory method calculations up to the fourth approximation. Multiple effective temperatures were considered, and a linear weighing system was used to create mobility vs. reduced field strength plots. The results suggest that the structural enlargement due to ion heating plays a significant role in mobility at high fields, aligning better with experimental data. FAIMS' dispersion plots also show improved agreement with experimental results. However, the contribution of inelastic collisions and energy transfer to rotational degrees of freedom in gas molecules remains a complex and challenging aspect.

AVERAGED CIRCULAR SPATIAL RESTRICTED THREE-BODY PROBLEM: INTERNAL CASE, NEW RESULTS

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).

Various Methods of Realization for Fractional-Order Elements

Fractional-order circuits are finding applications in control systems, signal processing, and other allied fields. As a result, realizing fractional-order elements or fractance devices is an essential research topic. A unique fractance device is realized in this study using continued fraction expansion (CFE) and partial fraction expansion (PFE) formulas. Continued fraction expansion is used to calculate the rational approximation. For simulation, the fourth order rational approximation for fractional order, =-1/2, -1/3, -1/4 is used. All simulations are carried out with the help of the MATLAB and TINA software packages. The theoretical and simulation results are in agreement.

Optimal control and global stability of the SEIQRS epidemic model

Medical treatment, vaccination, and quarantine are the most efficacious controls in preventing the spread of contagious epidemics such as COVID-19. In this paper, we demonstrate the global stability of the endemic and disease-free equilibrium by using the Lyapunov function. Moreover, we apply the three measures to minimize the density of infected people and also reduce the cost of controls. Furthermore, we use the Pontryagin Minimum Principle in order to characterize the optimal controls. Finally, we execute some numerical simulations to approve and verify our theoretical results using the fourth order Runge-Kutta approximation through Matlab.

Interpolation of 3D data streams with C2 PH quintic splines

The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic splines is proposed. To facilitate real-time computations, an efficient local data stream interpolation algorithm is introduced to successively construct each spline segment as a quintic PH biarc interpolating second- and first-order Hermite data at the initial and final end-point, respectively. A C2 smooth connection between successive spline segments is obtained by taking the locally required second-order derivative information from the previous segment. Consequently, the data stream spline interpolant is globally C2 continuous and can be constructed for arbitrary C1 Hermite data configurations. A simple and effective selection of the free parameters that arise in the local interpolation problem is proposed. The developed theoretical analysis proves its fourth approximation order while a selection of numerical examples confirms the same accuracy for the spline extension of the scheme. In addition, the performances of the method are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.

Predicting ion mobility as a function of the electric field for small ions in light gases

High resolution mobility devices such as Field Asymmetric Waveform Ion Mobility Spectrometry (FAIMS) and Differential Mobility spectrometers (DMS) use strong electric fields to gas concentration ratios, E/N, to separate ions in the gas phase. While extremely successful, their empirical results show a non-linear, ion-dependent relation between mobility K and E/N that is difficult to characterize. The one-temperature theory Mason-Schamp equation, which is the most widely used ion mobility equation, unfortunately, cannot capture this behavior. When the two-temperature theory is used, it can be shown that the K−E/N behavior can be followed quite closely numerically by equating the effect of increasing the field to an increase in the ion temperature. This is attempted here for small ions in a Helium gas environment showing good agreement over the whole field range. To improve the numerical characterization, the Lennard-Jones (L-J) potentials may be optimized. This is attempted for Carbon, Hydrogen, Oxygen and Nitrogen at different degrees of theory up to the fourth approximation, which is assumed to be exact. The optimization of L-J improves the accuracy yielding errors of about 3% on average. The fact that a constant set of L-J potentials work for the whole range of E/N and for several molecules, also suggests that inelastic collisions can be circumvented in calculations for He. The peculiar K−E/N hump behaviors are studied, and whether mobility increases or decreases with E/N is shown to derive from a competition between relative kinetic energy and the interaction potentials.

Application of variational iteration method to find the solution of coupled system of first order differential equations

Present work is devoted to the application of variational iteration method to obtain the solutions of coupled system of first order differential equations. Numerical examples are taken to test the efficiency of this method. We have shown that the successive approximations of each problem are converging to their exact solution. Further we have shown graphically the third, fourth, fifth and sixth approximation values and exact values.

A system reliability estimation method by the fourth moment saddle point approximation and copula functions

AbstractDespite many advances in the field of computational system reliability analysis, estimating the joint probability distribution of correlated non‐normal state variables on the basis of incomplete statistical data brings great challenges for engineers. To avoid multidimensional integration, system reliability estimation usually requires the calculation of marginal failure probability and joint failure probability. The current article proposed an integrated approach for estimating system reliability on the basis of the high moment method, saddle point approximation, and copulas. First, the statistic moment estimation based on the stochastic perturbation theory is presented. Thereafter, by constructing CGF (concise cumulant generating function) for the state variable with its first four statistical moments, a fourth moment saddle point approximation method is established for the component reliability estimation. Second, the copula theory is briefly introduced and extensively utilized two‐dimensional copulas are presented. The best fit copula for estimating the probability of system failure is selected according to the AIC (Akaike Information Criterion). Finally, the derived method is applied to three numerical examples for the sake of a comprehensive validation.

Parallel FFT algorithms for high-order approximations on three-dimensional compact stencils

The recent development of multicore technologies on modern desktop computers makes parallelization of the proposed numerical approaches a priority in algorithmic research. The main performance improvement of personal computers in the upcoming years will be made based on the increasing number of cores on modern CPUs. This shifts the focus of algorithmic research from the development of sequential numerical methods to parallel methodology. This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations (Turkel et al., 2013) with the problem coefficient depending on only one of the coordinate directions. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. Similar direct solvers for the lower-order approximations of the two and three-dimensional Helmholtz equation were considered in several previous publications by the authors and other researchers (see, e.g. Gryazin et al. (2000); Gryazin (2014); Elman and O’Leary (1998); Elman and O’Leary (1999); Toivanen and Wolfmayr (2020)). The authors also consider a generalization of the presented algorithm to the solution of a wide class of linear systems obtained from approximation on the compact 27-point three-dimensional stencils on the rectangular grids with similar requirements on the stencil coefficients. The general restrictions on the coefficients in the considered class of compact schemes are developed and presented. This class includes the second, fourth and sixth-order compact approximation schemes for the three-dimensional Helmholtz equation considered in this paper and our previous publications (Gryazin et al., 2000; Gryazin, 2014; Gryazin, 2014). As an example of the diversity of applications of the developed general method, the direct parallel implementation of a compact fourth-order approximation scheme for a convection–diffusion equation is considered. Another goal of this paper is to investigate the scalability of the proposed technique in the case of a large linear system using different parallel programming extensions. The results of the implementation of this method in OpenMP, MPI and hybrid programming environments on the multicore computers and multiple node clusters are presented and discussed. The results demonstrate the high efficiency of the proposed direct solvers for many important applications on the structured grid with the corresponding 27-diagonal matrices of sizes up to 1011 by 1011.

A new efficient fourth order collocation scheme for solving Burgers’ equation

In present work a new fourth order modified cubic B-spline (mCB) based upon collocation technique (mCBCT4) has been developed to evaluate new numeric results of the nonlinear Burgers’ equation, appear in rigorous real-world physical phenomena like - sound & shock waves in viscous medium, waves in fluid filled viscous elastic tubes, magneto-hydrodynamic-waves in medium with finite electrical-conductivity, in modeling of turbulent fluid, and in continuous stochastic processes. At first, the Burgers’ equation is remodeled into a set of 1st order ordinary differential equations (ODE), in which fourth order accurate approximation of the unknown functions, and its spatial derivatives obtained via mCBCT4. In this way a set of first-order ODE is obtained, which we solve via SSP-RK(ℓ+1,ℓ) scheme (ℓ=3,4). The accuracy, efficiency and effectiveness of the developed technique mCBCT4 is demonstrated in terms of six different test examples of nonlinear Burgers’ equation by computing the error norms: L2 and L∞ errors. The proposed mCBCT4 scheme is also tested for nonlinear Burger’s equation with very small kinematic viscosities. The proposed mCBCT4 is shown unconditionally stable scheme. The numerical findings demonstrate that the developed mCBCT4 performs better than some recently developed good techniques and enables to produces comparably more accurate solutions than some recently developed reliable techniques.

Application of variational iteration method to find the solution of coupled system of first order differential equations

Present work is devoted to the application of variational iteration method to obtain the solutions of coupled system of first order differential equations. Numerical examples are taken to test the efficiency of this method. We have shown that the successive approximations of each problem are converging to their exact solution. Further we have shown graphically the third, fourth, fifth and sixth approximation values and exact values.

Direct comparison of SVM and LR classifier for SEMG signal classification using TFD features

The performance of any bio-medical signal based robotic device depends upon the classification accuracy of the system. In this context, surface electromyography (SEMG) signals acquisition was done from 15 right hand dominated healthy subjects. After the data acquisition, some pre-processing steps like smoothing, rectification and normalization was performed followed by Daubechies 4 (db 4 ) discrete wavelet transform (DWT) de-noising. DWT was also utilized for SEMG signals decomposition up to four levels. DWT decomposed the SEMG signal in approximation and detailed coefficients and finally fourth level approximation and detailed coefficients were exploited for time-frequency domain features extraction. The performance of individual time-frequency domain features was critically compared with support vector machine (SVM) classifier and linear regression (LR) model. The results indicated that the performance of SVM classifier was found better than the LR classifier. SVM classifier achieved the 95.8% classification accuracy with all combined time-frequency domain features in the form of the feature vector.

Stiffeners Mechanical Effect Analysis by Numerical Coupling

Given any structure, we seek to find the solution of mechanical problem as precisely and efficiently as possible. Within this condition, the BEM is robust and promising development, standing out in the analysis of cases with occurrence of large stress gradients, as in problems of fracture mechanics. Moreover, in BEM the modeling of infinite means is performed naturally, without the use of approximations. For methods involving domain integration, such as FEM, this is not possible, as models with high number of elements are usually adopted and their ends are considered flexible supports. This paper deals with the development of numerical models based on the BEM for mechanical analysis of stiffened domains. In the modeling of hardeners, immersed in a medium defined by the BEM, we tried to use both the FEM, already present in the literature, and the BEM 1D, being a new formulation. The objective was to perform the coupling of BEM with FEM and BEM 1D for elements of any degree of approximation, evaluating both isotropic and anisotropic medium. In addition, a complementary objective was to verify the effects of the adoption of different discretization and approximation degrees on the stiffeners. However, the coupling with the BEM 1D leaded to more stable results and better approximations. It was observed that the FEM results were instable for many results, mainly in the quadratic approximations. When the approximation degree rises, the methods tend to converge to equivalent results, becoming very close in fourth degree approximation. Lastly, it was observed stress concentration in the stiffeners ends. In these regions, the discretization and the approximation degree have large influence to the numerical response.

A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration

In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly by the MIB scheme to generate fictitious values near the interface. For the exterior part, two layers of zero-padding solutions are introduced beyond the original rectangular domain so that the FFT inversion becomes feasible. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed via the MIB scheme to generate fictitious values near boundaries. Based on fictitious values at both interfaces and boundaries, the augmented MIB method reconstructs Cartesian derivative jumps as auxiliary variables. Then, by treating such variables as unknowns, an enlarged linear system is obtained. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. Therefore, the FFT-based augmented MIB not only achieves a fourth order of accuracy in dealing with interfaces and boundaries, but also produces an overall complexity of O(n2log⁡n) for a n×n uniform grid. Moreover, the augmented MIB scheme can provide fourth order accurate approximations to solution gradients and fluxes.

Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients

In this work, a new fourth order compact approximation is derived for Riemann–Liouville space fractional derivatives. Modified wave numbers are obtained for various approximations of fractional derivatives. Considering these modified wave numbers, Fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations. A novel fourth order compact scheme is developed for space fractional advection–diffusion reaction equations with variable coefficients in one and two dimensions. In the proposed compact scheme, the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations. This splitting of second derivative approximations allows us to obtain a similar system of linear equations as in Zhao and Tian (2017) which presents only second order accurate finite difference method for advection–diffusion reaction equations in one-dimension. The stability of the proposed compact scheme is demonstrated numerically. Furthermore, the proposed compact scheme is employed to space fractional Black–Scholes equation for pricing European options as an application of fractional derivatives in mathematical finance. Numerical illustrations are presented to validate the theoretical claims.

The approach to optimization of finite-difference schemes for the advective stage of finite-difference-based lattice Boltzmann method

The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

PROPAGATION OF OPTICAL FEMTOSECOND PULSES THROUGH QUARTZ AND GLASS ВК-7

In the approximation of dispersion theory to the fourth order inclusive, a theoretical study of the propagation of optical femtosecond pulses of a Gaussian profile and a profile proportional to the hyperbolic secant through quartz and BK-7 glass was carried out. Based on the solutions of the equations for the propagation of pulses of various initial durations through optical materials, dependencies “material length − pulse duration” are obtained. It is shown that, compared with the second approximation of dispersion theory, the fourth approximation gives a slightly lengthy pulse duration. Peculiarities of various time pulse profiles on the pulses duration are also shown. Analytical consideration is confirmed by numerical calculations.

Investigation of oxygen vacancy migration energy in yttrium doped cerium

Using the statistical moment method, the analytic expression of vacancy migration energy in yttria doped cerium (YDC) is derived within the fourth order moment approximation. This expression depends strongly on the interaction potentials between the diffusing oxygen ion and the surrounding cations. Our results show the predominant direction of oxygen vacancy migration and the influence of cation barriers on oxygen vacancy diffusion. The dependences of the migration energy on temperature and dopant concentration are evaluated in detail. Calculated results are compared with other theoretical results.

Анализ устойчивости неявных конечно-разностных решеточных схем Больцмана с направленными разностями

Статья посвящена анализу устойчивости неявных конечно-разностных схем для системы кинетических уравнений, применяемых для проведения гидродинамических расчетов в рамках метода решеточных уравнений Больцмана. Представлены семейства двухслойных и трехслойных схем с направленными разностями первого-четвертого порядков аппроксимации по пространственным переменным. Важной особенностью схем является то, что конвективные слагаемые аппроксимируются одной конечной разностью. Показано, что в выражении для аппроксимационной вязкости схем высоких порядков отсутствуют фиктивные слагаемые, что позволяет применять их во всем диапазоне значений времени релаксации. Анализ устойчивости проводится по линейному приближению с использованием метода Неймана. Получены приближенные условия устойчивости в виде неравенств на значения параметра Куранта. При расчетах показано, что площади областей устойчивости в пространстве параметров у двухслойных схем больше, чем у трехслойных. Исследованные схемы могут применяться при расчетах как непосредственно, так и в методах типа предиктор-корректор. The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.