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The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods. We assume that the equation has a hypothetical soliton solutions. By reorganizing the resulting equations, we obtain a system of equations. Using Maple software, we get unknown coefficients in the system and writing them in the original equation, we obtain new solition solutions of the equation. The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright, kink type, bright periodic and dark solutions. We provided 3-D figures to illustrate the solutions. Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number ( R 0 ) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R 0 = 1.4575 . For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Matrix calculations are often required for the analysis of any big-data cloud computing system. It is quite common to process big-data associated matrices possessing the sparsity and low-rank properties. In order to efficiently deal with big-data matrices, we propose a new divide-and-iterate framework, which can be invoked to solve an enormously large linear system of equations by taking advantage of factored matrices. The Kaczmarz algorithm (KA) is utilized here to design the parallel iterative algorithms which are capable of solving a large system of equations by iteratively updating the solution through the reduction into the factorized subsystems in parallel. The convergences of our proposed new iterative algorithms are justified by the rigorous proofs. Besides, the time- and memory-complexities are studied to demonstrate the resource efficiency of the proposed algorithms. Numerical experiments are also presented to illustrate the effectiveness of this proposed new framework.

In this work, an efficient novel technique, namely, the q -homotopy analysis transform method ( q -HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrodinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviours which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q − HATM, two cases each of two time-fractional problems are considered. One important feature of the q − HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behaviour of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q − HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.

The solution of a system of simultaneous linear equations is a fundamental problem in numerical linear algebra and is a basic ingredient of many scientific simulations. Examples are scientific or engineering problems modeled by ordinary or partial differential equations. The numerical solution is often based on discretization methods leading to a system of linear equations.

This chapter presents Gaussian elimination and Gauss-Jordan row reduction that are important techniques for solving linear systems. Attempts to solve systems of linear equations inspired much of the development of linear algebra. The study of linear systems leads to the examination of further properties of matrices including row equivalence, rank, and the row space of a matrix. A linear equation is an equation involving one or more variables in which only the operations of multiplication by real numbers and summing of terms are allowed. When several linear equations involving the same variables are considered together, a system of linear equations is obtained. Many methods are available for finding the complete solution set for a given linear system in which the Gaussian elimination is most important that begins with the augmented matrix for the given system and examines each column in turn from left to right. There are three operations that we are allowed to use on the augmented matrix in the Gaussian elimination method: (1) multiplying a row by a nonzero scalar, (2) adding a scalar multiple of one row to another row, and (3) switching the positions of two rows in the matrix.

This chapter discusses systems of ordinary differential equations. It presents a review of matrix algebra and calculus. It defines the characteristic equation and characteristic polynomial and shows how to calculate characteristic polynomial and eigenvalues of matrices through examples. Mathematica can compute both the characteristic polynomial and eigenvalues directly with the commands CharacteristicPolynomial and Eigenvalues. The chapter illustrates the homogeneous system of first-order linear differential equations. It provides solution of homogeneous linear systems with constant coefficients through some examples. It also discusses some theorems and terminology used in establishing the fundamentals of solving systems of differential equations. To apply variation of parameters, one first calculates a fundamental matrix for the associated homogeneous equation. In many cases, Laplace transforms can be used to solve a system of linear differential equations. The chapter illustrates the techniques needed to accomplish this with Mathematica through examples. It further explains nonlinear systems, linearization, and equilibrium points.

This work presents a novel scheme to couple the Discrete Element Method (DEM) and the Boundary Element Method (BEM) for the multi-scale modelling in the time domain. The DEM can model discontinuous material at micro scale very well, but it cannot represent infinite domains. Hence, coupling with the BEM is proposed. A formulation employing the DEM and BEM in different subdomains of the same body is presented. There is no overlap between the sub-domains, and the system of equations is derived based on strong equilibrium and compatibility conditions at the interface. The proposed coupling scheme is based on monolithic time integration. The conducted numerical experiments of one-dimensional wave propagation show excellent agreement with the analytical solution. Some spurious wave reflections are observed at the interface, but their effect is quantified and deemed not critical for infinite domains, which are of main interest. Even though the applications for one-dimensional wave propagation are of limited practical engineering interest, this work represents a significant theoretical breakthrough. This paper establishes a reference for future coupling schemes for two- and three-dimensional multi-scale analysis.

Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the Patlak–Keller–Segel model. This macroscopic system of equations is derived from the microscopic model when limiting behaviour is studied. However, on taking the limit and passing from the microscopic equations to the macroscopic equations, fluctuations are neglected. Perturbing the system by a Gaussian random field restitutes the inherent randomness of the system. This gives us the motivation to study the classical Patlak–Keller–Segel system perturbed by random processes. We study a stochastic version of the classical Patlak–Keller–Segel system under homogeneous Neumann boundary conditions on an interval O = [ 0 , 1 ] . In particular, let W 1 , W 2 be two time-homogeneous spatial Wiener processes over a filtered probability space A . Let u and v denote the cell density and concentration of the chemical signal. We investigate the coupled system { d u − ( r u Δ u − χ div ( u ∇ v ) ) d t = u ∘ d W 1 , d v − ( r v Δ v − α v ) d t = β u d t + v ∘ d W 2 , with initial conditions ( u ( 0 ) , v ( 0 ) ) = ( u 0 , v 0 ) . The positive terms r u and r v are the diffusivity of the cells and chemoattractant, respectively, the positive value χ is the chemotactic sensitivity, α ≥ 0 is the so-called damping constant. The noise is interpreted in the Stratonovich sense. Given T > 0 , we will prove the existence of a martingale solution on [ 0 , T ] .

The randomized extended Gauss-Seidel method is a popular representative among the iterative algorithm due to its simplicity for solving the inconsistent and consistent systems of linear equations, which builds the connection between the randomized Kaczmarz and Gauss-Seidel methods. In this work we develop a general version of the randomized extended Gauss-Seidel method, as well as some new iterative schemes. We prove that our algorithm can exponentially converge in expectation to the solutions of the consistent or inconsistent linear systems under two different sampling strategies. Numerical examples show that the proposed algorithm is feasible and effective, where the block method performs significantly better than the corresponding original form.

Time-series interferograms acquired on a deep pit with a Ground-Based Synthetic Aperture Radar (GB-SAR) system showed that the atmospheric phase (AP) could be complexly space variant due to rapid changes of the weather conditions and steep topography. Conventional compensation methods that simulate the AP with typical parametrical models are no longer applicable. Based on the theoretical path integral model of the AP, a grid partition (GP) method is proposed. By dividing one interferogram into a certain number of small grids, the refraction variation inside each grid is assumed to be a constant. A system of linear equations is first built based on sufficient permanent scatterers (PSs). Then, bounds and inequality constraints are set to limit the refraction variation of each grid. A constrained linear least-square problem is solved with two-step process to estimate and compensate the AP. To fully validate the feasibility of the GP method, simulated phase interferograms based on four conventional AP models and with the consideration of deformation areas and noise phase are first processed. Then, four experimental interferograms with different types of AP components are processed and made comparisons with the conventional parametrical methods. The quantitative comparisons of the simulated and experimental data sets both proved that the GP method can well reduce the AP errors.

The Gauss Jordan method is used in this study to equalize chemical reactions using a system of linear equations. One of the most common topics in Chemistry is balancing chemical reaction equations. Students typically struggle with balancing the chemical equation and find it difficult to understand; sometimes teachers struggle as well to balance chemical equations. The results of the equation balancing comply with the law on the conservation of matter and confirm that the existing methods for balance of chemical equations do not contradict each other. To solve the mathematical problem, the Gauss-Jordan method was used. Any chemical reaction can be handled using this method by certain reactants and products. we can add several other features to the python application which can check whether the chemical equation is already balanced or not, we can also categorise the type of the chemical reaction.

The present article deals with the reflection and transmission of waves at an interface of piezoelectric (ALN & PZT-5T) half-spaces with microstructures. Unlike the classical piezoelectric material, in the considered media, five coupled waves (2 bulk and 3 surface) [namely the Quasi-longitudinal wave (QP), Quasi-transverse wave (QSV), Electric-acoustic wave (EA), P-type surface wave (SP) and S-type surface wave (SS)] are generated in response to an oblique incident Quasi plane wave. The main objective of this study is to investigate the influence of the characteristic length of microstructure, the inertial characteristic length and flexoelectric coefficients on the reflection and transmission coefficients. For this purpose, the linear algebraic equation, continuity conditions at the common interface are contracted. The linear system of equations comprising the reflection and transmission coefficients is derived, which is solved by using the Cramer's rule. The consequence of this electromechanical phenomenon may be advantageous in certain engineering applications that involve smart nano-composites.

By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.

The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: point, line and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.

This paper studies the potential of using the successive over-relaxation iteration method with polynomial preconditioner (P(m)-SOR) to solve variably saturated flow problems described by the linearized Richards’ equation. The finite difference method is employed to numerically discretize and produce a system of linear equations. Generally, the traditional Picard method needs to re-evaluate the iterative matrix in each iteration, so it is time-consuming. And under unfavorable conditions such as infiltration into extremely dry soil, the Picard method suffers from numerical non-convergence. For linear iterative methods, the traditional Gauss-Seidel iteration method (GS) has a slow convergence rate, and it is difficult to determine the optimum value of the relaxation factor w in the successive over-relaxation iteration method (SOR). Thus, the approximate optimum value of w is obtained based on the minimum spectral radius of the iterative matrix, and the P(m)-SOR method is extended to model underground water flow in unsaturated soils. The improved method is verified using three test examples. Compared with conventional Picard iteration, GS and SOR methods, numerical results demonstrate that the P(m)-SOR has faster convergence rate, less computation cost, and good error stability. Besides, the results reveal that the convergence rate of the P(m)-SOR method is positively correlated with the parameter m. This method can serve as a reference for numerical simulation of unsaturated flow.

Purpose The aim of this paper is to propose the model for analyzing the electromagnetic performances of permanent magnet vernier machines (PMVMs) under healthy and faulty conditions. Design/methodology/approach The model uses interconnected reluctance network formed based on the geometrical approximations to predict magnetic performances of the machine. The network consists of several types of reluctances for modeling different parts of machine. Applying Kirchhoffs laws in the network and the machine windings, magnetic and electrical equations are obtained, respectively. To construct the model system of equations, the electrical equation is converted into algebraic form by using the trapezoidal technique. Moreover, the system of equations must be solved by Newton–Raphson method in each step-time because of considering the core saturation effect. Findings The proposed model is developed based on the modified magnetic equivalent circuit (MEC) method, in which the number of flux paths in different parts of the machine can be arbitrary selected. The saturation effect, skewed slots, the desired machine geometrical parameters and various winding arrangements are included in the proposed model; therefore, it can evaluate the time and space harmonics in modeling the PMVMs. Furthermore, a pattern for inter-turn fault detection is extracted from the stator current spectrum. Finally, 2 D-finite element method (FEM) and 3 D-FEM analysis are carried out to evaluate and verify the results of the proposed MEC model. Originality/value Generally, the element numbers have important role in modeling the machine and calculating its performance. Hence, the proposed MEC model’s capability to choose desired number of flux paths is advantage of this paper. Moreover, the developed MEC can be used for analyzing several electrical machines, including other types of vernier machines, with simple modification.

Linear programming (LP) has long proved its merit as the most flexible and most widely used technique for resource allocation problems in various fields. To solve an LP problem, we have traditionally considered crisp values for the parameters, which are unrealistic in real-world decision-making under uncertainty. The fuzzy set theory has been used to model the imprecise parameter values in LP problems to overcome this shortcoming, resulting in a fuzzy LP (FLP) problem. This paper proposes a new method for solving fuzzy variable linear programming (FVLP) problems in which the decision variables and resource vectors are fuzzy numbers. We show how to use the standard simplex algorithm to solve this problem by converting the fuzzy problem into a crisp one once a linear ranking function is chosen. The novelty of the proposed model resides in that it requires less effort on fuzzy computations as opposed to the existing fuzzy methods. Furthermore, to solve the FVLP problem using the existing methods, fuzzy arithmetic operations and the solution to fuzzy systems of equations are required. By contrast, only arithmetic operations of real numbers and the solution to crisp systems of equations are required to solve the same problem with the method proposed in this study. Finally, a transportation case study in the coal industry is presented to demonstrate the applicability of the proposed algorithm.

In this work it proposes a mathematical model for ion channels based on two concepts, the Hodgkin and Huxley's as well as the Law of Mass Action in addition, we consider the kinetics of channels as a dynamic process of Markov`s chain. With the previous premises, a system of differential equations is proposed that when it is solved, all properties of the macroscopic currents are determined. The activation, deactivation, inactivation, and recovery of the inactivation concepts remain as processes that are part of a chemical reaction. With this system of equations, all the experimental protocols used in electrophysiology to characterize macroscopic currents can be modeled. Another advantage is that the model allows, with the same system of equations, to determine the properties of voltage-dependent channels regardless of the type of ion that pass through in the channel.

We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.

In this paper, a method for computing the dynamics of a geared system excited by its static transmission error is proposed. The method is based on the iterative spectral method (ISM) and on the harmonic balance method (HBM). It is shown that the dynamic transmission error (DTE) can be obtained in the frequency domain by solving a linear system of equations, which in turn allows the computation of the modal and physical coordinates of the system.

A novel time-domain (TD) integral-equation (IE) technique for analyzing the pulsed electromagnetic (EM) scattering by a 2-D metasurface, represented by a thin highly contrasting screen with combined dielectric and magnetic properties, is presented. The problem under consideration is formulated with the aid of the EM reciprocity theorem of the time-convolution time. Subsequently, upon discretizing the space–time solution domain, the reciprocity relation is cast into two independent matrix forms whose elements are derived analytically using the classic Cagniard–DeHoop (CdH) technique. The thus obtained convolution-type systems of equations are solved for induced electric and magnetic currents through a stable step-by-step updating procedure. The presented numerical method is finally validated using alternative TD solutions.

International audience; Both compressible and incompressible porous medium models are used in the literature to describe the mechanical properties of living tissues. These two classes of models can be related using a stiff pressure law. In the incompressible limit, the compressible model generates a free boundary problem of Hele-Shaw type where incompressibility holds in the saturated phase. Here we consider the case with a nutrient. Then, a badly coupled system of equations describes the cell density number and the nutrient concentration. For that reason, the derivation of the free boundary (incompressible) limit was an open problem, in particular a difficulty is to establish the so-called complementarity relation which allows to recover the pressure using an elliptic equation. To establish the limit, we use two new ideas. The first idea, also used recently for related problems, is to extend the usual Aronson-Bénilan estimates in $L^\infty$ to an $L^2$ setting. The second idea is to derive a sharp uniform $L^4$ estimate on the pressure gradient, independently of space dimension.

The authors improve the numerical method of calculating multi-wave models to increase the speed and monotonize (reduce the oscillations of numerical calculations) the numerical solution of problems of multi-wave dynamics for lengthy systems such as space tethers tens of kilometers long; pipelines in air and in liquid; underwater towed systems; airlifts 5 to 10 km long for the extraction of minerals from the bottom of the oceans, etc. The method is based on the decomposition of the numerical algorithm by wave types and wave velocities. It is shown that due to quantization in calculating longitudinal and transverse waves, it is possible to achieve a further increase in the computation speed compared to the wave factorization algorithm and compared to solving the full system of equations, without reducing the range of sustainable calculation. The final increase in the productivity of the program code is at least 50 to 200% when performing calculations, depending on the required accuracy and options for the decomposition of multi-wave models. This modification of the wave factorization method is relevant in solving the problems of controlling a distributed system, operative analysis of transient motion modes, etc., where the pace of calculations is critically important. A comparative evaluation of the accuracy of the full algorithm, the of wave factorization method, and of the decomposition by wave types and wave velocities has been carried out. Comparative analysis of the calculated data showed the monotonicity of the numerical solution profile on the basis of factorized algorithms, their lower sensitivity to errors in the initial data. A finite-difference scheme factorized by perturbation directions and wave types with variable dispersion-diffusion properties is constructed.

This article presents a new approach to construct reduced-order models for large nonlinear circuits excited by periodical or quasi-periodical sources. The key feature in the constructed reduced model is that it enables, through solving a much smaller system, tracing the variations in the periodical steady-state response of the original circuit with regards to variations in selected design parameters. The proposed approach is based on the notion of using discrete empirical interpolation to project the nonlinear operator of the circuit equation onto a smaller subspace. In addition, the original system equations are projected onto the subspace spanned by the first few derivatives of the solution with respect to the selected design parameters. Computational savings are made possible through solving a reduced system of equations instead of the full system. Numerical examples are presented to validate the accuracy and efficiency of the proposed approach.

In this paper, we examine a one-dimensional system of equations for a discrete gas model (the Godunov–Sultangazin system). The Godunov–Sultangazin system is the Boltzmann kinetic equation for a model one-dimensional gas consisting of three groups of particles. In this model, the momentum is preserved whereas the energy is not. We prove the existence of a unique global solution to the Cauchy problem for a perturbation of the equilibrium state with periodic initial data. For the first time, we find the rate of stabilization to the equilibrium state (exponential stabilization).

This work calculates the reliability of protective systems of industrial facilities, such as nuclear, to analyze the case of equipment subject to aging, important in the extension of the qualified life of the facilities. By means of the method of supplementary variables, a system of partial and ordinary integral-differential equations was developed for the probabilities of a protective system of an aging channel. The system of equations was solved by finite differences. The method was validated by comparison with channel results with exponential failure times. The method of supplementary variables exhibits reasonable results for values of reliability attributes typical of industrial facilities.

The Henrici’s transformation is a method of extrapolation used for solve non-linear equations and systems of equations. By combining the optimal gradient method with a modified Henrici’s transformation, we have already developed a non-linear optimization method. This method is based on the resolution of a linear system at each iteration. In this paper, we improve this method by giving a new formulation that allows us to go from one iteration to the other recursively, which does not require the resolution of an linear system. The demonstration of this new formulation is established and numerical examples are given.

In this paper, an effective numerical technique based on Lucas and Fibonacci polynomials coupled with finite differences is developed for the solution of nonlinear reaction–diffusion Brusselator system. The system arises in modeling of chemical processes such as enzymatic reactions, plasma and laser physics in multiple coupling between modes and in the formation of ozone by atomic oxygen via a triple collision. The proposed scheme first converts the problem to discrete form and then with a collocation approach to a system of linear equations which is easily solvable. Performance of the method is checked by solving one- and two-dimensional test problems. Validation of the results is examined in terms of L ∞ , L 2 and relative error L R norms. In case of no exact solution, quality of computed solution is examined for small value of diffusion coefficient η . It is observed that when 1 - ξ + β > 0 , the solutions converges to equilibrium points ( β , ξ / β ) .

K-nearest neighbor (KNN) based weighted multi-class twin support vector machines (KWMTSVM) is a novel multi-class classification method. In this paper, we propose a novel least squares version of KWMTSVM called LS-KWMTSVM by replacing the inequality constraints with equality constraints and minimized the slack variables using squares of 2-norm instead of conventional 1-norm. This simple modification leads to a very fast algorithm with much better results. The modified primal problems in the proposed LS-KWMTSVM solves only two systems of linear equations whereas two quadratic programming problems (QPPs) need to solve in KWMTSVM. The proposed LS-KWMTSVM, same as KWMTSVM, employed the weight matrix in the objective function to exploit the local information of the training samples. To exploit the inter class information, we use weight vectors in the constraints of the proposed LS-KWMTSVM. If any component of vectors is zero then the corresponding constraint is redundant and thus we can avoid it. Elimination of redundant constraints and solving a system of linear equations instead of QPPs makes the proposed LS-KWMTSVM more robust and faster than KWMTSVM. The proposed LS-KWMTSVM, commensurate as the KWMTSVM, all the training data points into a “1-versus-1-versus-rest” structure, and thus our LS-KWMTSVM generate ternary output {-1,0,+1} which helps to deal with imbalance datasets. Numerical experiments on several UCI and KEEL imbalance datasets(with high imbalance ratio) clearly indicate that the proposed LS-KWMTSVM has better classification accuracy compared with other baseline methods but with remarkably less computational time. This work was supported by Science and Engineering Research Board (SERB) under Early Career Research Award Grant No. ECR/2017/000053 and Ramanujan fellowship Grant No. SB/S2/ RJN-001/2016. This work was also supported by Council of Scientific & Industrial Research (CSIR), New Delhi, INDIA under Extra Mural Research (EMR) Scheme Grant No. 22(0751)/17/ EMR-II.

Neural networks as well as the related neural dynamics have been widely exploited to conduct online computing operations for solving various problems in recent years. This article makes progress along this direction by proposing a saturation-allowed neural dynamics (SAND) model for solving the perturbed time-dependent system of linear equations with noise-tolerant capacity. Specifically, by elaborately constructing a new general framework enhanced by the error-integration information and nonlinear projection functions (PFs), a SAND model is proposed and investigated under various additive noises. In addition, theoretical analyses reveal that the proposed SAND model is of global convergence with zero theoretical error. Moreover, when the value of PF is strictly limited by bounds, i.e., a saturation function, the upper bound of the residual errors of the proposed SAND model is subject to the noises and the bounds of PF. Computer simulation results, as well as robot experiments, verify the superior property of the proposed SAND model for solving the perturbed time-dependent system of linear equations, compared with the state of the prior art.

On the basis of the general relativistic statistical and kinetic theory, a consistent closed cosmological model is formulated. It is based on a statistical system of scalar charged fermions interacting by means of classical and phantom scalar fields. Based on the study of the microscopic dynamics of scalar charged particles, within the framework of the Lagrangian as well as Hamiltonian formalism, a function of the dynamic mass of scalar charged particles was constructed and it was shown that for the consistency of the theory, it is necessary to remove the nonnegativity condition for this function. On the basis of the Lagrangian formalism, equations of gravitational and scalar fields with singular sources are formulated and microscopic conservation laws are obtained. Within the framework of the general relativistic kinetic theory, macroscopic equations of gravitational and scalar fields are formulated and macroscopic conservation laws are obtained. These equations' full correspondence to microscopic equations with singular sources is shown. Further, on the basis of the obtained equations, a cosmological model for a degenerate system of scalarly charged fermions is formulated. An exact solution of the constitutive equations for a degenerate scalar-charged plasma in the cosmological model is obtained, which made it possible to significantly simplify the original system of equations. On the basis of the obtained solution of the constitutive equations, two fundamentally different cosmological models are formulated, one of which has two types of singly scalarly charged fermions, while the second has one kind of fermions charged with two charges of various nature. A qualitative analysis of the obtained 6-dimensional dynamic system for a two-component model is carried out., 31 pages, 70 references

A meshfree method on fixed grids is devised for simulation of Poisson's equation on 3D image-based cellular materials. The non-boundary fitted discretization of such jagged voxel models of complex geometries is accomplished through embedding the micro-CT scan image in a Cartesian grid of nodes. The computational nodes inside the solid voxels are found by a simple point-in-membership test. Using a set of modified singular functions around the voids, along with the library-rational-exponential basis functions (EBFs) satisfying the governing differential equation, an enriched spatial solution is locally constructed on a generic computational cloud/cell (GCC) of nodes containing the voids. Within each GCC, the boundary conditions are satisfied through a weighted least-squares approximation. Finally, by establishing point-wise compatibility between the solutions of the GCCs a well-conditioned small linear system of equations is resulted. The results are compared with those of the finite element method (FEM) using extremely fine meshes with an excessive number of nodes.

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $\lambda\beta e^{\beta u }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $\lambda$ and the size of $\beta^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.), Comment: 70 pages. To appear in Comm. Math. Phys. Minor typos corrected

This paper addresses a first numerical simulation to the nonlinear dynamic system of equations that describes the prey-predator model at the predator mating period. Some male species accompany the females during the mating period. In this case, both male and female feed on the same prey. The presented work shows the numerical solution for this specific case of the prey-predator mathematical model via an exponential time differencing method. In addition, the paper provides the biological implication of the solution.

Aim. To modify the classical method [1, 4] that causes incorrect estimation of the required size of SPTA in cases when the replacement rate of failed parts is comparable to the SPTA replenishment rate. The modification is based on the model of SPTA target level replenishment. The model considers two situations: with and without the capability to correct requests in case of required increase of the size of replenishment. The paper also aims to compare the conventional and adjusted solution and to develop recommendations for the practical application of the method of SPTA target level replenishment. Methods. Markovian models [2, 3, 5] are used for describing the system. The flows of events are simple. The final probabilities were obtained using the Kolmogorov equation. The Kolmogorov system of equations has a stationary solution. Classical methods of the probability theory and mathematical theory of dependability [6] were used. Conclusions. The paper improves upon the known method of estimating the required size of the SPTA with a safety stock. The paper theoretically substantiates the dependence of the rate of backward transitions on the graph state index. It is shown that in situations when the application is not adjusted, the rates of backward transitions from states in which the SPTA safety stock has been reached and exceeded should gradually increase as the stock continues to decrease. The multiplier will have a power-law dependence on the transition rate index. It was theoretically and experimentally proven that the classical method causes SPTA overestimation. Constraint (3) was theoretically derived, under which the problem is solved sufficiently simply using the classical methods. It was shown that if constraint (3) is not observed, mathematically, the value of the backward transition rate becomes uncertain. In this case, correct problem definition results in graphs with a linearly increasing number of states, thus, by default, the problem falls into the category of labour-intensive. If the limits are not observed, a simplifying assumption is made, under which a stationary solution of the problem has been obtained. It is shown that, under that assumption, the solution of the problem is conservative. It was shown that, if the application is adjusted, the rate of backward transition from the same states should gradually decrease as the stock diminishes. The multiplier will have a hyperbolic dependence on the transition rate index. This dependence results in a conservative solution of the problem of replenishment of SPTA with application adjustment. The paper defines the ratio that regulates the degree of conservatism. It is theoretically and experimentally proven that in such case the classical method causes SPTA underestimation. A stationary solution of the problem of SPTA replenishment with application adjustment has been obtained. In both cases of application adjustment reporting, a criterion has been formulated for SPTA replenishment to a specified level. A comparative analysis of the methods was carried out.

A new solution for advective-dispersive solute transport of sequentially decaying species is developed for a three-dimensional aquifer of finite thickness, finite width, semi-infinite extent along the direction of flow, dual porosity, equilibrium linear sorption, and first-order decay of both the sorbed and dissolved mass with coefficients that may differ in the mobile and immobile domains, and an arbitrary time-dependent source expressed as a first- or third-type boundary condition over a rectangular patch perpendicular to groundwater flow. The system of equations is solved in closed form in the Laplace domain. The model is benchmarked against a single-species analytical model, MT3DMS, and RT3D. The intended use of the model is to guide the selection of transport parameters for and check the accuracy of more complex numerical solute transport models.

Nonlinear feedback shift registers (NFSR) are widely used in cryptography as the source of pseudo-random sequences used in ciphers. The nature of the feedback in a given NFSR affects its output sequence and its statistical quality. The complexity of the problem of restoring the initial state of an NFSR by partially known values of the output is one of the signs of a "good" NFSR. In this regard, we note that the presence of the output affine functions lowers the quality of an NFSR since the corresponding problem is reduced to the solution of a system of linear equations. This paper is concerned with the conditions providing the absence of nontrivial affine functions among NFSR output functions. The obtained theoretical results can be used to improve the effectiveness of experimental methods of finding NFSRs with no affine output functions.

In this contribution two algorithms are developed to solve non-linear system of equations which can contain a large number of measurements. These algorithms are based on nonlinear block least-squar...

In this paper, improved algorithms are proposed for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. These algorithms are more stable compared to conventional alternatives. Further, a stopping criterion changeover is proposed for use with these improved algorithms. This results in higher accuracy (lower true relative error) compared to the case where no changeover is done. Numerical results confirm the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover. These improvements could potentially be applied to other preconditioned algorithms based on bi-Lanczos-type methods.

A method is proposed to construct the -ω-regular expression that specifies the intersection of sets of -ω-words represented by -ω-regular expressions R1 and R2. The sought-for expression is constructed without passing to ω-automata, i.e., by directly transforming the expression R = R1 ∩ R2. The process of constructing the -ω-regular expression specifying the intersection R1 ∩ R2 is represented in the form of a tree of intersections whose vertices correspond to intersections of simple -ω-regular expressions obtained during transforming the intersection R1 ∩ R2 . The constructed tree of intersections defines a system of linear equations with variables whose values are sets of -ω-words. One of these variables R corresponds to the set of -ω-words specified by the intersection R1 ∩ R2,i.e., the expression specifying the intersection of -ω-regular expressions R1 and R2 is the value of the variable R in the solution of this system of linear equations.

A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

The Hodgin-Huxley model is one of the most widely studied biological systems of nonlinear differential equations that is applied to explore nerve cells activities via electrical communications. In this paper we consider some numerical aspects of a simplified version of this model known as the FitzHugh-Nagumo (FHN) equation. Dynamical experiments conducted herein not only confirm those obtained from earlier studies but also facilitate a better understanding of the qualitative features of the FHN model especially those that initiate the behavior a threshold-triggered excitation media. To this end, methods of dynamical system analysis such as bifurcation and linear stability analysis are deployed to investigate the general qualitative features of an inhibitor-activator system which characterizes the FHN system of equations.

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

In this paper, an improved node moving technique was developed for adaptive solution of some flow problems. Collocated Discrete Least Squares Meshless (CDLSM) method was applied for simulation. This method is a truly meshless method and also enjoys the symmetric and positive-definite property. Then, an improved node moving technique was developed based on the spring analogy. In this mechanism, each node was assumed to be connected to its neighboring nodes via some virtual springs. Then, higher values of stiffness were allocated to the springs located at higher error areas. In the previously published works, the corresponding system of equations was assumed to be linear. In this work, first, it was shown that this assumption may lead to inaccurate results in some cases and then, an improved method was proposed considering the nonlinear nature of the system of equations. Some numerical examples were used to illustrate the ability of the proposed node moving technique for some simple examples and benchmark problems in the Computational Fluid Dynamics (CFD) context. The Results of the study showed a considerable improvement in comparison with those of previously published works.

The development of a mathematical model of the filtration flow during dissociation of gas hydrates for the two-dimensional case is researched considering the motion of both components of the gas hydrate (water and gas). Non-isothermal effects and the gas being not ideal while filtering liquid and gas are considered; the hydrate dissociation process is assumed to be in equilibrium. A method is developed for solving the system of equations of the mathematical model using an implicit difference scheme, tridiagonal matrix algorithm and simple iterations, as well as a developed method for calculating hydrate saturation. This method allows to find the spatial distributions of the main parameters of the gas-liquid filtration flow (temperature, pressure and phases saturations) for each moment in time, as well as position of the boundary of phase transitions.

In this letter, the homotopy continuation (HC) approach for a problem of coupling matrix (CM) extraction is presented. HC method is used to solve the system of polynomial equations that are formulated by comparing polynomial coefficients of given filter scattering parameters ${S_{11}}$ and ${S_{21}}$ and those being described based on well-known relations between scattering parameters and CM. HC requires an initial ( $t=0$ ) system of equations with a known solution. During the HC algorithm run, the system and solution are transformed to the final ( $t=1$ ) system and solution that is our goal. In our approach, the initial system ( $t=0$ ) and its solution are formulated by comparing polynomial coefficients of S-parameters obtained for CM including entries of any, even random, values. To validate the method a numerical experiment has been presented, i.e., CM extraction for measured sixth-order (two regular and two parasitic cross-couplings) filter. The experiment demonstrated excellent results obtained with the proposed method.

This paper deals with proposing an approximate solution for the well-known Burgers equation as a canonical model of various fields of science and engineering. Our novel combined approximation algorithm is based on the linearized Taylor approach for the time discretization, while the spectral Chebyshev collocation method is utilized for the space variables. This implies that in each time step, the proposed combined approach reduces the one- and two-dimensional model problems into a system of linear equations, which consists of polynomial coefficients. The error analysis of the present approach in 1D and 2D is discussed. Through numerical simulations, the utility and efficiency of the combined scheme are examined and comparisons with exact solutions as well as existing available methods have been performed. The comparisons indicate that the combined approach is efficient, practical, and straightforward in implementation. The technique developed can be easily extended to other nonlinear models.

Numerical simulation of a supersonic flow of a viscous heat-conducting gas around the HB-2 model for various angles of attack, α = 0°, 8°, and 16°, and temperatures of the model surface is carried out based on a compact hyperbolic system of quasi-gas dynamic (CHQGD) equations. The three-dimensional features of the flow and dependence of the heat flux on the angle of attack and wall temperature are investigated. The presence of the angle of attack leads to the formation of two longitudinal vortices on the leeward side of the model near the symmetry plane. These separation regions significantly affect the distribution of the heat flux on the leeward surface.