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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.

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.

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.

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.

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.

We employ the multi-wavelet Galerkin method for the numerical solution of the telegraph equation with initial and boundary conditions. The problem becomes a sparse system of linear equations and the GMRES method is used to solve this system. The advantages of this scheme are complexity reduction, simplicity, and less grid selection. The convergence analysis is investigated and numerical experiments guarantee it. To show the ability of the method, we compare it with other methods and it can be confirmed that our results are better than others.

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.

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).

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 ) .

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

In the article, computer design of routes of linear structures is considered as a spline approximation problem. A fundamental feature of the corresponding design tasks is that the plan and longitudinal profile of the route consist of elements of a given type. Depending on the type of linear structure, line segments, arcs of circles, parabolas of the second degree, clothoids, etc. are used. In any case, the design result is a curve consisting of the required sequence of elements of a given type. At the points of conjugation, the elements have a common tangent, and in the most difficult case, a common curvature. Such curves are usually called splines. In contrast to other applications of splines in the design of routes of linear structures, it is necessary to take into account numerous restrictions on the parameters of spline elements arising from the need to comply with technical standards in order to ensure the normal operation of the future structure. Technical constraints are formalized as a system of inequalities. The main distinguishing feature of the considered design problems is that the number of elements of the required spline is usually unknown and must be determined in the process of solving the problem. This circumstance fundamentally complicates the problem and does not allow using mathematical models and nonlinear programming algorithms to solve it, since the dimension of the problem is unknown. The article proposes a two-stage scheme for spline approximation of a plane curve. The curve is given by a sequence of points, and the number of spline elements is unknown. At the first stage, the number of spline elements and an approximate solution to the approximation problem are determined. The method of dynamic programming with minimization of the sum of squares of deviations at the initial points is used. At the second stage, the parameters of the spline element are optimized. The algorithms of nonlinear programming are used. They were developed taking into account the peculiarities of the system of constraints. Moreover, at each iteration of the optimization process for the corresponding set of active constraints, a basis is constructed in the null space of the constraint matrix and in the subspace – its complement. This makes it possible to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations. As an objective function, along with the traditionally used sum of squares of the deviations of the initial points from the spline, the article proposes other functions taking into account the specificity of a particular project task.

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.

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.

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 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 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.

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.

This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.

We consider questions concerning the solvability and the number of solutions of systems of Boolean equations. Many mathematical models arising in both operations research and cryptography are described in the language of such systems. This is due, in particular, to the fact that, in general, the problem of checking the compatibility of such systems of equations is NP-complete; therefore, the study of the qualitative properties of a system of Boolean equations provides additional information that permits one either to single out polynomially solvable particular cases or to increase the efficiency of enumeration schemes. The focus is on two aspects. The first one concerns the study of the existence and number of solutions of a Boolean equation and a system of equations when parametrizing the problem by the right-hand sides. Formulas and estimates are given for calculating this number both in general and in particular cases. Its maximum is also investigated depending on the specified parameter. The second aspect is devoted to a special case of the problem when the equation is given by the so-called continuous linear form. The properties of such forms and various criteria of continuity are studied.

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

Es frecuente encontrar en la investigación en las ciencias e ingenierías, que se requiere de factorizaciones de matrices para facilitar o simplificar los cálculos, ya sea para resolver sistemas de ecuaciones lineales, o no lineales, provenientes de análisis de datos o de modelado o simulación de procesos industriales. Cuando las matrices son cuadradas y de rango completo existen factorizaciones que permiten hacer este trabajo eficientemente; sin embargo, cuando las matrices son rectangulares y/o deficientes en rango, las extensiones de estas factorizaciones se hacen necesarias y deben ser tratadas cuidadosamente para que los resultados sean confiables. Objetivo: En este trabajo se describe la descomposición de rango completo de Cholesky y su uso en algunas aplicaciones en áreas como la Estadística, donde a menudo se requiere resolver sistemas sobredeterminados por el método de los mínimos cuadrados o extensiones del mismo. Metodología. Se describen los pasos para generar esta descomposición y su uso en la estimación máximo verosímil de parámetros por el método de los mínimos cuadrados reponderados. Resultados: Se presenta experimentación preliminar en datos sintéticos, con resultados que sustentan el uso de esta descomposición para matrices rectangulares o deficientes en rango. Conclusión: La factorización propuesta para resolver el problema de estimación de parámetros con la función de verosimilitud es satisfactoria y pudiera ser utilizada para optimizar el tiempo de cómputo de la solución de problemas similares.

The user-centric cloud radio access network (C-RAN) is promising for significantly reducing the channel training overhead because only the intra-cluster channel state information (CSI) is required. However, the inter-cluster interference may degrade the network performance. To address this problem, we present a novel framework for the uplink of user-centric C-RANs where the interference cancellation is posed as a system of bi-unknown vectors linear equations. To solve this unusual system of equations, we propose a quasi-least squares (QLS) algorithm and analyze its robustness by exploiting the random matrix theory. We reveal the fact that QLS is very sensitive to the channel estimation error due to involving the inverse of an ill-conditioned matrix. It is well known that truncated singular value decomposition (TSVD) is an effective regularization scheme that can mitigate this ill-conditioning effect. Accordingly, we employ TSVD to further improve the robustness of QLS against the imperfect channel estimation. In addition, since the performance of the TSVD based algorithm strongly depends on the truncation parameter, a parameter-choice method using the constant modulus (CM) feature is also provided. Finally, simulation results are presented to examine the effectiveness and robustness of the proposed method.

We construct classes of systems of equations whose sets of solutions are invariant under a group of mappings isomorphic to a one-parameter group of rotations. It is shown that unbounded solutions of these systems are unstable. We also present the applications of obtained results to the problems of nonlinear mechanics.

We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

Multi-scale models enriched with the second gradient of the displacement field and based on the concept of representative volume element (RVE) are examined and discussed in detail in this paper. The development of new models relies on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived employing variational arguments. Two classes of enhanced multi-scale constitutive models are analysed: standard second-order homogenisation and fully second-order homogenisation. On the computational side, the numerical treatment of second gradient equilibrium problems with a mixed approach is addressed systematically within an implicit finite element discretisation in conjunction with the full Newton–Raphson scheme for the iterative solution of the corresponding non-linear systems of equations. Different formulations available in the literature for standard second-order homogenisation are presented, along with a critical appraisal of their assumptions and limitations. A macroscopic second-gradient constitutive law with an orthotropic length scale is developed based on numerical findings. The impact of finite strains, the micro constituents’ behaviour, and the RVE morphology on the resulting macroscopic length scale parameter’s evolution is assessed. A fully second-order homogenisation-based model at finite strains is also presented. The suitability of this class of models to capture size effects due to the micro-constituent’s size is illustrated by an FE2 simulation.

Homogeneous linear systems integrate special properties that differentiate them from other linear systems and allow to simplify the search for solutions that, under certain conditions, promote general solutions of even heterogeneous systems and non-linear systems, hence their crucial importance in Mathematics, related sciences and in Engineering. From the Nine Chapters on the Mathematical Art of Ancient China to authors such as Seki Kowa, Leibniz, Cayley, Silvester, Bocher, the resolutions of linear systems started to rely on matrix methods based on theoretical results such as the Gauss-Jordan elimination algorithm, Cramer's theorem, the Kronecker-Capelli theorem. Classic iterative methods such as Jacobi-Richardson, Gauss-Seidel, Cholesck factorization, the SOR method, conjugate gradient iterative methods, as well as graphical methods were also introduced. However, this article presents an innovative methodological alternative called systematic dimensional contraction, which is not based on matrices: it aims, among other dynamics, to systematically reduce the number of unknowns until the respective resolution is viable. In this view, the objective is to analyze the operability of this method of systematic dimensional contraction in the study of equations and linear systems, using homogeneous techniques. For this purpose, this article makes use of a theoretical-methodological research, of an explanatory typology, with bibliographic technical procedures and that uses the inductive-deductive method. Thus, the methodologic proposal for a systematic dimensional contraction method was constructed and applied to obtain original and exact solutions of homogeneous systems of linear equations, because solutions of this nature are a necessary condition for the construction of the homogeneous vector product and, in general, of the homogeneous theory of Vector Spaces.

Convexity is discussed for several free boundary value problems in exterior domains that are generally formulated as Δ u = f ( u ) in Ω ∖ D , | ∇ u | = g on ∂ Ω , u ≥ 0 in R n , \begin{equation*} \Delta u = f(u) \ \text {in } \Omega \setminus D, \quad |\nabla u | = g \ \text { on } \ \partial \Omega , \quad u\geq 0 \ \text { in } \ \mathbb {R}^n, \end{equation*} where u u is assumed to be continuous in R n \mathbb {R}^n , Ω = { u > 0 } \Omega = \{u > 0\} (is unknown), u = 1 u=1 on ∂ D \partial D , and D D is a bounded domain in R n \mathbb {R}^n ( n ≥ 2 n\geq 2 ). Here g = g ( x ) g= g(x) is a given smooth function that is either strictly positive (Bernoulli-type) or identically zero (obstacle type). Properties for f f will be spelled out in exact terms in the text. The interest is in the particular case where D D is star-shaped or convex. The focus is on the case where f ( u ) f(u) lacks monotonicity, so that the recently developed tool of quasiconvex rearrangement is not applicable directly. Nevertheless, such quasiconvexity is used in a slightly different manner, and in combination with scaling and asymptotic expansion of solutions at regular points. The latter heavily relies on the regularity theory of free boundaries. Also, convexity for several systems of equations in a general framework is discussed, and some ideas along with several open problems are presented.

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

Being a continuation of the paper published in Mathematics and Statistics, vol. 7, No. 5, 2019, this article describes the algorithm for the first stage of spline- approximation with an unknown number of elements of the spline and constraints on its parameters. Such problems arise in the computer-aided design of road routes and other linear structures. In this article we consider the problem of a discrete sequence approximation of points on a plane by a spline consisting of line segments conjugated by circular arcs. This problem occurs when designing the longitudinal profile of new and reconstructed railways and highways. At the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation for optimizing the spline parameters using a special nonlinear programming algorithm. The dynamic programming algorithm is practically the same as in the mentioned article published earlier, with significant simplifications due to the absence of clothoids when connecting straight lines and curves. The need for the second stage is due to the fact that when designing new roads, it is impossible to implement dynamic programming due to the need to take into account the relationship of spline elements in fills and in cuts, if fills will be constructed from soils of cuts. The nonlinear programming algorithm is based on constructing a basis in zero spaces of matrices of active constraints and adjusting this basis when changing the set of active constraints in an iterative process. This allows finding the direction of descent and solving the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. As an objective function, instead of the traditionally used sum of squares of the deviations of the approximated points from the spline, the article proposes other functions, taking into account the specifics of a specific project task.

We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOCCVI). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality (VI), which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Especially, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments, illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that, in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.

A technique for setting boundary conditions on solid surfaces based on the method of near-wall functions is presented. The technique is based on solving the Reynolds-averaged Navier-Stokes equations with the Spalart-Allmaras (SA) closure model in the boundary layer approximation. The obtained solution is used to formulate flow boundary conditions that compensate the insufficient mesh resolution of the boundary layers. For a simplified system of equations, discretization is performed and a solution algorithm is constructed. A parallel software implementation of the method is carried out in a finite-volume computational code. Based on the test cases, which are canonical turbulent flows, a series of calculations are carried out to demonstrate the capabilities of the developed technique. Recommendations and limitations related to the practical application of the proposed technique are developed.

We consider a class of hyperbolic systems of linear inhomogeneous partial differential equations with one spatial variable. As a rule, in the case of systems of partial differential equations, when solving particular problems, additional conditions are immediately used to ensure the uniqueness of the solution. However, this greatly complicates the construction of a solution in the case of additional conditions of nonstandard form. For a similar situation in the case of ordinary differential equations, one tries to find a general solution, for which one can then try to use the given additional conditions. However, for systems of partial differential equations, this approach is difficult, since, as a rule, in this case it is not possible to construct the general solution. For the class of systems of linear inhomogeneous partial differential equations considered in the present paper, it was possible to find an algorithm for constructing a general solution. A distinctive feature of the considered systems of equations is the multiplicity of the corresponding characteristics. As an application of the proposed algorithm, a general solution of the Kolmogorov system of equations for the probabilities of the states of a process is obtained, which describes the behavior of a popular model of a stochastic system of the type $$k $$ -out-of- $$ n: F $$ with a general distribution of the repair time for failing components. The indicated system of Kolmogorov equations is a system of partial differential equations of the mentioned class. Therefore, it is possible to construct a general solution for this system.