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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.

The accumulated large amount of satellite magnetic data strengthen our capability of resolving the electrical conductivity of Earth’s mantle. To invert these satellite magnetic data, accurate and efficient forward modeling solvers are needed. In this study, a new finite-element based forward modeling solver is developed to accurately and efficiently compute the induced electromagnetic field for a realistic 3D Earth. Firstly, the nodal-based finite element method with linear shape function on tetrahedral grid is used to assemble the final system of linear equations for the magnetic vector potential and electric scalar potential. The FGMRES solver with algebraic multigrid (AMG) preconditioner is used to quickly solve for the final system of linear equations. The weighted moving least-square method is employed to accurately recover the electromagnetic field from the numerical solutions of magnetic vector and electric scalar potentials. Furthermore, a local mesh refinement technique is employed to improve the accuracy of estimated electromagnetic field. At the end, two synthetic models are used to verify the accuracy and efficiency of our newly developed forward modeling solver. A realistic 3D Earth model is used to simulate the induced magnetic field at 450 and 200 km altitudes which are the planned flying altitudes of Macau’s geomagnetic satellites. The simulation indicates that (1) amplitude of the mantle-induced magnetic field can reach 10–30 nT at 450 km altitude, which is 10–30% of the primary magnetic field. The induced magnetic field at 200 km altitude has larger amplitudes. These mantle-induced magnetic fields can be measured by Macau geomagnetic satellites; (2) amplitude of the ocean-induced magnetic field can reach 5–30 nT at satellite altitudes, which needs to be carefully considered in the interpretation of satellite magnetic data. We are confident that our newly developed forward modeling solver will become a key tool for interpreting satellite magnetic data.

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.

In our recent work on iterative computation in hardware, we showed that arbitrary-precision solvers can perform more favourably than their traditional arithmetic equivalents when the latter’s precisions are either under- or over-budgeted for the solution of the problem at hand. Significant proportions of these performance improvements stem from the ability to infer the existence of identical most-significant digits between iterations. This technique uses properties of algorithms operating on redundantly represented numbers to allow the generation of those digits to be skipped, increasing efficiency. It is unable, however, to guarantee that digits will stabilise, i.e. never change in any future iteration. In this article, we address this shortcoming, using interval and forward error analyses to prove that digits of high significance will become stable when computing the approximants of systems of linear equations using stationary iterative methods. We formalise the relationship between matrix conditioning and the rate of growth in most-significant digit stability, using this information to converge to our desired results more quickly. Versus our previous work, an exemplary hardware implementation of this new technique achieves an up-to 2.2x speedup in the solution of a set of variously conditioned systems using the Jacobi method.

A solution of the system of equations for the basic operators of the spin-fermion model is obtained in the form of harmonic oscillations of Fermi operators at three frequencies and single-frequency harmonic oscillations of the spin-fermion coupling operator (spin polaron).

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.

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

In this paper, we focus on solving a class of two-by-two block complex system of linear equations arising from finite element discretization of a distributed optimal control constrained by a time-harmonic parabolic equations. We propose a new iterative method for solving the obtained system. In each iteration of the method a two-by-two block system of linear equations with real coefficient matrix should be solved. We solve this system inexactly using the generalized minimal residual (GMRES) and the Chebyshev acceleration methods in conjunction with the real-valued preconditioned square block (PRESB) preconditioner. The convergence and spectral properties of the method are discussed. Numerical results in 2-dimensional case are presented to demonstrate the efficiency of the method.

Mesh refinement may give rise to variable dimension system of linear equations in adaptive analysis. In each adaptive step, the matrix decomposition methods are adopted to solve linear algebraic equations, which is obviously time-consuming and computationally expensive for large-scale problems. A new algorithm for solving a variable dimension system of linear equations is proposed in this paper, which combines matrix deformation with the Sherman-Morrison formula. This method not only retains the advantages of the Sherman-Morrison formula, but also is particularly suitable for adaptive analysis. Compared with the matrix decomposition method, our method has outstanding performance in promoting computational efficiency. Several numerical examples are used to illustrate the superiority of the proposed method when applying to solve 2D elasticity problems.

We study some properties of integro splines. Using these properties, we design an algorithm to construct splines $S_{m+1}(x)$ of neighbouring degrees to the given spline $S_m(x)$ with degree m. A local integro-sextic spline is constructed with the proposed algorithm. The local integro splines work efficiently, that is, they have low computational complexity, and they are effective for use in real time. The construction of nonlocal integro splines usually leads to solving a system of linear equations with band matrices, which yields high computational costs.

The localized method of fundamental solutions is a recent domain-type meshless collocation method with the fundamental solutions of governing equations as the radial basis functions. This approach forms a sparse system matrix and has a higher efficiency than the traditional method of fundamental solutions. In this paper, a modified version of the localized method of fundamental solutions is developed for bending analysis of simply supported and clamped thin elastic plates. Some auxiliary nodes on the boundary are firstly introduced to provide additional weight coefficients, which contribute to the construction of the determined system of equations and avoid the over-determined equation system of the localized method of fundamental solutions for bending analysis of thin elastic plates. Several numerical experiments with simply supported or clamped boundary conditions are provided, and numerical results are in good agreement with the analytical or COMSOL Multiphysics solutions.

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12] . The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16] .

U ovome radu je opisan razvoj web aplikacije za računanje sustava linearnih jednadžbi putem matrica. Na početku samoga rada je opisano koje su tehnologije se koristile prilikom izrade samoga rada te je ukratko nešto rečeno o pojedinoj tehnologiji. Opisana je cijela teorijska podloga koja stoji iza rješavanja problematike kojom se rad bavi a to je rješavanje sustava linearnih jednadžbi putem matrica. Prikazana je i programska implementacija rada.Korišten je jedan od najpopularnijih programskih okvira to je React prilikom izrade cijeloga rada. Sve tehnologije koje su korištene u radu se koriste i u stvarnoj industriji koja se bavi razvojem web aplikacija. This work shows the development of web applications for solving systems of linear equations using matrices. At the beginning of this work is described which technologies are used inside of this work and in a few sentences it tells something about all technologies that are used. This work describes all theoretical background for solving all problems about solving systems of linear equations using matrices. Also is shown program implementation of this whole work. In this work is using one of the most popular frameworks for solving client side problems that is React. All technologies which are used in this work are in full application in the field which is working with web applications.

The article considers the possibilities of developing the combined discrete-continuous vibratory systems, in which the disturbing member is designed in the form of the uniform elastic rod with distributed inertia and stiffness parameters. The forced oscillations of the continuous member of the three-mass vibratory system are analyzed. Based on the Krylov-Duncan functions (circular and hyperbolic functions), the system of equations describing the motion of the continuous rod is derived. The novelty of the present paper consists in deriving the mathematical model of the discrete-continuous vibratory system, in which the model of the discrete subsystem is combined with the model of the continuous subsystem by applying the reactions in the supports holding the uniform elastic rods. The inertia-stiffness parameters of the vibratory system are determined and the analytical dependencies for calculating the reactions in supports are derived. The frequency-response curves of the considered discrete-continuous vibratory system are constructed. The deflection (bending) diagram of the continuous members is plotted for the case of forced oscillations of the combined discrete-continuous vibratory system.

This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products. We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

The support vector classification-regression machine for K-class classification (K-SVCR) is a novel multi-class classification method based on the “1-versus-1-versus-rest” structure. In this paper, we propose a least squares version of K-SVCR named LSK-SVCR. Similarly to the K-SVCR algorithm, this method assesses all the training data into a “1-versus-1-versus-rest” structure, so that the algorithm generates ternary outputs {− 1,0,+ 1}. In LSK-SVCR, the solution of the primal problem is computed by solving only one system of linear equations instead of solving the dual problem, which is a convex quadratic programming problem in K-SVCR. Experimental results on several benchmark, MC-NDC, and handwritten digit recognition data sets show that not only does the LSK-SVCR have better performance in the aspects of classification accuracy to that of K-SVCR and Twin-KSVC algorithms but also has remarkably higher learning speed.

We study the system of equations which bases on the one-dimensional Schrodinger equation and connects the potential, amplitude, and phase functions. Using the methods of compatibility theory of systems of partial differential equations, we obtain the completely integrable systems that connect only two functions of the above three. As a corollary, we construct some exact solutions of the Schrodinger equation.

The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are contaminated by some noise. For ill-posed TLS poblems, regularization techniques are necessary to stabilize the computed solution; otherwise, TLS produces a noise-dominant output. In this paper, we show that the regularized total least squares (RTLS) problem can be reformulated as a nonlinear least squares problem and can be solved by the Gauss–Newton method. Due to the nature of the RTLS problem, we present an appropriate method to choose a good regularization parameter and also a good initial guess. Finally, the efficiency of the proposed method is examined by some test problems.

This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving identification problems with the VFM is first to derive parametric stress field, where the stress components at any point depend on the unknown constitutive parameters, across the volume of interest from the measured deformation fields. Applying the principle of virtual work to the parametric stress fields, one can write scalar equations of the unknown parameters and solve the obtained system of equations to deduce the values of unknown parameters. However, no rules have been proposed to select the virtual fields in identification problems related to nonlinear elasticity and there are multiple strategies possible that can yield different results. In this work, we propose a systematic, robust and automatic approach to reconstruct the systems of scalar equations with the VFM. This approach is well suited to finite-element implementation and can be applied to any problem provided that full-field deformation data are available across a volume of interest. We also successfully demonstrate the feasibility of the novel approach by multiple numerical examples. Potential applications of the proposed approach are numerous in biomedical engineering where imaging techniques are commonly used to observe soft tissues and where alterations of material properties are markers of diseased states.List of symbols

Purpose.The main purpose of the work is to determine and classify the heated cars’ boxes based on the probability of appearance of roller and cassette type boxes in the classes of heated and overheated boxes, as well as the laws of probability density distribution of the recognition signs of normally heated and overheated roller and cassette type boxes.Methodology.The operation features of freight cars with cassette type axle boxes with increased operating heating have been investigated. The methodology of assessing the probability of recognition errors was proposed, which takes into account the fact that sets of normally heated and overheated boxes consist of subsets of boxes with different types of bearings. A system of equations is obtained, the roots of which represent еру values that minimize the recognition probability of the errors of the heated boxes.Findings.It was found out that with some methods of determining the bearing type, for example, by the average value of the ranges of thermal image for each car, the probability of erroneous selection may depend on the probability density distribution of the sign for bearings of different types and the threshold value of this sign. The optimal thresholds for detecting the overheated roller boxes in comparison with the optimal thresholds for detecting overheated cassette boxes were determined. It has been established that the pass of an overheated cassette bearing, provided that the type of bearing is determined correctly, is less likely to lead to an accident than if the cassette box is classified as a roller box. The rejection criteria of axle boxes according to their heating temperature difference on one of the wheel set axis for three variants of settings of the alarm system according to an arrangement of multipurpose complexes of technical means (CTM) were formulated. The practical implementation of this method of adjusting the CTM settings for the Minsk branch of the Belarusian Railways was demonstrated.Originality.A system of equations is obtained, which allows finding the optimal values of temperature thresholds for the detection of overheated roller and cassette boxes under the assumption that the error probabilities in the selection of boxes by their types are known and constant.Practical value.The developed method of adjusting the alarm settings of CTM makes it possible to significantly reduce unjustified train delays and the number of car uncouplings.

This study carries the novel applications of the Sinc collocation method to investigate the numerical computing paradigm of Schrodinger wave equation and Transport equation as a great level of accuracy and precision. A global collocation-based Sinc function is embedded with a cardinal expansion to discretize initially time derivatives by finite difference method and secondly spatial derivatives are approximated with $$\theta $$ -weighted scheme. Sinc collocation method (SCM) is found to be a more robust approach in order to avoid singularities in proposed problems and for yielding accurate numerical results. The governing PDEs are transformed with the help of Sinc function into algebraic system of equations, and further, these algebraic equations are solved with the help of computational iteration scheme to obtain the numerical results. The scheme provides a reliable and excellent procedure for adaptation of finding unknown in Sinc function for these problems. An extensive stability analysis is done to validate the convergence, accuracy and exactness of the proposed scheme.

In this paper, a class of absolute value equations (AVE) $ Ax-B|Cx| = d $ with $ A, B, C\in \mathbb{R}^{n\times n} $ is considered, which is a generalized form of the published works by Wu [1], Wu and Shen [2] and Mezzadri [3]. Some conditions to guarantee the unique solution of the above AVE are gained. The corresponding results of the above published works are generalized.

We study a system of equations of dynamics of a thermoviscoelastic continuous medium with an Oldroyd-type rheological relation, which generalizes a Navier–Stokes–Fourier system. In the planar case, we prove the unique existence of strong solutions. The proof is based on the construction of Galerkin approximations and their strong estimates, which provide the corresponding limit passage.

An Empirical-Markovian approach is proposed to estimate the pavement structural capacity as a function of key stochastic and design parameters. The key stochastic parameters are the initial and terminal deterioration transition probabilities typically estimated from pavement distress records. These two transition probabilities have a major impact on the pavement performance trend predicted using Markovian processes. In addition, typical pavement design factors are included as related to traffic loadings, materials properties, and climate conditions. In particular, two distinct Empirical-Markovian models are developed to estimate the pavement structural capacity in terms of relative strength indicators such as the structural number (SN) and gravel equivalent (GE). The first model can estimate the structural capacity based on the initial transition probability and relevant design parameters, while the second one deploys the terminal transition probability along with other design parameters. The recommendation is to use the higher of the two structural capacity values estimated for a particular pavement project. The sample models presented based on the California Department of Transportation (Caltrans) design method have resulted in good model fittings as demonstrated by the various deployed statistics and error analysis, thus indicating the usefulness of the proposed Empirical-Markovian approach in estimating the pavement structural capacity for rehabilitation and design purposes. The model exponents can be obtained from solving a linear system of equations using data from a small project sample or solving a multi-variable linear regression model when a large road sample is available. The latter case provided sample generalized models with statistics indicating their high significance.

Given the critical role played by natural gas in providing electricity, heat, and other essential services, better models are needed to understand the dynamics of natural gas networks during extreme events. This paper aims at establishing appropriate and fast simulation models to capture the slow dynamics of linepack depletion for ideal isothermal natural gas pipeline networks. Instead of solving partial differential equations (PDE) on a large scale, three alternative implicit ordinary differential equation (ODE) simulation techniques are derived and discussed. The first one is commonly used in the literature with a slack node assumption. We show that the system of equations associated with this model is degenerate when flux injections are controlled (i.e. specified) at all nodes. To recover regularity under such a condition, two novel implicit ODE models are proposed, both with different techniques for specifying boundary conditions. They are easy to derive and efficient to simulate with standard ODE solvers. More importantly, they present useful frameworks for analyzing how networks respond to system-wide mass flux imbalances. These techniques offer different alternatives for simulating system dynamics based on how sources and loads are chosen to be modeled, and they are all proven to be regular (non-degenerate) in tree-structured networks. These proposed techniques are all tested on the 20-node Belgium network. The simulation results show that the conventional model with the slack node assumption cannot effectively capture linepack depletion under long term system-wide mass flux imbalance, while the proposed models can characterize the network behavior until the linepack is completely depleted., Comment: 40 pages, accepted for publication in the Applied Mathematical Modeling (AMM) Journal

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.

In this work, phase-field modeling of hydraulic fractures in porous media is extended towards a Global–Local approach. Therein, the failure behavior is solely analyzed in a (small) local domain. In the surrounding medium, a simplified and linearized system of equations is solved. Both domains are coupled with Robin-type interface conditions. The fractures inside the local domain are allowed to propagate and consequently, both subdomains change within time. Here, a predictor–corrector strategy is adopted, in which the local domain is dynamically adjusted to the current fracture pattern. The resulting framework is algorithmically described in detail and substantiated with some numerical tests.

The occurrence of buckling during cooling of an injection-molded fiber-reinforced plastic (FRP) object is a complex phenomenon which can result in unacceptably large deformations. The present work focuses on the thermal buckling of injection-molded thin disks, as they are representative for the local behavior in parts of a more complex object. An analytical approach based on the Foppl-von Karman theory for thin elastic plates is presented and used to analyze its buckling stability and initial post-buckling behavior. The system of equations is formulated in a coordinate-free approach for a thin anisotropic linear thermo-elastic body with residual thermal stresses. Large out-of-plane displacements are assumed. The particular case of a free disk with cylindrical orthotropy and uniform thermo-elastic properties through its thickness is considered. A general technique to study the bifurcation in the solutions (i.e., buckling) based on a perturbation approach and the Fourier-Galerkin method to determine post-buckling deflections is proposed. Experimental and numerical results available in the literature are used as validation cases. The results of the present model show qualitative agreement with the experimental data. It is found that the periodicity of the first buckling mode for orthotropic disks is fully determined by the ratio between the elastic moduli in the tangential and radial directions. The predictions are also in excellent quantitative agreement with the numerical results of FEM models for the same cases. The effect of the disk thickness on the buckling temperature and warpage magnitude is correctly captured.

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix A k of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD method, by replacing A by a matrix that is closer to A than A k in a unitarily invariant matrix norm and has the same spectral condition number as A k . Approximations of the SVD of a large matrix A can be computed quite efficiently by using a randomized SVD (RSVD) method. This paper presents a novel modified truncated randomized singular value decomposition (MTRSVD) method for computing approximate solutions to large-scale linear discrete ill-posed problems. The rank, k, is determined with the aid of the discrepancy principle, but other techniques for selecting a suitable rank also can be used. Numerical examples illustrate the effectiveness of the proposed method and compare it to the truncated RSVD method.

This manuscript is related to compute approximate solutions for a class of fractional distributed order differential equations (FDODEs). The corresponding derivative of fractional order is taken in Caputo sense. The adopted numerical scheme is based on collocation method together with Haar wavelet. This is an efficient numerical algorithm which convert the consider problem to a some system of equations of algebraic type. Upon utilizing the Broyden tools the obtain nonlinear system is solved for the intended numerical results. Further in case of linear equations the obtained system is solved by Gauss elimination procedure. To strengthen our results we testify several problems corresponding to different collocation points. We present also maximum absolute and root mean square errors. The graphical presentations are also given.

In this article, a temporal coupled-mode theory (CMT) formalism is proposed to design graphene-based metasurface (MTS) structures. Such structures may be nonuniform and synthesized by numerous graphene resonators. To obtain a desired functionality, such MTSs may be designed, by performing multiple parametric analyses of the structure’s geometric parameters. An accurate calculation of their response may be attained by a full-wave vectorial finite-element method (VFEM), by modeling graphene as a surface boundary condition; however, it requires high computational resources, especially for larger configurations. CMT offers an appealing time- and memory-efficient alternative, by solving a sparse system of equations, of particularly small size, fed by the results of simpler and smaller simulations, with the ability to take into account graphene dispersion. CMT may be used to design more complex MTS structures, by providing approximate and preliminary results, which shall be used as a starting point for the final FEM fine-tuning.

This paper proposes two complex-valued zeroing neural network (Cv-ZNN) models for solving dynamic complex time-variant linear equations. The models involve two complex-valued nonlinear processing methods and adapt two real-valued activation functions . The convergence and robustness of the two Cv-ZNN models are analyzed comprehensively. Firstly, the convergence discussion shows that the corresponding stable error can rapidly converge to zero in finite time, with the upper limit of constriction time calculated. The upper limit of stable error is successfully obtained when the model implementation error is injected into the Cv-ZNN models, which displays their superior robustness. Besides, the results of numerical simulations reveal that the Cv-ZNN models effectively address the time-variant linear system of equations over complex field, even in noisy environments . The simulation results also demonstrate that two novel nonlinear activation functions perform much better than the Linear, Power, Bipolar Sigmoid Power-Sigmoid and Sign-bi-power activation functions.

Neste artigo são apresentados resultados parciais de uma pesquisa realizada com estudantes de um curso de Licenciatura em Matemática do Rio Grande do Sul, com o objetivo investigar as contribuições de uma sequência didática para o ensino e aprendizagem de Sistemas de Equações Lineares, à luz da Teoria de Registros de Representações Semióticas. Para isso, foi aplicada uma sequência didática sobre Sistemas de Equações Lineares, dividida em quatro blocos. O primeiro bloco foi destinado a uma familiarização com sistemas possíveis e impossíveis 2x2, por meio de problemas; o segundo bloco foi composto por problemas envolvendo sistemas possíveis 3x3 e 5x4; o terceiro bloco trouxe problemas referentes a sistemas impossíveis 3x3 e, por fim, o quarto bloco desenvolveu-se a representação gráfica de sistemas 2x2. Concluímos que a maior parte dos estudantes conseguiram transitar ao menos entre dois tipos distintos de representações de um mesmo objeto matemático, no entanto apresentaram dificuldades na representação gráfica de sistemas lineares, de modo que fizeram o uso do software GeoGebra para tais representações.

Free-surface seepage problems are one of the most challenging issues in the practice of geotechnical engineering since the determination of the free surface commonly demands sophisticated numerical...

The article deals with issues related to the quantitative assessment of the potential accuracy of measuring the coordinates of point, long and area tangent and non-tangent objects using matrix CENS. The analyzed objects can be used as navigation landmarks, the coordinates of which in step-by-step sighting can be determined by a matrix CENS with high accuracy. Quantitative assessment of the potential accuracy of CENS is as follows: A model of the current image is built, in the process of inspecting the earth's surface, in the course of movement at a certain speed and at a certain angle to the surface. The frame sizes, parameters of the distinguished elements, laws of movement of the distinguished elements are defined. The representation of the intended object and the surrounding background is set. Assumptions are made about the presence or absence of a priori information about the parameters being evaluated and interfering parameters. Based on these assumptions, the criterion of optimality is selected. There is a system of equations that describes the algorithm for processing signals CENS. The solution of the system of equations allows to find the global extremum of the criterion of optimality and variance of estimates of the analyzed parameters. As the model of the analyzed image the zone model in which object with one luminous filling is located on a homogeneous background with other luminous filling is accepted. In the absence of a priori information about the parameters evaluated, the criterion of maximum plausibility is chosen as the criterion of optimality. It is shown that there is a strict symmetry of the error value before the appearance of the upper object in the frame and after its exit from the frame. The best accuracy is observed at a high signal-to-noise ratio of two objects. Decreasing this ratio to one in the auxiliary object slightly impairs the accuracy of binding to the main object. With a small signal-to-noise ratio in the main object, the accuracy is determined by the accuracy of the auxiliary object until it leaves the frame, and then the accuracy (low) of the main object. Determining the coordinates of non-tangential objects is carried out with less SLE than tangential, as in this case, the binding of CENS is on objects surrounded by a homogeneous background. The developed technique allows to carry out quantitative estimations of potential accuracy of measurement of coordinates of point, extended and area objects by means of matrix CENS. The analyzed objects can be used as navigational landmarks, the coordinates of which in step-by-step sighting can be determined by the matrix CENS with high accuracy.

The method for processing of the measurement results obtained from Comite International des Poids et Measures (CIPM) Key, Regional Metrology Organizations (RMO) or supplementary comparisons, from the proficiency testing by interlaboratory comparisons and the calibrations is proposed. It is named by authors as adjustment by least square method (LSM). Additive and multiplicative parameters for each measuring standard of every particular laboratory will be the results of this adjustment. As well as the parameters for each artifact. The parameters of the measurements standards are their additive and multiplicative degrees of equivalence from the comparison and the estimations of the systematic errors (biases) from calibrations. The parameters of the artifacts are the key comparisons reference value from the comparison and the assigned quantity values from the calibrations. The adjustment is considered as a way to solving a problem of processing the great amount of homogeneous measurements with many measuring standards at a different comparison levels (CIPM, RMO or supplementary), including connected problems. Four different cases of the adjustments are considered. The first one is a free case of adjustment. It was named so because of the fact that none of participants has any advantage except their uncertainties of measurements. The second one is a fixed case of adjustment. Measuring results of RMO and supplementary comparisons are rigidly linked to additive and multiplicative parameters of measuring standards of particular laboratories participated in CIPM key comparisons. The third one is a case of adjustment with dependent equations. This one is not so rigidly linked of the new comparisons results to previous or to some other comparisons as for fixed case. It means that the new results of comparisons are influenced by the known additive and multiplicative parameters and vice versa. The fourth one is a free case of adjustment with additional summary equations. In that case certain checking equations are added to the system of equations. So, the sum of parameters multiplied by their weights of all measurement standards for particular laboratories participated in comparisons should be equal to zero.