System Of Linear Algebraic Equations Research Articles

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Micromechanical analysis of magnetoelectric coupling of composites containing nonlinear magnetostrictive and piezoelectric constituents

This paper is a reformulation of the simplified unit-cell and Mori-Tanaka micromechanics theories in their applications to nonlinear magnetoelectric analysis of two-phase composite materials with 0–3, 1–3, and 2–2 connectivity types, respectively. We elucidate the similarity between the two models insofar as concentration-factor matrices are concerned. The representations of the bulk magnetostrictive and piezoelectric phases are based on nonlinear constitutive equations. Due to material nonlinearity, the mathematical frameworks are accomplished via incremental formulation that provides a system of linear algebraic equations at each increment which is obviously a great advantage over nonlinear one. The derived nonlinear composite constitutive relations that govern the hysteresis behavior of composites are implemented to study the composite composed of Terfenol-D and PZT constituents. The responses of this composite to a complete cycle of magnetic field are shown and the micromechanics predictions are compared in light of existing experimental data.

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.

Electrodynamic modeling of slot gratings by the generalized scattering matrix method – spherical wave decomposition

An algorithm is presented for the electrodynamic analysis of a two-dimensional waveguide slot array of finite dimensions. To solve the boundary value problem, the generalized scattering matrix method is used. The complex problem for a structure with large electrical dimensions is divided into two subproblems: wave scattering on one lattice element and the interaction of waves within the lattice. In accordance with this method, the electromagnetic field of a solitary lattice element is represented in the form of an expansion in incident and scattered spherical waves. The solution to the first subproblem is given by the scattering operator, which relates the amplitudes of the incident and scattered waves. The solution to the second subproblem yields an interaction matrix that relates the amplitudes of waves incident on the mth array element with the amplitudes of waves scattered by the nth element. Application of the scattering operator and interaction matrix to the analysed lattice leads to a system of linear algebraic equations for the amplitudes of the scattered waves. A non-periodic slot grating, focused in the Fresnel zone, containing up to a thousand elements is analysed. The obtained numerical results are in good agreement with the known behaviour of focused leaky wave gratings. Possible areas of application of the method are discussed.

Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.

Viscous Current-Induced Forces.

We study the motion (translational, vibrational, and rotational) of a diatomic impurity immersed in an electron liquid and exposed to electronic current. An approach based on the linear response time-dependent density functional theory combined with the Ehrenfest dynamics leads to a system of linear algebraic equations, which account for the competing and counteracting effects of the current-induced force (electron wind) and the electronic friction. We find and emphasize the coupling between the center of mass motion and that of the nuclei relative to each other, the feature due to the mediation of the two-body interaction by the environment. The current-induced forces, by means of the dynamic exchange-correlation (xc) kernel f_{xc}(r,r^{'},ω), include the electronic viscosity contribution. Starting from the ground state at the equilibrium internuclear distance and applying a current pulse, we observe three phases of the motion: (i)acceleration due to the prevalence of the current-induced force, (ii)stabilization upon balancing of the two forces, and (iii)deceleration due to the friction after the end of the pulse. At lower, but still metallic, electron densities, the dynamic xc contribution to the force significantly affects the acceleration (deceleration) at the first (third) phase of the process. For the Cs density (r_{s}≈6 a.u.), this correction amounts up to 40% in the rotation regime.

Massless spin 2 field in 50-component approach: exact solutions with cylindrical symmetry, eliminating the guage degrees of freedom

We begin with some known results of the 50-component theory for a spin-2 field described in cylindrical coordinates. This theory is based on the use of a 2nd-rank symmetric tensor and a 3rd-rank tensor symmetric in two indices. In the massive case, this theory describes a spin-2 particle with an anomalous magnetic moment. According to the Fedorov – Gronskiy method, which is based on projective operators, all 50 functions involved in the description of the spin-2 field for the case of the free particle can be expressed in terms of only 7 different functions constructed from Bessel functions. This leads to a homogeneous system of linear algebraic equations for 50 numerical parameters. We have found 6 independent solutions to these equations. Additionally, we have obtained explicit expressions for 4 guage solutions defined in accordance with the Pauli – Fierz approach. These solutions are exact and correspond to non-physical states that do not affect observable quantities, such as the energy-momentum tensor. Finally, we have constructed two classes of solutions that represent physically observable states.

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

Construction of compositional geological models using the variational grid method of geomapping and the object-hierarchical approach

The variational grid method of geomapping is based on approximation by bicubic splines, the permissible grid size of which is limited by the amount of computer RAM. The paper describes an approach to solving problems of constructing maps of geological parameters, using this method, when computer resources are insufficient. The approach is based on the sequential mapping for a set of fragments of the entire mapping area. The implementation of the approach is considered for two options. The first, if the data density is sufficiently high, consists of sequential calculations over partially overlapping “bands”. If the data distribution is significantly uneven, the second option is implemented, which consists in calculating a detailed grid for local areas and smoothly pasting it into the general map. Smoothness at the boundaries of bands and insets is ensured through the condition of equality of spline coefficients for coinciding nodes, which is set by adjusting the system of linear algebraic equations when solving the mapping problem. Based on these approaches, a multiscale structural-geological model of the sedimentary cover over the territory of Western Siberia was implemented. The formation of a compositional model is carried out on the basis of an object-hierarchical approach, which ensures consistency in the calculations of all its elements, as well as automation of constructions.

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .

ОБЩИЙ ПОДХОД К РЕШЕНИЮ ЗАДАЧИ ИЗГИБА ПРЯМОУГОЛЬНОЙ ПЛАСТИНКИ С ЛЮБЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИ

In the work under consideration, an approach is proposed that makes it possible to calculate rectangular plates with any continuous types of boundary conditions from a unified position. The solution for plate deflections is constructed as a product of two functions, each of which represents its own function of the differential equation of bending vibrations of the rod with boundary conditions corresponding to the support of the rectangular plate. The problem reduces to an infinite system of linear algebraic equations for the coefficients of the eigenfunctions, which can be solved by the truncation method. Two examples of the calculation of rectangular plates with rigidly clamped and hingedly supported sides under the action of a concentrated force and with two clamped and two hingedly supported sides under the action of a uniformly distributed load are given.

Orthonormal discrete Legendre polynomials for stochastic distributed‐order time‐fractional fourth‐order delay sub‐diffusion equation

In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples.

Явные решения в проблеме звукоизоляции с использованием многослойных структур при нормальном прохождении волн

The problem of sound isolation at ultra-low frequencies (30–200 Hz) by using multilayered structures is con-sidered, taking into account the normal wave propagation and an arbitrary number of layers. It is assumed that the multilayered structure consists of parallel acoustic layers, connected to each other along vertical boundary lines; acoustic half-spaces (air) are located to the left and right of the structure. For the problem, we define the acoustic parameters of the structure. The wave equations of the harmonic process are described for the left and right half-spaces of the structure, using the Helmholtz equation and a system of linear algebraic equations with given boundary conditions. Besides, explicit analytical solutions are obtained to find the transmission coefficient (transparency coefficient), with an arbitrary number of layers. Examples of solution for low-frequency sound absorption for building structures with different numbers of layers of alternating building materials and air spaces are considered. The effectiveness of using mixed structures, in which air layers are sandwiched between high impedance layers, for low-frequency sound absorption has been confirmed by assessing the level of noise reduction.

Solution of the foam-drainage equation with cubic B-spline hybrid approach

This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2, and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.

Joint application of the Monte Carlo method and computational probabilistic analysis in problems of numerical modeling with data uncertainties

Abstract In this paper, we suggest joint application of computational probabilistic analysis and the Monte Carlo method for numerical stochastic modeling problems. We use all the capabilities of computational probabilistic analysis while maintaining all the advantages of the Monte Carlo method. Our approach allows us to efficiently implement a computational hybrid scheme. In this way, we reduce the computation time and present the results in the form of distributions. The crucial new points of our method are arithmetic operations on probability density functions and procedures for constructing on the probabilistic extensions. Relying on specific numerical examples of solving systems of linear algebraic equations with random coefficients, we present the advantages of our approach.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

On Constructing Magnetic and Gravity Images of Mercury from Satellite Data

A new technique for simultaneous reconstruction of “gravity” and “magnetic” images of Mercury from satellite data based on the regional version of S-approximations is proposed. The mathematical statement of the inverse problem on finding the images of a planet from the data on the potential fields recorded at different times with different accuracy is reduced to solving ill-conditioned systems of linear algebraic equations (LAES) with approximate right-hand sides. Based on the analytical approximations of the Mercury’s magnetic and gravity fields determined from the solution of the ill-conditioned SLAE, the distributions of the equivalen sources on the spheres are determined. The results of the mathematical experiment on constructing the magnetic image of Mercury from the radial component of the magnetic induction vector analytically continued towards the field sources are presented.