System Of Linear Algebraic Equations Research Articles

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

Собственные колебания упругой прямоугольной CFCF-пластинки при одноосном растяжении-сжатии ее плоскости

The influence of a uniaxial distributed load in the plane of a thin rectangular plate on the value of the first natural frequency of its oscillations has been studied. We considered a plate, two opposite faces of which are clamped, and the other two are free (CFCF-plate, C-clamped edge, F-free edge). The load is applied to the clamped edges. The main differential equation for the coordinate component of the deflection function and all the boundary conditions of the problem are fulfilled exactly using two hyperbolic-trigonometric series in two coordinates and an additional function depending on one variable x. The problem was reduced to an infinite system of linear algebraic equations with respect to one sequence of uncertain coefficients, containing as parameters the magnitude of the load and the frequency of oscillations. For a number of load values, the natural frequencies of oscillations were found by enumerating frequency values in combination with the method of successive approximations when solving a reduced system of linear algebraic equations. To ensure acceptable accuracy of calculations, the number of terms in the series (the size of the reduced system), the number of iterations and the number of significant digits in the mantissa when calculating non-trivial coefficients of the system were changed. A square plate was considered as an example. Based on the calculation results, a graph was constructed of the dependence of the first natural frequency of oscillations on the magnitude of tension-compression forces, which is a curve close to a parabola. Under Eulerian compressive load, the oscillations stop. The shapes of natural vibrations changed slightly and were similar to the shape of the curved surface of a plate under the action of a uniform transverse load. The purpose of this study is to create an effective algorithm for calculating the natural frequencies of oscillations of a CFCF-plate when the uniaxial tensile-compression load of its clamped faces changes.

A comprehensive study of subdivision collocation method for Burgers’ equation

This study explores the use of subdivision schemes to efficiently solve Burgers’ equation. Burgers’ equation is a fundamental fluid dynamics equation that describes the nonlinear behavior of fluid flow. This type of nonlinear equation is difficult to solve analytically, which makes the numerical solution an important tool. The subdivision collocation method (SCM) converts Burgers’ equation into a system of algebraic linear equations using the quasilinearization technique. The results of this study demonstrate that the proposed approach yields accurate numerical solutions for Burgers’ equation. Additionally, the subdivision approach is computationally efficient and requires fewer computational resources than existing numerical methods, making it a promising tool for solving Burgers’ equation in practical applications. Overall, this study provides valuable insights into the approximate solution of Burgers’ equation by implementing subdivision schemes.

Исследование отражения и прохождения волны через планарный слой кирального метаматериала, расположенного в прямоугольном волноводе с учетом материальной дисперсии

The work analyzes the diffraction of the fundamental wave of a rectangular waveguide on a planar layer of a chiral metamaterial located in a transverse plane. The mathematical model of the metamaterial takes into account its chirality and dispersion of material parameters. To connect the electromagnetic field vectors through the parameters of the chiral layer, two-sided approximate boundary conditions are used. Cross-polarization of the field is taken into account in the work. A system of linear algebraic equations for the reflection and transmission coefficients of the fundamental and cross-polarized waves was obtained in the work. It is shown that at the cutoff frequencies of the fundamental and cross-polarized waves are also excited in the waveguide. It is shown that at different levels of attenuation in the waveguide, situations arise when the degeneracy of resonant frequencies arises and disappears. It has been proven that the reflection and transmission of the fundamental wave at low attenuation values is of a frequency-selective resonant nature.

Mathematical Modeling of Elastically Deformed States of Thin Isotropic Plates Using Chebyshev Polynomials

Abstract. In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor ($RL$), Resistor-Capacitor ($RC$), Resonant (Inductor-Capacitor) ($LC$), and Resistor-Inductor-Capacitor ($RLC$) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models.

Mathematical simulation of transverse cut in the fiber of a composite reinforced with unidirectional fibers with linear cohesion cracks

In the reviewed paper, the case of a transverse displacement plane with two pairs of biperiodic (corresponding to the X and Y axes) cohesive cracks of unequal size, weakened by biperiodic circular holes, is considered. The circular holes are filled with reinforcing fibers and the surface is covered with a thin homogeneous non-metallic material of the same thickness. In this case, boundary issues between filler and coating, coating and matrix, and boundary issues along cohesion cracks are determined. For both cases, the solution to the problem is sought in the form of an analytical function with complex variables. According to the boundary conditions of the case, a system of unequal algebraic equations is found along the holes, and singular integral equations are constructed along the cohesion cracks. Currently, the singular integral equations are brought to the system of finite linear algebraic equations with the help of mathematical transformations. Both systems are solved together using the Gaussian method and the crack growth is determined using the stress intensity factor formulas at the crack tips. During the solution of the problem, the stress intensity coefficients at the end are found by the variation of the length of the cracks based on the radius of the circular holes.

Option pricing in the illiquid markets under the mixed fractional Brownian motion model

This paper deals with the option pricing in the illiquid markets under the mixed fractional geometric Brownian motion model with jump process. We propose a general long memory financial model, where its featuring parameters are related to demand and supply by showing also the existence of some restrictions on them. Moreover, by using the delta Hedging strategy and replicating portfolio, we obtain an integro partial differential equation (PIDE) for the option price which is solved by the spectral numerical method with suitable diagonal functions and an infinite series. In particular, by using some operational matrices and Gauss–Hermite quadrature rule, we derive a linear system of algebraic equations solved by a standard collocation method. Moreover, we study the existence and uniqueness of the solution of PIDE and prove the convergence of the numerical scheme. The applicability and efficiency of the collocation method are shown on some nontrivial numerical examples.

ON THE UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATIONS WITH PARAMETER

A linear boundary value problem for a differential equation with a parameter is investigated on a finite interval by the parameterization method. The studied boundary value problem with parameter is reduced to an equivalent multipoint boundary value problem with parameters by splitting the interval, introducing additional parameters at the points of splitting and new functions. The obtained equivalent boundary value problem contains Cauchy problems for ordinary differential equations with respect to new functions. By substituting the solution representation of the Cauchy problem into the boundary conditions and continuity conditions of the solution, a system of linear algebraic equations with respect to the introduced parameters is compiled. An algorithm for finding a solution to the boundary value problem with parameters is constructed. The formulation of the theorem on sufficient conditions of unique solvability of the boundary value problem with parameters is given. Sufficient conditions of its unique solvability are obtained in terms of the data of the original boundary value problem. An example showing the fulfillment of the conditions of the theorem is given.

Solving least-squares problems in directed networks: A distributed approach

In this paper, we introduce a distributed algorithm that is specifically designed to tackle the least-squares problem within a network system of linear algebraic equations. Our focus is on static directed multi-agent networks, where each agent possesses knowledge of a distinct subset of the linear equations. Furthermore, we examine a scenario where agents lack information about their “out-degrees” at any given time. By imposing the strong connectivity condition on the communication network, we establish that the local estimated solution of each agent exhibits exponential convergence towards the least-squares solution of the corresponding network system of linear algebraic equations.

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

A spectral iterative algorithm for solving constrained optimal control problems with nonquadratic functional

The numerical approximation of the solution to Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) in constrained optimal control problems (COCPs) with nonquadratic functional is studied in this investigation. Discretizing both of the space and time of HJB PDE by the Legendre collocation method, a nonlinear system of algebraic equations is obtained to find the expansion coefficients of the approximate solution. However, solving the system of nonlinear algebraic equations is generally relatively difficult and time-consuming. To improve the implementation of the method, we combine the Legendre collocation method with the policy iteration (PI) algorithm. This process leads to a sequence of linear systems of algebraic equations to find the expansion coefficients. The convergence results for the proposed method are provided. Moreover, it is shown that the PI algorithm for a class of constrained optimal control problems with odd nonquadratic functionals is mathematically equivalent to the quasi-Newton's iteration. Finally, the efficiency and accuracy of the method are demonstrated by solving a number of examples.

Application of flatlet oblique multiwavelets to solve the fractional stochastic integro‐differential equation using Galerkin method

This study introduces a new numerical approach named flatlet oblique multiwavelets (FOMW) to solve fractional‐order stochastic integro‐differential equation (FSI‐DE). The FOMW is used to create an operational matrix of the stochastic integral, which helps transform the FSI‐DE into a linear system of algebraic equations. This method requires only a small number of basis functions to achieve satisfactory results. The paper also includes a comprehensive discussion on error estimation and convergence analysis, which provides an upper bound of error for the proposed method. Furthermore, numerical experiments have been conducted and compared with other methods, demonstrating the superiority of FOMW in terms of accuracy and efficiency.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial

In this paper, we extend fractional-order derivative for the shifted Vieta-Lucas polynomial to generalized-fractional integro-differential equations involving non-local boundary conditions using Galerkin method as transformation technique and obtained N - \delta + 1 system of linear algebraic equations with \lambda i, i = 0, . . . , N unknowns, together with \delta non-local boundary conditions, we obtained (N + 1)- linear equations. The accuracy and effectiveness of the scheme was tested on some selected problems from the literature. Judging from the table of results and figures, we observed that the approximate solution corresponding to the problem that has exact solution in polynomial form gives a closed form solution while problem with non-polynomial exact solution gives better accuracy compared to the existing results.

Reconstruction techniques for complex potentials

An approach for solving a variety of inverse coefficient problems for the Sturm–Liouville equation −y″ + q(x)y = ρ2y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectrum Sturm–Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives

In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

Computational analysis of a class of singular nonlinear fractional multi-order heat conduction model of the human head

The subject of the article is devoted to the development of a matrix collocation technique based upon the combination of the fractional-order shifted Vieta–Lucas functions (FSVLFs) and the quasilinearization method (QLM) for the numerical evaluation of the fractional multi-order heat conduction model related to the human head with singularity and nonlinearity. The fractional operators are adopted in accordance with the Liouville–Caputo derivative. The quasilinearization method (QLM) is first utilized in order to defeat the inherent nonlinearity of the problem, which is converted to a family of linearized subequations. Afterward, we use the FSVLFs along with a set of collocation nodes as the zeros of these functions to reach a linear algebraic system of equations at each iteration. In the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} norm, the convergence analysis of the FSVLFs series solution is established. We especially assert that the expansion series form of FSVLFs is convergent in the infinity norm with order O(1K3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}(\\frac{1}{K^3})}$$\\end{document}, where K represents the number of FSVLFs used in approximating the unknown solution. Diverse computational experiments by running the presented combined QLM-FSVLFs are conducted using various fractional orders and nonlinearity parameters. The outcomes indicate that the QLM-FSVLFs produces efficient approximate solutions to the underlying model with high-order accuracy, especially near the singular point. Furthermore, the methodology of residual error functions is employed to measure the accuracy of the proposed hybrid algorithm. Comparisons with existing numerical models show the superiority of QLM-FSVLFs, which also is straightforward in implementation.

Shear buckling of ship structures rectangular elements

In the work to determine the spectrum of critical loads and the corresponding forms of a rectangular clamped panel (plate) buckling under the action of balanced tangential forces on its contour, the Bubnov-Galerkin method using polynomials in two coordinates is proposed. This problem of the ship skin element pure shift does not have an exact closed solution, and the known approximate solutions require an analysis of their accuracy and reliability. The aim of the work is to obtain and analyze analytical solutions using polynomials of various degrees. Approximating deflection functions satisfying all the boundary conditions of the problem are represented sequentially by polynomials of 10th, 12th, 14th, 16th and 18th degrees in two coordinates with undefined coefficients. The solution of the main differential equation of the problem is found approximately in the integral sense, as a result of which homogeneous systems of linear algebraic equations with respect to unknown coefficients of polynomials are obtained. These systems contain a shear load as a parameter. To obtain eigenvalues (critical loads), the determinants of the systems are equated to zero. Numerical results are obtained in the Maple analytical computing system. For each approximation (polynomial), a power equation with respect to the critical load, the solution of which is paired values differing in signs is obtained. The forms of buckling are oblique waves. For a ship skin square panel, the first form of buckling is a single bulge along the diagonal of the panel. The second form is obtained in the form of two bulges directed in opposite directions (symmetrically-antisymmetrically with respect to the diagonals), etc. The numerical results are compared with the results of other authors. It is established that with an increase in the number of the polynomial terms, the initial critical loads and forms of buckling are specified, first of all.