System Of Algebraic Equations Research Articles

The use of OFDM signals in pseudorandom tuning of the operating frequency technology (FHSS)

The paper investigates the ways of organizing FHSS communication system using OFDM signals under the conditions of the measured parameters of the channel used. It is assumed that "protective gaps" are not used in the structure of the group signal FHSS technology. A variant of the formation of the OFDM symbol structure with a "test combination" is considered, the knowledge of which allows the regularization method to solve the problem of estimating the samples of the impulse response of the communication channel with memory. In this case, estimates are obtained from solving a system of algebraic equations, the order of which is determined by the size of the channel memory. To solve the problem of demodulating the signal from the "working package" of the OFDM symbol, estimates of the envelope counts of this part of the OFDM symbol are formed, according to which decisions on the transmitted discrete symbols are recorded at the output of the FFT calculation unit. Data on computer modeling of the problems under consideration are presented, the results of which confirm the possibility of using the proposed solutions for special purpose networks.

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

Solving systems of homogeneous linear algebraic equations in pricing problems of an alternative financial model

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Application of holography method for the restoration of the Williams series near the crack tip

The article describes the processing of the results of a series of experiments performed by the interference-optical method of holographic interferometry (holographic photoelasticity) aimed at computing the amplitude coefficients of the M. Williams series constituting the stress and displacement fields at the crack tip for several cracked configurations. The main objective of this study is the experimental and computational determination of the coefficients of the M. Williams series for the stress, strain and displacement fields associated with the crack tip in an isotropic linearly elastic medium taking into account regular (non-singular) terms in the multiparameter Williams series. These coefficients are named generalized stress intensity factors. The method of holographic interferometry is shown to be a convenient and efficacious tool for reconstructing the stress field near the tip of the crack, because during the experiment it is possible to obtain two families of interference fringe patterns: absolute retardation fringes (isodromics) for vertical and horizontal polarizations. Experimental outcomes were thoroughly processed using the developed digital application allowing us to accumulate the isodromics orders and coordinates of points belonging to absolute retardations. In this work, absolute retardation fringes in a plate with a central horizontal crack and a crack inclined at different angles are obtained. For each type of experimental sample, the coefficients of the Williams series were calculated taking into account non-singular terms (in the representation of M. Williams, fifteen terms were preserved). A procedure for linearization of nonlinear algebraic equations following from the relations of Favre’s law is proposed. By solving the obtained overdeterministic system of linear algebraic equations, the generalized stress intensity factors (coefficients of the M. Williams series) are estimated. Conjointly, the finite element analysis of the specimens with same geometry was effectuated. The experimentally determined values of the Williams series are compared with the results of the finite element calculation of the stress-strain state performed in the SIMULIA Abaqus software.The results of the numerical and experimental studies were found to be quite consistent. It is lucidly shown that it is imperative to keep the higher order terms in the Williams series expansions for the fields associated with the crack tip.

A discrete model for force-based elasticity and plasticity

The article presents a mathematical model that simulates the elastic and plastic behaviour of discrete systems representing isotropic materials. The systems consist of one lattice of nodes connected by edges and a second lattice with nodes placed at the centres of the existing edges. The derivation is based on the assumption that the kinematics of the second lattice is induced by the kinematics of the first, and uses stored energies in edges of both lattices to derive a edge forces in the first lattice. This leads to a non-linear system of algebraic equations describing elasticity and plasticity in lattices. A numerical solution to the non-linear system is proposed by providing a matrix formulation necessary for software implementation. An illustrative example is given to justify the formulation and demonstrate the system behaviour.

A space-time spectral approximation for solving two dimensional variable-order fractional convection-diffusion equations with nonsmooth solutions

The study focuses on the numerical solutions of two-dimensional variable-order fractional convection-diffusion equations, which combine the principles of diffusion and convection to describe the movement of particles, energy, other physical quantities within a system. The numerical solution is obtained using shifted Legendre Gauss–Lobatto and shifted Chebyshev Gauss–Radau collocation techniques. The convection-diffusion equation is transformed into a system of algebraic equations utilizing shifted Chebyshev Gauss–Radau and shifted Legendre Gauss–Lobatto nodes. Additionally, numerical test examples are presented to demonstrate the method’s efficacy to handle nonsmooth solutions to the given problems.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

A mixed-method to numerical simulation of variable order stochastic advection diffusion equations

The study of stochastic problems is very important and there is an increasing demand for investigating the behavior of a number of sophisticated dynamical systems in different areas of science as well as in engineering and finance. In this paper, we present a method which is based on the collocation method (CM) and the operational matrix method (OMM) by using Hosoya polynomials (HPs). This mixed-method helps us to convert the stochastic problems into a system of algebraic equations to access a solution of such problems. One of the most important advantages of this method is the simulation of the noise terms at the collocation points. Finally, two types of the VO–SADEs are simulated and tested via the proposed technique to confirm the superiority and accuracy.

An Efficient Numerical Method Based on Bell Wavelets for Solving the Fractional Integro-Differential Equations with Weakly Singular Kernels

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples.

A projection method based on the piecewise Chebyshev cardinal functions for nonlinear stochastic ABC fractional integro‐differential equations

In this study, the Atangana–Baleanu fractional derivative in the Caputo type (as a kind of non‐local and non‐singular derivative) is used to define a new class of stochastic fractional integro‐differential equations. A projection method (more precisely, a Galerkin approach) based on the piecewise Chebyshev cardinal functions is developed to solve these stochastic fractional equations. To construct this method, the operational matrices of fractional and stochastic integrals of these basis functions are obtained and used in the established method. By approximating the solution of the problem with a finite expansion of the expressed basis functions (in which the expansion coefficients are unknown), a system of algebraic equations is obtained. By solving this system, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are obtained. The convergence analysis of the proposed method is investigated, theoretically and numerically. The accuracy of the established procedure is illustrated by solving several numerical examples.

Novel Approximate Solutions for Nonlinear Blasius Equations

The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

Numerical Integration of Highly Oscillatory Functions with and without Stationary Points

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems.

Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials

In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.

Precision and efficiency of an interpolation approach to weakly singular integral equations

PurposeThis study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.Design/methodology/approachThe authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.FindingsApproaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.Research limitations/implicationsThe convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.Practical implicationsThere is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.Social implicationsThis work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.Originality/valueTo the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

Optimization of passive acoustic bedload monitoring in rivers by signal inversion

Abstract. Recent studies have shown that hydrophone sensors can monitor bedload flux in rivers by measuring the self-generated noise (SGN) emitted by bedload particles when they impact the riverbed. However, experimental and theoretical studies have shown that the measured SGN depends not only on bedload flux intensity but also the propagation environment, which differs between rivers. Moreover, the SGN can propagate far from the acoustic source and be well measured at distant river positions without bedload transport. It has been shown that this dependency of the measured SGN data on the propagation environment can significantly affect the performance of monitoring bedload flux by hydrophone techniques. In this article, we propose an inversion model to solve the problem of the SGN propagation and integration effect. In this model, we assume that the riverbed acts as SGN source areas with intensity proportional to the local bedload flux. The inversion model locates the SGN sources and calculates their corresponding acoustic power by solving a system of linear algebraic equations, accounting for the actual measured cross-sectional acoustic power (acoustic mapping) and attenuation properties. We tested the model using data from measured bedload SGN profiles (acoustic mapping with a drift boat) and bedload flux profiles (direct sampling with an Elwha sampler) acquired during two field campaigns conducted in 2018 and 2021 on the Giffre river in the French Alps. Results confirm that the bedload flux measured at different verticals on the river cross-section correlates more with the inversed acoustic power than measured acoustic power. Moreover, it was possible to fit data from the two field campaigns with a common curve after inversion, which was not possible with the measured acoustic data. The results of the inversion model, compared to measured data, show the importance of considering the propagation effect when using the hydrophone technique and offer new perspectives for the calibration of bedload flux with SGN in rivers.

Substitutional Based Gauss-Seidel Method For Solving System of Linear Algebraic Equations

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

Romanovski-Jacobi spectral schemes for high-order differential equations

We develop direct solution techniques for solving high-order differential equations with constant coefficients using the spectral tau method. The spatial approximation is based on Romanovski-Jacobi polynomials {Rnα,β(x)}n=0N with α>−1, β<−2N−α−1, x∈(0,∞) and n is the polynomial degree. Then, a hybrid approach combining the Romanovski-Jacobi tau method with the Romanovski-Jacobi collocation technique is introduced for the numerical solution of high-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces the problems to those of solving systems of algebraic equations, which greatly simplifies the problem. Finally, the Romanovski-Jacobi collocation method is presented for solving nonlinear high-order initial value problems. Numerical examples are included to demonstrate the validity and applicability of the techniques, and comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

ЕКВІВАВАЛЕНТНІСТЬ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ ТА ФУНКЦІЇ ГРІНА ДЛЯ ПОБУДОВИ ТОЧНОГО СКІНЧЕННОВИМІРНОГО АНАЛОГА КРАЙОВОЇ ЗАДАЧІ ДЛЯ ЗВИЧАЙНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ 4-ГО ПОРЯДКУ

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines, formed by the fundamental system of solutions of the Cauchy problems for the homogeneous equation. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes which are formed by partial sums of series in even powers of the grid step for solving the Cauchy problems. Keywords: boundary-value problem, ordinary differential equation of the 4th order, Cauchy problem, Wronskian, local spline, superposition of solutions, exact discrete analog, system of linear algebraic equations, 5-diagonal matrix.

A Novel Shifted Jacobi Operational Matrix Method for Linear Multi-terms Delay Differential Equations of Fractional Variable-order with Periodic and Anti-periodic Conditions

This paper investigates the generalized linear multi-terms delay fractional differential equation of variable order with periodic and anti-periodic conditions. In this work, a novel shifted Jacobi operational matrix technique is applied to solve a class of these equations, so that the original problem becomes a system of algebraic equations that can be solved by numerical methods. The proposed technique is successfully applied to the aforementioned problem. Sufficient and complete numerical tests are presented to demonstrate the accuracy, generality, efficiency of presented technique and the flexibility of this scheme. The numerical results of this method are compared with other existing methods such as fractional backward differential formulas (FBDF). Comparing the outcomes of these schemes, as well as comparing the current technique (NSJOM) with the exact solution, demonstrates the efficiency and validity of this method. It should be noted that the implementation of current method is considered very easy and general for many numerical techniques. Furthermore, the error and its bound are estimated.

Kink soliton solution of integrable Kairat-X equation via two integration algorithms

&lt;p&gt;In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.&lt;/p&gt;