System Of Algebraic Equations Research Articles

Modeling of suspended weightlessness on the cables of the beam system, by changing the tension forces

The problem of weightlessness simulation of beam systems suspended on inextensible cables is considered. Imitation of weightlessness means zeroing or reducing any selected force factor (for example, the reaction of the support or the moment in the support or joint), and the kinematic factor (deflection or angle of rotation). It is required to select the forces in the cables such that the sum of the squares of the deflections at the points of the elastic line of the beam is minimal. The problem is formulated as a nonlinear programming problem; the search for the minimum of the objective function with constraints, in the form of equilibrium equations, is carried out. In general, all equations written out for a geometrically variable system are linearly dependent. Parameters are selected from the system of equations, the vectors at which are entered into the basis, and the remaining parameters are considered free and are the coordinates of the objective function. The problem was reduced to the problem of quadratic programming without restrictions. Partial derivatives of coordinates give a system of linear algebraic equations that allows you to determine the coordinates taken as free parameters, and then calculate the coordinates entered into the basis. The reference plan of nonlinear optimization problems can have local minima; it is shown that for any initial basis, the optimal plan is the only one. To calculate the deflections of the beam, the method of initial parameters is used. Deflection, angle of rotation, additional angles of rotation in articulated joints are considered as initial parameters; as well as the reaction and bending moment. The continuum problem is transformed into a discrete one by limiting the number of points at which deflections are calculated. The objective function has a finite number of variables. It is determined which number of selected points on the elastic line of the beams is sufficient to ensure the convergence of the functions of deflections, angles of rotation, bending moments and transverse forces for the purpose of application to practical calculations. Optimization of deflections of a beam pivotally fixed, suspended on two cables with verification of solutions, change of basic variables and convergence study depending on the choice of the number of points at which deflections are calculated is performed. The deformation of systems of I-beams connected by hinges to each other, having linear weight in gravity conditions, is analyzed. To simulate weightlessness, the system is supported by six cables. The boundary conditions are considered: – rigid pinching; – hinge-fixed support, – sliding sealing; – free edge. Models of three-beam systems in the simulation of weightlessness, to a certain extent equivalents. The type of boundary condition affects the first beam to a greater extent; the tension forces of the cables equalize the deformed and stressed state in subsequent beams. Any of the considered systems with the presented boundary conditions can be converted into an equivalent one by changing the boundary force factors, setting torques or installing a spring with a given stiffness and adjusting the tension of the cables.

Weakly singular Volterra integral equation with combined logarithmic-power-law kernel: Analytical and computational consideration

In this article, we consider weakly singular Volterra integral equation (WSVIE) with the kernel combining the power-law and logarithmic singularitiesf(ζ)=Au(ζ)+∫0ζlog(ζ−η)u(η)(ζ−η)βdη,ζ∈[0,1],β∈[0,1), where, f(ζ) is a smooth function. Equations of this kind, introduced by V. Volterra himself within the framework of the theory of functional compositions, are also directly related to the modern problems of fractional dynamics. Our approach is based on employing the operational matrix technique with the shifted Legendre polynomials (SLP) as a basis. This approach reduces WSVIE to a system of linear algebraic equations with coefficients, which contain closed-form analytical expressions. This allows the use of computer algebra systems for obtaining approximate solutions based on the finite terms truncation of series. We provided a number of examples highlighting such a workflow in addition to establishing the error bound, convergence analysis and stability analysis.

Numerical Treatment for the Solution of Stochastic Fractional Differential Equation Using Lerch Operational Matrix Method

Abstract The article aims to propose the Lerch operational matrix method to solve a stochastic fractional differential equation. In this approach, the Lerch polynomials have been used as a basis function. Then, the product operational matrix, integral operational matrix, stochastic operational matrix, and operational matrix of fractional integral based on the Lerch polynomials have been constructed. The main characteristic of this method is to reduce the stochastic fractional differential equation into a system of algebraic equations by using derived operational matrices and suitable collocation points. Moreover, the convergence and error analysis of the presented method is also discussed in detail. Additionally, the applicability of the proposed technique is also demonstrated by solving some examples. To confirm the accuracy and effectiveness of the suggested technique, a comparison between the results produced by the proposed method and those obtained by other methods has been provided.

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

A novel method based on fractional order Gegenbauer wavelet operational matrix for the solutions of the multi-term time-fractional telegraph equation of distributed order

In this article, we propose an effective scheme based on a combination of the Tau method and fractional-order Gegenbauer wavelets for solving the multi-term time-fractional differential equations of distributed order. First, we define fractional order Gegenbauer wavelets and then obtain operational matrices of these orthogonal functions. Applying the Legendre–Gauss quadrature for the integral term, we use function approximations obtained by the presented wavelets and the Tau method for the solution of the distributed-order multi-term time-fractional telegraph equation. The proposed method reduces the numerical solution of multi order time-fractional equations to a system of algebraic equations. Then, the convergence analysis and error bounds of the proposed scheme are studied. Three illustrative examples are solved to justify the effectiveness of the proposed method compared with some previously published results.

Computational analysis of entropy generation in three-dimensional mixed convection flow with thermal dissipation and variable thermal conductivity

Entropy generation in steady three-dimensional thermal flow is investigated under the coupled influence of viscous dissipation and mixed convection. The stretching sheet with an exponential velocity profile induces the flow. In this examination, we presume that the fluid is incompressible and possesses a Newtonian behavior and temperature-dependent thermal conductivity. Using the appropriate classical similarity variables reduces the dimensional nonlinear dynamical system of governing equations to a dimensionless nonlinear dynamical set of ordinary differential equations. It’s important to note that the selection of an exponentially varying surface temperature is deliberate, as it enables precise similarity transformations to be established in situations involving both viscous dissipation and buoyancy forces. The simplified finite difference approach (SFDM) is used to solve the acquired nonlinear dynamical set of ordinary differential equations. The tridiagonal matrix approach is used to solve the resulting system of algebraic equations. In a particular situation, the present numerical simulation is numerically checked with the previously available results and observed a good agreement. The effects of physical parameters on velocity, temperature, surface stresses and heat tranfer rate, entropy generation, and the irreversibility parameter are investigated. The entropy generation rate exhibits a decreasing trend with respect to the mixed convection parameter in the vicinity of the stretching surface. Entropy reduction can be achieved by employing a working fluid with a low Prandtl number and by minimizing the fluid friction.

A New First Finite Class of Classical Orthogonal Polynomials Operational Matrices: An Application for Solving Fractional Differential Equations

In this paper, new operational matrices (OMs) of ordinary and fractional derivatives (FDs) of a first finite class of classical orthogonal polynomials (FFCOP) are introduced. Also, two algorithms are proposed for using the tau and collocation spectral methods (SPMs) to get new approximate solutions to the given fractional differential equations (FDEs). These algorithms convert the given FDEs subject to initial/boundary conditions (I/BCs) into linear or nonlinear systems of algebraic equations that can be solved using appropriate solvers. To demonstrate the robustness, efficiency, and accuracy of the proposed spectral solutions, several illustrative examples are presented. The obtained results show that the proposed algorithms exhibit higher accuracy compared to existing techniques in the literature. Furthermore, an error analysis is provided.

Исследование устойчивости химического реактора полимеризации этилена

The chemical reactor is the main and most complex object in the management of the technological process of polymerization. When designing chemical reactors, it is necessary to select design and operating parameters that will be able to provide the only or most stable state with the specified process characteristics. To study the stability of the ethylene polymerization process in an autoclave reactor with a stirrer, it is necessary to build a mathematical model based on the analysis of the physico-chemical laws of the process and search for the necessary steady-state conditions for the reactor. A solution based on a deterministic mathematical model of ethylene polymerization is proposed. The description of the process of polymer molecule formation, which includes the stages of initiation, growth and chain breakage; the mechanism of chemical kinetics of ethylene polymerization; a system of assumptions to simplify the construction of a mathematical model. The mathematical model of an ethylene polymerization reactor in an autoclave reactor with a stirrer is a system of differential equations. When calculating the static reactor mode, the mathematical description contains a system of nonlinear algebraic equations, the solution of which is implemented using numerical methods. It is noted that chemical reactors can have not one, but several stationary states at the same values of variables and parameters. Knowing the coordinates of the stationary states of the chemical process, you can choose the one that is more preferable from the point of view of productivity or the quality of the products produced. As a result of the simulation, the existence of a single solution under the conditions found that ensure a stable stationary state of the chemical ethylene polymerization reactor is confirmed.

On analyzing two dimensional fractional order brain tumor model based on orthonormal Bernoulli polynomials and Newton's method

Recently, modeling problems in various field of sciences and engineering with the help of fractional calculus has been welcomed by researchers. One of these interesting models is a brain tumor model. In this framework, a two dimensional expansion of the diffusion equation and glioma growth is considered. The analytical solution of this model is not an easy task, so in this study, a numerical approach based on the operational matrix of conventional orthonormal Bernoulli polynomials (OBPs) has been used to estimate the solution of this model. As an important advantage of the proposed method is to obtain the fractional derivative in matrix form, which makes calculations easier. Also, by using this technique, the problem under the study is converted into a system of nonlinear algebraic equations. This system is solved via Newton's method and the error analysis is presented. At the end to show the accuracy of the work, we have examined two examples and compared the numerical results with other works.

Transient analysis of a cracked functionally graded magneto‐electro‐elastic rectangular finite plane under anti‐plane mechanical and in‐plane electric and magnetic impacts

AbstractThe transient anti‐plane problem of a functionally graded magneto‐electro‐elastic (FGMEE) rectangular plane weakened by several embedded and edge cracks under various boundary conditions is solved. First, the solutions to the dynamic magneto‐electro‐elastic dislocation in the finite rectangular domain are derived by employing Laplace and finite Fourier cosine transforms. The solutions can be utilized to construct the singular integral equations in Laplace transform domain for FGMEE rectangular plane with multiple embedded and edge cracks. The integral equations arising from dislocation solutions have Cauchy‐type singularities with unknown dislocation density functions. These unknown functions may be obtained by reducing the integral equations into a system of linear algebraic equations, hereby, the dynamic stress intensity factors (DSIFs) for several arbitrarily embedded and edge cracks are obtained. The overshoots of dynamic stress intensity factors are shown to be intensely affected by the rectangular plane dimensions, length and position of the cracks, loading parameter, and negative or positive gradient index.

Orthonormal Chelyshkov polynomials for multi-term time fractional two-dimensional telegraph type equations

In this study, multi-term time fractional 2D telegraph type equations, as a new category of fractional differential equations, are introduced. The orthonormal Chelyshkov polynomials are used as basis functions to generate a collocation method for such equations. In this way, new results are obtained for the fractional and classical derivatives of these polynomials. In the proposed method, by representing the problem’s solution in terms of the orthonormal Chelyshkov polynomials (with unknown coefficients) and applying the extracted matrices, along with the collocation method, a solution is elicited for the problem under consideration. In fact, finding the solution of the initial problem turns into assigning the expansion coefficients through solving a system of algebraic equations. The accuracy of the proposed method is numerically checked with several examples.

SOLVING NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND WITH DISCONTINUOUS KERNELS BY USING THE OPERATIONAL MATRIX METHOD

A numerical method to solve the nonlinear Volterra integral equations of the first kind with discontinuous kernels is proposed. Usage of operational matrices for this kind of equation is a cost-efficient scheme. Shifted Legendre polynomials are applied for solving Volterra integral equations with discontinuous kernels by converting the equation to a system of nonlinear algebraic equations. The convergence analysis is given for the approximated solution and numerical examples are demonstrated to denote the precision of the proposed method.

Distributed Algorithms for Boolean Equations Over Networks

In this paper, we study systems of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private</i> equation known to this node only, and the aim of the paper is to develop distributed algorithms that allow all the nodes to obtain solutions to the network Boolean equations without exchanging their local Boolean equations. First, we observe that the Boolean equations can be locally lifted to a system of linear algebraic equations under a basis of Boolean vectors, which is distributedly solvable using existing distributed linear equation algorithms as a subroutine. Next, we construct a distributed Boolean equation solver by the nodes solving the lifted linear network equation for a number of randomly selected initial values, and then converting the algebraic solutions into solutions to the original Boolean equations by a novel Boolean vector search algorithm. We prove that for solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, such an algorithm returns the exact solution set of the Boolean equations at each node with high probability. We also present an algorithm for distributed verification of the satisfiability of Boolean equations when the solvability is not known beforehand, and prove its correctness. Finally, we show that by utilizing linear equation solvers with differential privacy to replace the in-network computing routines, the distributed Boolean equation algorithms can be made differentially private.

On Constructing Analytical Models of the Magnetic Field of Mercury from Satellite Data

Abstract—A new method is proposed for analytical description of the magnetic field of the Mercury from the data of satellite missions based on the local and regional versions of the linear integral representation method. The inverse problem on finding the sources of the field is reduced to solving ill-conditioned systems of linear algebraic equations with an approximately set right-hand part. The charts of the isolines of the z-component of the magnetic inductance vector in the Cartesian coordinate system rigidly connected with the planet, as well as the regional S-approximation of the field radial component are plotted. The results of the mathematical experiment on analytic continuation of the magnetic field towards the sources are presented.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

This research work deals with two spectral matrix collocation algorithms based on (novel) clique functions to solve two classes of nonlinear nonlocal elliptic two-points boundary value problems (BVPs) arising in diverse physical models. The nonlinearity together with nonlocality makes the models so difficult to solve numerically. In both matrix methods by expanding the unknown solutions in terms of clique polynomials the nonlocality in the models is handled. In the first and direct technique, the clique coefficients are determined in an accurate way through solving an algebraic system of nonlinear equations. To get rid of the nonlinearity of the models and in order to gain efficacy, the quasilinearization method (QLM) is utilized in the second approach. In the latter algorithm named QLM-clique procedure, the first directly clique collocation method is applied to a family of linearized equations in an iterative manner. In particular, we show that the series expansion of clique polynomials is convergent exponentially in a weighted L2 norm. Numerous numerical simulations validate the performance of two matrix collocation schemes and demonstrate the accuracy as well as the gain in computational efficiency in terms of elapsed CPU time. The proposed matrix algorithms for computation are easy to implement and straightforward, and provide more accuracy compared to other available computational values in the literature.

An efficient wavelet‐based approximation method for solving nonlinear fractional‐time long wave equations: An operational matrix approach

This research paper develops an efficient Gegenbauer wavelet‐based approximation method to solve nonlinear fractional‐time regularized long wave (RLW) equations arising in ocean engineering. The operational matrices of derivatives, as well as the fractional order integration, are engaged in the proposed method. Using the operational matrix method and collocation point, the nonlinear RLW equations are converted into a system of algebraic equations, and these equations are further solved. Some results regarding the error‐bound estimation of this method have been developed. The method's accuracy and efficiency are confirmed by error estimation. The obtained results are compared with other available results and exact solutions. Moreover, the use of Gegenbauer wavelets is found to be a simple, straightforward forward and efficient tool for solving nonlinear PDEs arising in ocean engineering.

Finite Element–Boundary Element Acoustic Backscattering with Model Reduction of Surface Pressure Based on Coherent Clusters

Computing backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shape is a problem of considerable practical significance. The finite element method is commonly applied to the discretization of the target; on the other hand, the boundary element method naturally incorporates the radiation boundary condition at infinity. The coupled model tends to be expensive, primarily due to the need to manipulate large, dense, and complex matrices and to repeatedly solve systems of complex linear algebraic equations of significant size for each frequency of interest. In this article, we develop a model reduction transformation based on the notion of coherence applied to the surface pressures, which considerably reduces the size of the systems to be solved. We found that the proposed model reduction approach delivers acceptably accurate results at a fraction of the cost of the full model. A typical speedup of an order of magnitude was realized in our numerical experiments. Our approach enables backscattering computations with considerably larger models than have been feasible to date.

A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

A numerical approach for an epidemic SIR model via Morgan-Voyce series

Abstract This study presents the problem of spreading non fatal disease in a population by using the Morgan-Voyce collocation method. The main aim of this paper is to find the exact solutions of the SIR model with vaccination. The problem may be modelled mathematically with a nonlinear system of ordinary differential equations. The presented method reduces the problem into a nonlinear algebraic system of equations by using unknown coefficient Morgan-Voyce polynomials and expanding approximate solutions. Morgan-Voyce polynomials are used. These unknown coefficients are calculated via the collocation method and matrix operation derivations. Two examples are given to show the feasibility of the method. To calculate the solutions, MATLAB R2021a is used. Additionally, comparing our method to the Homotopy perturbation method (HPM) and the Laplace Adomian decomposition method (LADM) proves the accuracy of the solution. The method studied can be seen as effective from these comparisons. So, it is essential to find solutions for the governing model. The study will contribute to literature since we also discuss the vaccination situation. The results of this study are valuable for controlling an epidemic.

A Comparative Study of Numerical Solution of Second-order Singular Differential Equations Using Bernoulli Wavelet Techniques

The main objective of this article is to discuss a numerical method for solving singular differential equations based on wavelets. Singular differential equations are first transformed into a system of linear algebraic equations, and then the linear system’s solution produces the unknown coefficients. Along with its estimated error, the convergence of the approximative solution is alsodetermined. Some numerical examples are thought to show that Bernoulli wavelet is better than Chebyshev and Legendre wavelet and other existing techniques.