System Of Nonlinear Algebraic Equations Research Articles

Iterative compact finite difference method for the numerical study of fully wet porous fins with different profile shapes

This paper investigates the temperature distribution of porous fins with different profiles (rectangular, triangular, convex parabolic and concave parabolic profiles) under fully wet conditions. The porous fin material is Al and the tips are insulated. An iterative discrete method is employed to solve the nonlinear governing equations. For this purpose, the governing equations are converted into a sequence of linear differential equations by the quasi-linearization method. Then, the numerical solutions of these linear equations are estimated using the compact finite difference scheme. The main advantage of the method is that it does not require to solve the system of non-linear algebraic equations. Moreover, unlike the Padé standard finite difference method, there is no need to calculate the derivatives of the solution from the governing differential equation in order to obtain the finite difference formulas with higher accuracy. The convergence analysis of the method is studied in detail. The effect of geometric and thermophysical parameters on temperature distribution and fins' efficiency is discussed using the plotted graphs. Comparing our numerical solutions with those obtained by the least squares method and the Chebfun package in Matlab shows the accuracy and efficiency of the proposed method.

Wavelets based computational algorithms for multidimensional distributed order fractional differential equations with nonlinear source term

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme.

Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in L2. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses.

A novel Lagrangian formulation to construct relativistic rotating stars: towards its application to their evolution calculations

ABSTRACT We present a new formulation to construct numerically equilibrium configurations of rotating stars in general relativity. Having in mind the application to their quasi-static evolutions on a secular time-scale, we adopt a Lagrangian formulation of our own devising, in which we solve force-balance equations to seek for the positions of fluid elements corresponding to the grid points, instead of the ordinary Eulerian formulation. Unlike previous works in the literature, we do not employ the first integral of the Euler equation, which is not obtained analytically in general. We assign a mass, specific angular momentum and entropy to each fluid element in contrast to the previous Eulerian methods, in which the spatial distribution of the angular velocity or angular momentum is specified. These distributions are determined after the positions of all fluid elements (or grid points) are derived in our formulation. We solve the large system of algebraic non-linear equations that are obtained by discretizing the time-independent Euler and Einstein equations in the finite-element method by using our new multidimensional root-finding scheme, named the W4 method. To demonstrate the capability of our new formulation, we construct some rotational configurations, both barotropic and baroclinic. As toy models, we also solve three evolutionary sequences that mimic the cooling, mass-loss, and mass-accretion.

Numerical Simulation for a High-Dimensional Chaotic Lorenz System Based on Gegenbauer Wavelet Polynomials

We provide an effective simulation to investigate the solution behavior of nine-dimensional chaos for the fractional (Caputo-sense) Lorenz system using a new approximate technique of the spectral collocation method (SCM) depending on the properties of Gegenbauer wavelet polynomials (GWPs). This technique reduces the given problem to a non-linear system of algebraic equations. We satisfy the accuracy and efficiency of the proposed method by computing the residual error function. The numerical solutions obtained are compared with the results obtained by implementing the Runge–Kutta method of order four. The results show that the given procedure is an easily applied and efficient tool to simulate this model.

On the Theory of Methane Hydrate Decomposition in a One-Dimensional Model in Porous Sediments: Numerical Study

The purpose of this paper is to present a one-dimensional model that simulates the thermo-physical processes for methane hydrate decomposition in porous media. The mathematical model consists of equations for the conservation of energy, gas, and liquid as well as the thermodynamic equilibrium equation for temperature and pressure (P−T) in the hydrate stability region. The developed model is solved numerically by using the implicit finite difference technique on the grid system, which correctly describes the appearance of phase, latency, and boundary conditions. The Newton–Raphson method was employed to solve a system of nonlinear algebraic equations after defining and preparing the Jacobean matrix. Additionally, the proposed model describes the decomposition of methane hydrate by thermal catalysis of the components that make up the medium through multiple phases in porous media. In addition, the effect of thermodynamic processes during the hydrate decomposition on the pore saturation rate with hydrates a7nd water during different time periods was studied in a one-dimensional model. Finally, in a one-dimensional model over various time intervals, t=1, 10, 50 s, the pressure and temperature distributions during the decomposition of methane hydrates are introduced and investigated. The obtained results include more accurate solutions and are consistent with previous models based on the analysis of simulations and system stability.

Development of Methods for Calculation the Load Angle of a Synchronous Generator Based on PMU Data

The development of the Wide Area Measurement Systems (WAMS) in the Unified Electric Power System (UES) of Russia and using phasor measurement technology (or PMU technology) open up new prospects in the electric power system (EPS) operational dispatch and automatic control. PMU devices installation on synchronous generators (SG) allows to obtain information not only about currents and stator voltages phasors, but also information about the excitation current and excitation voltage. This set of high-precision measurements of the network state parameters, in the presence of generators mathematical models, provides the possibility of calculating such parameters, the direct measurements of which are very difficult. One of these parameters, which can be determined using PMU, is the load angle of the SG (the angle between the generalized voltage phasor and the generator EMF phasor in the dq axis system) in the steady states and transients. The value of the load angle can be used to analyze the static and dynamic stability, as well as to quantify the damping properties of the SG at low-frequency oscillations (LFO) of the power systems electrical parameters. This study analyzes the possibilities for calculating the load angle based on a typical set of generator PMU signals: stator current and voltage vectors, excitation current and voltage. Two methods for calculating the load angle in steady states and transients have been developed and implemented. The first method is based on solving a system of nonlinear algebraic equations, and the basis of the second method is the mathematical model of the SG. The presented methods were tested using the SimInTech software package. Further, using the developed methods, the load angles of two turbogenerators installed at the plants of the Russian UES were calculated according to the PMU data of the parameters of the electromechanical transient process.

SOLUTION OF FRACTIONAL-ORDER REACTION-ADVECTION-DIFFUSION EQUATION ARISING IN POROUS MEDIA

In this article, the fractional-order nonlinear reaction-advection-diffusion equation describing contaminant transport in groundwater has been solved using shifted Legendre collocation method. The shifted Legendre polynomial is used to approximate the function. After that the operational matrix for fractional-order derivative in Caputo sense is applied on it. The shifted Legendre collocation points are employed to obtain a system of nonlinear algebraic equations which have been solved using Newton method. The application of the said methodology is demonstrated by applying it to two standard cases. The proposed method is validated by comparing the numerical results with those obtained using exact solutions through error analyses and the results are given in graphical as well as tabular forms. After the validation of its efficiency and effectiveness, the proposed numerical scheme is applied on a mathematical model related to porous media in a fractional order system. The salient feature of this article is the graphical exhibitions of the effects of fractional-order spatial and time derivatives, and also reaction and advection terms on the solution profile for different particular cases.

Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

<abstract><p>The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.</p></abstract>

Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

<abstract><p>The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.</p></abstract>

Analytical Study of Cubature Formulas on a Sphere in Computer Algebra Systems

The problem of finding the weights and nodes of cubature formulas of a given order on a unit sphere that are invariant under the icosahedral rotation groups (A.S. Popov’s problem) is studied analytically in computer algebra systems. Popov’s algorithm for reducing the problem to a system of nonlinear equations is implemented in the Sage computer algebra system. It is shown that, in Sage, difficulties with studying the resulting system of nonlinear algebraic equations arise starting from the order of approximation of 23. It is also shown that Popov’s problem of this order leads to a polynomial ideal whose Gröbner basis contains polynomials with extremely large integer coefficients, which makes it quite difficult to explore with the standard tools implemented in Sage. This basis was found in our computer algebra system GInv, the new version of which was made public by one of the authors of this article in 2021. This made it possible to fully describe the set of solutions of Popov’s problem in Sage. The exact solutions found in the article are compared with the solutions found numerically by Popov. The potential of using Popov’s problem as a test problem for systems specializing in computing the Gröbner basis is discussed.

A NOVEL NUMERICAL METHOD FOR SOLVING FUZZY VARIABLE-ORDER DIFFERENTIAL EQUATIONS WITH MITTAG-LEFFLER KERNELS

In the fuzzy calculus, the study of fuzzy differential equations (FDEs) created a proper setting to model real problems which contain vagueness or uncertainties factors. In this paper, we consider a class fuzzy differential equations (FFDEs) with non-integer or variable order (VO). The variable order derivative is defined in the Atangana–Baleanu–Caputo sense on fuzzy set-valued functions. The main problem under the fuzzy initial condition is converted to a new problem by the [Formula: see text]-cut representation of fuzzy-valued function. For solving the new problem, we use the operational matrices (OMs) based on the shifted Legendre polynomials (SLPs). By approximating the unknown function and its derivative in terms of the SLPs and substituting these approximations into the equation, the main problem is converted to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, an example is considered to confirm the accuracy of the proposed technique.

NUMERICAL SOLUTIONS OF NONLINEAR DELAY INTEGRO-DIFFERENTIAL EQUATIONS USING HAAR WAVELET COLLOCATION METHOD

In this paper, Haar wavelet collocation method (HWCM) for nonlinear delay Volterra, delay Fredholm and delay Volterra–Fredholm Integro-Differential Equations (IDEs) are studied numerically using HWCM. This method is very useful for solving nonlinear IDEs. The technique (HWCM) reduced the given equations into a system of nonlinear algebraic equations. The nonlinear system is then solved by Broydens technique. Some numerical examples are taken from literature for the validation purpose, computational efficiency and convergence of the proposed method. The approximate solution is compared with the exact solution and the maximum absolute and mean square root errors are presented for each example in tables. The comparison between exact and approximate solution is shown in figures for each example. The results are compared with existing methods from the literature. The results exhibit that the HWCM is simple, precise and efficient.

Особенности математического моделирования ГТД непрямой реакции

The article presents the developed mathematical model for calculating the parameters and characteristics of an indirect reaction gas turbine engine, which is implemented in asoftware environment and embedded in the general algorithm of the pro-gram "Calculation of traction-economic and mass-specific characteristics of a power plant and aircraft motion parameters". At the same time, special attention is paid to the features ofmathematical modeling of indirect reaction gas turbine engines with respect to turbojet engines. A description of a mathematical procedure for solving systems of nonlinear algebraic equations in determining the parameters and characteristics of a power plant in off-design modes of operation using a modified Newton's method is presented. The results of verification of the modified program are also shown by comparing the parameters and char-acteristics of the power plant based on the AI-24VT turboprop engine calculated with its help, with the characteristics taken from the technical description, with a detailed analysis of the qualitative and quantitative flow of throttle characteristics in accordance with the generally accepted theory of aviation engines

NEW APPROACH BASED ON COLLOCATION AND SHIFTED CHEBYSHEV POLYNOMIALS FOR A CLASS OF THREE-POINT SINGULAR BVPS

In the recent decades, variety of real-life problems arises in astrophysics have been mimic using the class of three-point singular boundary value problems (BVPs). Finding an effective and accurate approach for a class of three-point BVPs is still a difficult problem, though. The goal of this paper is to design a numerical strategy for approximating a class of three-point singular boundary value problems using the collocation technique and shifted Chebyshev polynomials. Utilizing shifted Chebyshev polynomials, the problem is reduced to a matrix form, which is then converted into a system of nonlinear algebraic equations by employing the collocation points. The key advantages of the new approach are (a) it is a straightforward mathematical formulation, which makes it effortless to code, and (b) it is easily adaptable to solve various classes of three-point singular boundary value problems. The convergence analysis is carried out to ensure the viability of the proposed scheme. Various examples are considered and tested in order to illustrate its applicability and efficiency. The results show excellent accuracy and efficiency compared to the other existing methods.

A NUMERICAL METHOD FOR TWO-DIMENSIONAL DISTRIBUTED-ORDER FRACTIONAL NONLINEAR SOBOLEV EQUATION

This study introduces the distributed-order fractional version of the nonlinear two-dimensional Sobolev equation. The orthonormal Chebyshev cardinal polynomials are used to construct a numerical method for this equation. To this end, some derivative matrices related to these polynomials are obtained. The proposed approach turns to solve this equation into solving a nonlinear system of algebraic equations by approximating the unknown solution using the expressed polynomials and employing their derivative matrices. The applicability and validity of this method are examined by solving three examples.

Pell-Lucas collocation method for solving a class of second order nonlinear differential equations with variable delays

In this study, the approximate solution of the nonlinear differential equation with variable delays is investigated by means of a collocation method based on the truncated Pell-Lucas series. In the first stage of the method, the assumed solution form (the truncated Pell-Lucas polynomial solution) is expressed in the matrix form of the standard bases. Next, the matrix forms of the necessary derivatives, the nonlinear terms, and the initial conditions are written. Then, with the help of the equally spaced collocation points and these matrix relations, the problem is reduced to a system of nonlinear algebraic equations. Finally, the obtained system is solved by using MATLAB. The solution of this system gives the coefficient matrix in the assumed solution form. Moreover, the error analysis is performed. Accordingly, two theorems about the upper limit of the errors and the error estimation are given and these theorems are proven. In addition, the residual improvement technique is presented. The presented methods are applied to three examples. The obtained results are displayed in tables and graphs. Also, the obtained results are compared with the results of other methods in the literature. All results in this study have been calculated by using MATLAB.

AN OPERATIONAL MATRIX APPROACH TO SOLVE A 2D VARIABLE-ORDER REACTION ADVECTION DIFFUSION EQUATION WITH VIETA–FIBONACCI POLYNOMIALS

A reaction-advection-diffusion equation describes many physical phenomena, such as the transportation of particles, groundwater pollution, viscoelasticity, and many others. In this study, a well-known fractional operator of variable order is used to present the space-time variable-order reaction-advection-diffusion equation. The operational matrix of the variable order derivative is developed with the aid of shifted Vieta-Fibonacci polynomials. This operational matrix is used in the approximation of derivatives of variable order to construct residual associated with the considered problem, and then it is collocated at some points in the domain, which generates a system of non-linear algebraic equations. Newton's method is applied to solve the obtained system of non-algebraic equations. To validate the precision of the proposed scheme, some problems are solved by the proposed scheme, and its comparisons are made with the existing analytical solution, which clearly indicates the improved accuracy of the proposed method. The convergence of the scheme and error analysis are also discussed in this paper.

Approximate solution of the nonlinear Fredholm equation of the second kind by the Bubnov-Galerkin method with bases in the form of Legendre wavelets

In this paper, a review of new works on approximate methods for solving nonlinear Fredholm integral equations of the second kind is carried out. Legendre wavelets are used for the numerical solution of nonlinear Fredholm integral equations of the second kind by the Galerkin-Bubnov projection method. The Newton-Kantorovich method for a system of nonlinear algebraic equations is considered and numerical calculations are carried out. The comparative analysis shows that the Bubnov-Galerkin method with basic functions in the form of Legendre wavelets is effective in terms of accuracy and convenient to implement.

Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ T cells model

In this study, we generated a novel functional matrix using Bernoulli wavelets. Also, we developed a novel technique called the Bernoulli wavelets collocation method to obtain reasonably accurate solutions for the HIV-infection model of CD4+ T cells. This mathematical model is in the form of a system of a nonlinear ordinary differential equation (ODE). This approach obtains the solution for this model by transforming it into a system of nonlinear algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. The collocation scheme is used to calculate these unknown coefficients. The consistency and proficiency of the developed approach are demonstrated through tables and graphs. Obtained results reveal that the current approach is more accurate than other methods in the literature. All computations have been made with the help of Mathematica software. Some properties of Bernoulli wavelets are discussed in terms of theorems.