System Of Nonlinear Algebraic Equations Research Articles

Advancements in numerical modeling of ideal multicomponent isotope separation Cascades: A comparative study

The complexity of operational conditions in stable isotope separation cascades necessitates the utilization of numerical methods for cascade design and performance analysis. This study focuses on solving a system of nonlinear algebraic equations derived from the mass and energy balance equations, which is a fundamental challenge in analyzing and designing cascades and developing numerical methods. Two existing numerical algorithms were selected from the literature and modified for quasi-ideal cascade systems. The performance of these algorithms was examined and compared, with a particular emphasis on influential parameters such as the number of stages, the number of isotopes, and the separation factor. The first algorithm solves the concentration and flow equations separately, while the second algorithm solves all equations simultaneously. The results indicate that the first algorithm exhibits significant superiority over the second algorithm. In most cases, the second algorithm either fails to converge or exhibits a slower convergence speed. Additionally, the results demonstrate that increasing the number of stages impacts the convergence time, while an increase in the separation factor leads to greater convergence complexity.

Numerical study of variable order model arising in chemical processes using operational matrix and collocation method

This article introduces the fractional variable order (VO) Gray-Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray-Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Simulating variable‐order fractional Brownian motion and solving nonlinear stochastic differential equations

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for . Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

On the parallel solution of hydro-mechanical problems with fracture networks and contact conditions

The paper presents a numerical method for simulating flow and mechanics in fractured rock. The governing equations that couple the effects in the rock mass and in the fractures are obtained using the discrete fracture-matrix approach. The fracture flow is driven by the cubic law, and the contact conditions prevent fractures from self-penetration. A stable finite element discretization is proposed for the displacement-pressure-flux formulation. The resulting nonlinear algebraic system of equations and inequalities is decoupled using a robust iterative splitting into the linearized flow subproblem, and the quadratic programming problem for the mechanical part. The non-penetration conditions are solved by means of dualization and an optimal quadratic programming algorithm. The capability of the numerical scheme is demonstrated on a benchmark problem for tunnel excavation with hundreds of fractures in 3D. The paper's novelty consists in a combination of three crucial ingredients: (i) application of discrete fracture-matrix approach to poroelasticity, (ii) robust iterative splitting of resulting nonlinear algebraic system working for real-world 3D problems, and (iii) efficient solution of its mechanical quadratic programming part with a large number of fractures in mutual contact by means of own solvers implemented into an in-house software library.

A New Iterative Broyden Legendre Wavelet Galerkin FEM Applied to Unsteady State Model of Two-Dimensional Elliptic Fin

Abstract The novelty of this paper is the investigation of numerical study of a mathematical model which deals with time dependent heat flow in elliptic fin ( dry, wet and partially wet). In this paper, we developed a nonlinear model of second-order heat equations in unsteady state condition. A new iterative Broyden Legendre Wavelet Galerkin Finite Element Method is used for the solution. The central difference approximation used for discretization of second order derivatives and then utilization of Hadamard, Khatri Rao and Face splitting matrices product with Legendre Wavelet Galerkin Method transfers our main problem in to system of non-linear algebraic equations. The iterative Broyden Method provides the solution for this system. In a particular case, present solution is compared with the exact solution and is approximately the same. Effect of different parameters such as Biot number, Latent heat, Kirpichev number, Fin thickness, μ, η and ξ on the temperature distribution are discussed in detail.

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.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

Finite element simulations of double diffusion in a staggered cavity filled with a power-law fluid

Double diffusion refers to a phenomenon where two different components of a fluid (such as heat and mass) exhibit distinct diffusive behaviors. In this study, we employ finite element-based numerical simulations to investigate the phenomenon of double diffusion in a non-Newtonian fluid within a staggered cavity. Mathematically, this system can be understood by coupling the two-dimensional continuity, momentum, energy, and concentration equations. Since the governing equations have been written in a dimensionless form, Galerkin's finite element method is used to find a solution. The velocity profile and temperature are calculated in a function space of quadratic polynomials (P2), while the pressure is calculated in a linear (P1) finite element function space. Discrete systems of nonlinear algebraic equations are resolved through the implementation of Newton's method with appropriate damping and PARDISO solver in the inner loops for solving the sparse linear systems. In this work, the data are presented graphically in the form of streamlines, isotherms, iso-concentration, average Nusselt numbers, average Sherwood numbers, and kinetic energy distribution. Code validation and grid independence study are also provided. Moreover, convective mass transfer is significantly correlated with the Lewis number, as demonstrated by the results. As the power-law index increases, convection also exhibits enhanced as a means of transmitting heat and mass.

Mapped Gegenbauer functions for solving Hammerstein generalized integral equations with Green’s kernels on the whole line

This paper introduces a collocation method for numerically solving Hammerstein generalized integral equations on the whole line. The approach utilizes mapped Gegenbauer functions to transform the Hammerstein integral equation into a system of nonlinear algebraic equations. Additionally, convergence results are obtained using the standard L2-norm. Finally, the effectiveness of the proposed method is demonstrated through several numerical experiments, which also serve to illustrate our theoretical findings.

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.

A radial basis function partition of unity method for steady flow simulations

A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles—rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities.

A study on Chlamydia transmission in United States through the Haar wavelet technique

In this study, a novel method known as the Haar wavelet collocation method (HWCM) is used to analyze the mathematical model of Chlamydia transmission in the United States. We use dependent variables with various parameters, such as birth rate, mortality rate, and recovery rate, etc., to explore the nature of the Chlamydia disease in susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals. The ordinary differential equations (ODEs) in this model are coupled nonlinear. In this method, the Chlamydia model is converted into a system of non-linear algebraic equations using properties of the operational matrix of Haar wavelets. Later, the Newton–Raphson approach is used to extract the unknown coefficients. Mathematica software has been used for all calculations. Tables and graphs contain tabulated results of calculations. As a result, it can be seen that the current approach is more precise than those used in the literature. Also, we discuss the effect of different parameters on susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals through graphs.

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 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.

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.

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.

Reliable analysis for obtaining exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation

<abstract><p>The (2+1)-dimensional Chaffee-Infante equation (CIE) is a significant model of the ion-acoustic waves in plasma. The primary objective of this paper was to establish and examine closed-form soliton solutions to the CIE using the modified extended direct algebraic method (m-EDAM), a mathematical technique. By using a variable transformation to convert CIE into a nonlinear ordinary differential equation (NODE), which was then reduced to a system of nonlinear algebraic equations with the assumption of a closed-form solution, the strategic m-EDAM was implemented. When the resulting problem was solved using the Maple tool, many soliton solutions in the shapes of rational, exponential, trigonometric, and hyperbolic functions were produced. By using illustrated 3D and density plots to evaluate several soliton solutions for the provided definite values of the parameters, it was possible to determine if the soliton solutions produced for CIE are cuspon or kink solitons. Additionally, it has been shown that the m-EDAM is a robust, useful, and user-friendly instrument that provides extra generic wave solutions for nonlinear models in mathematical physics and engineering.</p></abstract>

A reliable numerical algorithm based on an operational matrix method for treatment of a fractional order computer virus model

<abstract><p>A computer network can detect potential viruses through the use of kill signals, thereby minimizing the risk of virus propagation. In the realm of computer security and defensive strategies, computer viruses play a significant role. Understanding of their spread and extension is a crucial component. To address this issue of computer virus spread, we employ a fractional epidemiological SIRA model by utilizing the Caputo derivative. To solve the fractional-order computer virus model, we employ a computational technique known as the Jacobi collocation operational matrix method. This operational matrix transforms the problem of arbitrary order into a system of nonlinear algebraic equations. To analyze this system of arbitrary order, we derive an approximate solution for the fractional computer virus model, also considering the Vieta Lucas polynomials. Numerical simulations are performed and graphical representations are provided to illustrate the impact of order of the fractional derivative on different profiles.</p></abstract>