System Of Nonlinear Algebraic Equations Research Articles

A Computational Method Based on Shifted Jacobi Polynomials for the Fractional Differential Equations

In this paper, we present a co-local method based on shifted Jacobi polynomials to obtain approximate solutions to higher-order differential equations. To develop the desired approach by considering the expansion of the fractional problem solution using Jacobi polynomials and using derivative matrices, the solution of the fractional problem under consideration is transformed into a system of nonlinear algebraic equations. By solving the resulting system and obtaining the unknown expansion coefficients, an approximate solution to the original problem will be obtained. The accuracy of the proposed method was evaluated through various types of tested problems. Numerical results confirm the high accuracy of the proposed method in estimating the solution of the given problems.

Müntz–Legendre Wavelet Collocation Method for Solving Fractional Riccati Equation

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance.

An improved dual mesh control domain formulation for the unsteady flow of viscous incompressible fluids

The dual mesh control domain method (DMCDM) is applied, in conjunction with the reduced integration penalty formulation, to solve transient isochoric flow problems of Newtonian fluids, particularly at high Reynolds numbers. The DMCDM is a physically more intuitive method in that the global conservation laws are directly used (akin to the finite volume method), as opposed to the finite element method (FEM), where the weighted integral form of the governing equations is satisfied—an approach that bears little or no physical meaning in the context of fluid flow. However, the DMCDM exploits ideas from the FEM related to interpolation, duality, assembly, and numerical quadrature, allowing for the robust and efficient implementation of the former. In this paper, an element-based, parallelizable formulation for unsteady isochoric flows is investigated, in which the Navier–Stokes equations are semi-discretized using the DMCDM, achieving pressure–velocity coupling through the quadratic penalty method. Unexplored previously in the DMCDM literature, time-dependent problems are addressed, with the time dependence resolved using the alpha-family of approximations to arrive at a fully discretized system of nonlinear algebraic equations. This paper provides insights into implementation aspects, including numerical quadrature and the linearization of the system of equations using Newton's method—both of which have not been presented in detail previously. The ability of the method to capture fine-scale flow features, such as the formation of sub-eddies, is assessed using the lid-driven cavity problem at a high Reynolds number of 10 000. Additionally, the outflow boundary condition is examined using the flow over a backward-facing step problem, and finally, the method's ability to resolve instabilities is tested using the flow past a cylinder problem.

Convergence analysis of a novel fractional product integration–spectral collocation scheme for nonlinear weakly singular Volterra integro–differential systems

This paper intends to present a novel fractional basis function derived from Lagrange polynomials and to establish a new interpolation formula for approximating solutions of nonlinear fractional weakly singular Volterra integro–differential systems (VIDS) with Caputo fractional derivative. These systems are solved using a comprehensive approach that includes product integration, Nyström, and spectral collocation methods founded on sixth–kind Chebyshev polynomials. Utilizing a helpful connection between the Riemann–Lioville fractional integral operator and the Caputo derivative, weakly singular integro–differential equations and initial conditions are converted into an equivalent system of nonlinear algebraic equations. Approximate solutions and numerical derivatives of solutions are determined together using Newton's iterative method. The proposed scheme's convergence is examined, and an upper error bound in the L ∞ –norm and weighted L 2 –norm is acquired. Finally, the suggested approach is thoroughly evaluated for its simplicity and accuracy by including numerical examples.

Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits

Abstract In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage.

Fractional optimal control of systems using Bernoulli wavelets

In order to provide a new method for partial optimal control of systems, this work uses Bernoulli wavelets and uses Caputo's formula to solve partial optimal control problems (FOCPs) with inequality constraints. For the new optimization technique, the work uses partial order Bernoulli wave functions (F-BWFs) as basic functions. The answer using this method is expressed in terms of F-BWF where the coefficients have not yet been found. To simplify FOCPs in a system of nonlinear algebraic equations the procedure entails transforming inequality constraints into equality requirements by using operational matrices of fractional integration and F-BWFs and using the multipliers technique Lagrange. Numerical examples confirm the validity of the proposed strategy and demonstrate its correctness and efficiency when compared to the analytical or approximate answers provided by other methods.

Application of fractional shifted vieta-fibonacci polynomials in nonlinear reaction diffusion equation with variable order time-space fractional derivative

Abstract In this article, an accurate optimization algorithm based on new polynomials namely generalized shifted Vieta-Fibonacci polynomials (GSVFPs) is employed to solve the nonlinear variable order time-space fractional reaction diffusion equation (NVOTSFRDE). The algorithm combines GSVFPs, new variable order fractional operational matrices in the Caputo sense, and the Lagrange multipliers to achieve the optimal solution. First, the solution of the NVOTSFRDE is approximated as a series of GSVFPs with unknown coefficients and parameters. Then, the Lagrange multipliers method is adopted so that the NVOTSFRDE can be transformed into a class of nonlinear algebraic system of equations and we solve these equations using MATLAB and MAPLE software. Solving this system and substituting the coefficients and parameters into the approximation of the guessed functions, the solution of the NVOTSFRDE is obtained. The convergence analysis of the approach are discussed. The accuracy of the algorithm is verified through error analysis and mathematical examples. The accuracy of the new method is higher than that of the exciting method. The reconstruction results demonstrate that the proposed optimization algorithm is efficient for the NVOTSFRDE, and the algorithm is also convergent.

A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials

This paper proposes a numerical algorithm for the nonlinear fifth-order Korteweg–de Vries equations. This class of equations is known for its significance in modeling various complex wave phenomena in physics and engineering. The approximate solutions are expressed in terms of certain shifted Horadam polynomials. A theoretical background for these polynomials is first introduced. The derivatives of these polynomials and their operational metrics of derivatives are established to tackle the problem using the typical collocation method to transform the nonlinear fifth-order Korteweg–de Vries equation governed by its underlying conditions into a system of nonlinear algebraic equations, thereby obtaining the approximate solutions. This paper also includes a rigorous convergence analysis of the proposed shifted Horadam expansion. To validate the proposed method, we present several numerical examples illustrating its accuracy and effectiveness.

ELECTROOSMOTIC FLOW OF MICROPOLAR NANOFLUID THROUGH A NON-DARCY POROUS MEDIA WITH VON NEUMANN STABILITY CONDITION

This study investigates the action of time-periodic temperature and nanoparticle concentration divergence on electroosmotic micropolar Eyring-Powell nanofluid flow through a non-Darcy porous medium and over an infinite vertical plate. The effects of viscous and Ohmic dissipation, heat source, thermal radiation, Dufour trait, and chemical reaction are considered. The complicated system of differential equations which governs the problem is transformed into a system of nonlinear algebraic equations by using the finite difference method. Numerical results for the velocity, microrotation velocity, temperature, and nanoparticle concentration distributions, as well as the skin friction, reduced Nusselt number, and Sherwood number, are obtained. It is noted that the velocity becomes greater with an escalating Helmholtz-Smoluchowski velocity. Meanwhile, it elevates when rising in chemical reaction order. The enriching in thermophoresis parameter causes a dwindling influence on the nanoparticle concentration while also causing an increase in temperature. This study is significant in many diverse medical implementations as nanoparticles are utilized in the treatment of cancerous tumors.

Balancing Polynomial for Solution Nonlinear Stochastic Itô–Volterra Integral Equations

ABSTRACTThis paper introduces an innovative method for solving nonlinear stochastic Itô–Volterra integral equations using balancing polynomials and their associated operational matrices. This approach effectively transforms these complex stochastic equations into a system of nonlinear algebraic equations that can be solved using the Newton method. Balancing polynomials are chosen for their ability to enhance stability and convergence, providing a more reliable and manageable framework for tackling challenging stochastic problems. The paper also includes a convergence analysis and error estimation for the proposed method. Additionally, the effectiveness of this approach is demonstrated through four numerical examples. The results obtained from this method are compared with the exact solution and those from other established techniques, including the block‐pulse function method, the shifted Jacobi operational matrix (SJOM) method, and the shifted Jacobi spectral Galerkin (SJSG) method. These comparisons highlight the superior performance and accuracy of the proposed method. All numerical computations were performed using MATLAB.

A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations

Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. Objective: In this paper, a numerical method is presented for solving VO-FDEs. The proposed method is based on Chelyshkov polynomials (CPs). Methods: WRMs are used to obtain approximate solutions of the governing differential equations. In addition, a new weight function based on Mittag-Leffler functions is proposed. The proposed method is applied to a group of linear and non-linear examples with initial and boundary conditions. Results: Acceptable results are obtained in most of the examples. In addition, the effect of polynomials (such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials) on the accuracy of the approximate solution is studied, and their effect was found to be minimal in most tests. The effect of the proposed weight function is also studied in comparison with the weight functions presented in WRMs, and it was found that it has a strong effect in most examples. Conclusions: The results obtained indicate that the proposed method is effective for solving equations of variable order.

On Large Amplitude Vibrations of the Softening Duffing Oscillator at Low Excitation Frequencies—Some Fundamental Considerations

The Duffing equation containing a cubic nonlinearity is probably the most popular example of a nonlinear oscillator. For its harmonically excited, slightly damped, and softening version, stationary large amplitude solutions at subcritical excitation frequencies are obtained when standard semi-analytical methods like Harmonic Balance or Perturbation Analysis are applied. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary when applying the Harmonic Balance or other semi-analytical methods to make them converge. Some of these solutions are observed to be asymptotically stable, while in most cases, they are unstable. The current paper aims to give a descriptive explanation for this behavior of the nose solutions, which is mainly related to the exact solution of the free undamped vibrations. Based on this, approximations of the nose solutions are calculated with a procedure combining properties of Perturbation Analysis and Harmonic Balance. Therein, the exact solution of the free undamped vibrations is taken as the zeroth approximation, while higher-order solution parts are calculated by balancing the harmonics, and the phase shift of the zeroth approximation is calculated by a residuum minimization. This method just requires the solution of a system of linear algebraic equations, while systems of nonlinear algebraic equations have to be solved in the case of directly applying Harmonic Balance.

Eloquent numerical approach for solving generalized time fractional convection-diffusion-reaction problems

Abstract This paper attempts to develop a spectral method based on derivatives of orthogonal polynomials to solve the time fractional convection–diffusion-reaction equations. The method utilizes derivatives of fractional order orthogonal functions to approximate derivatives involved in the fractional differential equations. Specifically, the derivatives of fractional order Legendre functions and fractional order Chebyshev functions are used to represent both integer and non-integer derivatives of the solutions. These derivative representations are achieved through the use of operational matrices, which are matrices that encode the operations performed on the polynomials. An attempt is made to derive the operational matrix of Vieta-Fibonacci-like polynomials and used to solve the fractional differential equations. The derived operational matrix provides a systematic way to manipulate and work with these polynomials, facilitating their application in various mathematical and engineering problems. By employing these operational matrices, the original fractional convection–diffusion-reaction equation is transformed into a system of linear or nonlinear algebraic equations. However, if the system is nonlinear, a Newton-like solver is applied, which is capable of handling nonlinear systems. The estimation of error bounds of numerical solutions is also given. The numerical experiments have been performed over a few test examples to validate the proposed numerical method. The use of fractional order functions highlights their ability to solve fractional differential equations with non-smooth solutions accurately.

A study for analysing predator–prey ecological system using Legendre polynomial

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB.

A New Operational Matrices Based-Technique for Fractional Integro Reaction-Diffusion Equation Involving Spatiotemporal Variable-Order Derivative

In this article, a novel generalized model of the nonlinear fractional integro-reaction-diffusion equation (NFIRDE) with spatiotemporal variable-order (SVO) is introduced, where the variable order derivatives are equipped with the Atangana–Baleanu–Caputo (ABC) sense. This model represents a great generalization of a significant type of NFIRDE and their applications. Moreover, A novel, efficient and fully spectral shifted Legendre tau technique is developed to solve the proposed model. Despite the difficulty of applying this mechanism to solve this type of equations, due to the presence of nonlinear terms and the SVO functions that appear in the traditional differential and integral operational matrices. We deduce some new operational matrices that play the fundamental role in facilitating the implementation of the tau method. These operational matrices represent the SVO ABC-derivative, the integro term within the model, as well as the vector multiplications with the space-time Kronecker product. As a result, the proposed model is restructured into a system of nonlinear algebraic equations, which simplifying the solving process. We illustrate our method’s effectiveness and validity with numerical examples with both smooth and non-smooth solutions. Our findings show that the proposed tau method delivers accurate results and exhibits non-local properties.

An efficient discrete Chebyshev polynomials strategy for tempered time fractional nonlinear Schrödinger problems

IntroductionAn interesting type of fractional derivatives that has received widespread attention in recent years is the tempered fractional derivatives. These fractional derivatives are a generalization of the well-known fractional derivatives, such as Caputo and Riemann–Liouville. In fact, these derivatives are obtained by multiplying the expressed fractional derivatives by an exponential factor. These fractional derivatives have an additional parameter called λ such that in the case of λ=0, the classical Caputo or Riemann–Liouville fractional derivative is obtained. ObjectivesEmploying the Caputo tempered fractional derivative to define time fractional nonlinear Schrödinger equation and a coupled system of nonlinear Schrödinger equations. Applying the orthonormal discrete Chebyshev polynomials (ODCPs) to solve these problems. For this purposes, the operational matrices of ordinary and tempered fractional derivatives of the ODCPs are obtained. MethodsBy representing the problem’s solutions in terms of the ODCPs (with some unknown coefficients) and exploiting the expressed operational matrices, along with the collocation strategy, two systems of nonlinear algebraic equations are derived. By solving these systems, the expressed coefficients, and subsequently the solution of the main fractional problems are obtained. ResultsSome numerical examples are investigated to acknowledge the high accuracy of the designed approaches. ConclusionThe tempered fractional derivative in the Caputo form is utilized to define the time fractional nonlinear Schrödinger equation and a coupled system of nonlinear Schrödinger equations. The ODCPs are used to design a numerical strategy for these problems. To this purpose, some operational matrices for these polynomials are obtained. In the designed procedures, the problem’s solution are obtained by solving an algebraic system of equations. These systems are obtained by approximating the solution with the ODCPs and employing the expressed matrix relationships, along with the collocation technique. Some examples are presented to check the validity of the developed algorithms. The reported results acknowledged the high accuracy of the designed schemes.

A novel hybrid method with convergence analysis for approximation of HTLV-I dynamics model

This paper presents a novel numerical approach for approximating the solution of the model describing the infection of CD4+T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CD4^+T$$\\end{document}-cells by the human T-cell lymphotropic virus I (HTLV-I).The proposed method utilizes the operational matrix along with spectral method to convert the fractional model into a system of nonlinear algebraic equations. The Levenberg-Marquardt algorithm efficiently solves these equations. The study includes theoretical convergence analysis and error bounds to establish the validity of the proposed method. Through several test problems, we demonstrate the effectiveness and accuracy of the approach. We compare its performance and reliability to other existing methods in the literature. The results indicate that the proposed method is a reliable and efficient approach for solving the model.

Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

The Extended Weierstrass Transformation Method for the Biswas–Arshed Equation with Beta Time Derivative

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions.

Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

PurposeThis study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems.Design/methodology/approachDemonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method.FindingsThe numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method.Originality/valueUnlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.