Nonlinear Algebraic Equations Research Articles

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.

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method.

Planning Speed Mode of All-Wheel Drive Autonomous Vehicles Considering Complete Constraint Set

The study aims to improve the technique of motion planning for all-wheel drive (AWD) autonomous vehicles (AVs) by including torque vectoring (TV) models and extended physical constraints. Four schemes for realizing the TV drive were considered: with braking internal wheels, using a rear-axle sport differential (SD), with braking front internal wheel and rear-axle SD, and with SDs on both axles. The mathematical model combines 2.5D vehicle dynamics model and a simplified drivetrain model with the self-locking central differential. The inverse approach implies optimizing the distribution of kinematic parameters by imposing a set of constraints. The optimization procedure uses the sequential quadratic programming (SQP) technique for the nonlinear constrained minimization. The Gaussian N-point quadrature scheme provides numerical integration. The distribution of control parameters (torque, braking moments, SDs’ friction moment) is performed by evaluating linear and nonlinear algebraic equations inside of optimization. The technique proposed demonstrates an essential difference between forecasts built with a pure kinematic model and those considering the vehicle’s drive/control features. Therefore, this approach contributes to the predictive accuracy and widening model properties by increasing the number of references, including for actuators and mechanisms.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

A RELIABLE NUMERICAL ALGORITHM FOR TREATMENT OF FRACTIONAL MODEL OF CONVECTIVE STRAIGHT FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY

Fins serve the purpose of enhancing heat transfer between their surroundings and the underlying surface, resembling the function of heat exchangers in various scientific domains. This research undertakes an investigation into the temperature distribution within convective straight fins, considering the impact of both constant and thermally conductive temperature-dependent properties. The chosen approach utilizes a numerical technique known as the Laguerre operational matrix collocation method, which relies on the Laguerre operational matrix and Laguerre collocation method for Caputo fractional derivatives of arbitrary orders. This methodology effectively converts the issue into a set of nonlinear algebraic equations (NLAEs). To evaluate this arbitrary order model, we employed the collocation method to obtain an approximate solution. The numerical results obtained through this approach demonstrate its ease of application and computational efficiency. In addition, we performed a comparative analysis with the Homotopy Perturbation Sumudu Transform Method (HPSTM) and the Variational Iteration Method (VIM). The numerical and graphical presentation of the results underscores the efficacy and precision of the methodologies employed within the scope of this investigation.

A Frequency Domain Approach to Predict Power System Transients

The dynamics of power grids are governed by a large number of nonlinear differential and algebraic equations (DAEs). To safely operate the system, operators need to check that the states described by these DAEs stay within prescribed limits after various potential faults. However, current numerical solvers of DAEs are often too slow for real-time system operations. In addition, detailed system parameters are often not exactly known. Machine learning approaches have been proposed to reduce the computational efforts, but existing methods generally suffer from overfitting and failures to predict unstable behaviors. This paper proposes a novel framework to predict power system transients by learning in the frequency domain. The intuition is that although the system behavior is complex in the time domain, there are relatively few dominant modes in the frequency domain. Therefore, we learn to predict by constructing neural networks with Fourier transform and filtering layers. System topology and fault information are encoded by taking a multi-dimensional Fourier transform, allowing us to leverage the fact that the trajectories are sparse both in time and spatial frequencies. We show that the proposed approach does not need detailed system parameters, greatly speeds up prediction computations and is highly accurate for different fault types.

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>

ВОЛНЫ В СРЕДЕ ВЯЗКОСТНОЙ КВАРК-ГЛЮОННОЙ ПЛАЗМЫ

Since quark-gluon plasma was discovered, it has been called the new state of matter, prompting the beginning of the study of collective effects. Continuing this topic, the oscillation modes of internal waves in a quark-gluon plasma have been investigated. The medium of a quark-gluon plasma is considered an ideal fluid with a small viscosity. The viscosity is taken into account using a dielectric function with a viscosity parameter, which is related to the entropy, and this parameter can be chosen in accordance with different theoretical approaches to describing a quark-gluon plasma. The problem was solved in the long-wavelength limit, but this solution corresponds to the physical case described in the article. This approach reduces the problem to non-linear algebraic equations that were solved using software for numerical calculations. For illustration, the oscillation modes of a viscous quark-gluon plasma are compared with the oscillation modes of a collisional quark-gluon plasma - this will show the effect of the parameter of the dielectric function, which takes into account only viscosity. The solutions are presented in the form of graphs of frequency versus wavenumber, in which you can clearly see the oscillatory waves. Also, these vibrational modes can be used to judge the nature of the waves, and for physical interpretation.

Propagation patterns of dromion and other solitons in nonlinear Phi-Four ($ \phi^4 $) equation

<abstract><p>The Phi-Four (also embodied as $ \phi^4 $) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.</p></abstract>

A third-order numerical method for solving fractional ordinary differential equations

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

Analytical discovery of dark soliton lattices in (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation

<p>This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of $ (\frac{G'}{G}) $-expansion method were employed, namely, extended $ (\frac{G'}{G}) $-expansion method and the generalized $ (r+\frac{G'}{G}) $-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics.</p>

A compact, equality-based weighted residual formulation for periodic solutions of systems undergoing frictional occurrences.

A very compact weighted residual formulation is proposed for the construction of periodic solutions of oscillators subject to frictional occurrences. Coulomb's friction is commonly expressed as a differential inclusion which can be cast into the complementarity formalism. When targeting periodic solutions, existing algorithms rely on a procedure alternating between the frequency domain, where the dynamics is solved, and the time domain, where friction is dealt with. In contrast, the key idea of the present work is to express all governing equations including friction as equalities, which are then satisfied in a weak integral sense through a weighted residual formulation. The resulting algebraic nonlinear equations are solved numerically using an adapted trust region nonlinear solver and basic integral quadrature schemes. To increase efficiency, the Jacobian of the friction forces is calculated analytically in a piecewise linear fashion. The shape functions considered in this work are the classical Fourier functions. It is shown that periodic solutions with clear multiple sticking and sliding phases can be found with a high degree of accuracy. The equality-based formulation is shown to be effective and efficient, convergence being achieved in all cases considered with low computational cost, including for large numbers of harmonics. Importantly, this new friction formulation does not suffer from the typical limitations or hypotheses of existing frequency-time domain methods for non-smooth systems, such as regularization, penalization, or massless frictional interfaces.

A Modified Trapezoidal Technique with Improved Accuracy of Nonlinear Algebraic and Transcendental Equations

A Modified Trapezoidal Technique with Improved Accuracy of Nonlinear Algebraic and Transcendental Equations

Generalized Legendre Polynomial Configuration Method for Solving Numerical Solutions of Fractional Pantograph Delay Differential Equations

This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.

Haar wavelet approach to study the control of biological pest model in Tea plants

n this study, we consider a novel approach called the Haar wavelet collocation method (HWCM) toexamine the mathematical model of pest propagation in tea plants and how biological enemies might controlthem. This model is in the form of a system of coupled ordinary differential equations (ODEs). When studyingthe system, we consider tea plants, pests that harm the plants, biological enemies that are their reasonablecompetitors of pests, self-reproduction of the tea plants, natural death of pests and natural enemies, etc. Byturning the Mathematical model into a system of non-linear algebraic equations, we use the properties ofthe Haar wavelets. The opted method can solve the biological pest management problem in tea plants. Thevalues of the unknown coefficients are recovered using the collocation method and Newton Raphson method.The Mathematica program acquires the numerical results, nature, and uniformity. The acquired findings showthat the current method is more accurate than those indicated in tables and graphs.

Numerical treatment of the radiated and dissipative power-law nanofluid flow past a nonlinear stretched sheet with non-uniform heat generation

The main aim of this paper is to investigate the effect of non-uniform heat generation and viscous dissipation on the boundary layer flow of a power-law nanofluid over a nonlinear stretching sheet. Within the thermal domain, the analysis considers both thermal radiation and variable thermal conductivity. Through the use of similarity transformations, the governing boundary layer equations are transformed into a system of ODEs. The spectral collocation method (SCM) with shifted Vieta-Lucas polynomials (VLPs) is implemented to give an approximate expression for the derivatives and then use it to numerically solve the proposed system of equations. By employing this technique, the system of ODEs is converted into a system of nonlinear algebraic equations. The dimensionless temperature, concentration, and velocity are graphically presented and analyzed for various values of the relevant governing parameters. Through the presented graphical solutions, we can see that the main outcomes indicate that an increase in the power law index, thermal conductivity parameter, and radiation parameter leads to a noticeable decrease in the local Nusselt number, with reductions of around 0.05 percent, 0.23 percent, and 0.11 percent, respectively. In contrast, the Prandtl parameter demonstrates an opposing effect, elevating the local Nusselt number by about 0.1 percent. We validated the accuracy of the numerical solutions by comparing them in some special cases with existing literature.

Comparison between Calculated, Simulated and Actual Plant Data of Multiple Effect Evaporators to Concentrate Sugarcane Juice

Multiple effect evaporators systems operating at steady state can be described by a series of nonlinear algebraic equations that include total and solute mass balances, energy balances, heat transfer rate equations, and the composition and temperature dependence of related thermodynamic properties such as vapor pressures and enthalpies. The main objective is to compare between the manual calculations, simulation and real data collected from Asalaya plant for validation the best design of multiple effect evaporators system in forward feed used in Asalaya Sugar Factory. In this design a short tube vertical evaporator, Calandria type has been used. The models include overall equations of materials balance, equations of energy balance and equations of heat transfer rate for measurement area for all the effects. These equations were solved using manual calculations methods and MATLAB coded program. The results of the present investigations have been compared with the data obtained from the Asalaya Sugar Factory with five evaporators. The results showed that the concentration of sugar cane juice at the final stage is equal to 60 Brix for the manual calculations, simulation and actual plant data with zero error. The temperature of solution in the pre- effect evaporator was found to be about (114.94, 114.94, 115)ºC, with a relative error equal to 0.05% and the steam temperature in the pre effect was found about (120)ºC for the manual calculations, simulation and actual plant data, with zero error. The area and number of tubes in the (pre, second, third and fourth) effect, are in a good agreement, with the standard value (Er=3.26%). The above results show great similarity with the real behavior of evaporator unit. This can be used as a reliable tool\for the simulation of different operating conditions.

A numerical study on the nonlinear fractional Klein–Gordon equation

This article helps to develop a numerical approach based on Fibonacci wavelets for solving fractional Klein-Gordan equations (FKGEs). The FKGEs are solved with Caputo-type fractional derivative. Using the definition of Fibonacci wavelets, we constructed the operational matrices of integration. These operational matrices of integration led to the development of the collocation method called the Fibonacci wavelet collocation method (FWCM). This method transforms the given nonlinear partial differential equation into a system of nonlinear algebraic equations using collocation points to determine the unknown coefficients. By substituting the unknown coefficients in the method, we obtained the numerical solution of the present approach. We furnish the different error norms for the present technique. The obtained results are compared with the Clique polynomial method. These findings demonstrate the computational attractiveness, efficiency, effectiveness, reliability, and robustness of the proposed method for addressing a variety of physical models in science and engineering.