System Of Nonlinear Equations Research Articles

Invariant solutions, lie symmetry analysis, bifurcations and nonlinear dynamics of the Kraenkel-Manna-Merle system with and without damping effect

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics.

On the Periodicity Solutions of Five Systems of Rational Systems of Difference Equations of Order Five

The primary aim of this paper is to explore the behavior of several nonlinear systems of difference equations following T_{n+1}=(E_{n}E_{n-4})/(T_{n-3}(\pm 1 \pm E_{n}E_{n-4})), E_{n+1}=(T_{n}T_{n-4})/(E_{n-3}(pm 1 \pm T_{n}T_{n-4})), and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulare the dynamics and validate our results.

Employing the Sadik Residual Power Series Method to Analyze a System of Nonlinear Caputo Time-Fractional Partial Differential Equations

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.

Analytical Solutions and Instability Analysis of truncated $\mathrm{M}$-Fractional Coupled Dispersionless Equations

Abstract This paper systematically investigates the truncated M-fractional coupled dispersionless equations, a nonlinear system of partial differential equations characterized by its M-fractional derivative. We employ the Jacobi elliptic function expansion method to derive analytical solutions for the coupled system. Additionally, the modulation instability of the solutions is thoroughly explored, providing a detailed exposition of the mathematical framework governing the system. The analytical solutions are graphically illustrated and analyzed to highlight their physical significance. These findings hold significant applications in nonlinear optics, offering new insights into wave propagation and stability within such systems.

Numerical Assessment of MHD Tri-Hybrid Nanoparticles by Tangent Hyperbolic Model with Chemical Reaction: Thermodynamic Analysis

This paper investigates the influence of magneto-tangent hyperbolic nanofluid on the flow of a tri-hybrid nanoliquid consisting of [Formula: see text], and [Formula: see text] particles suspended in [Formula: see text]. The entropy production is encountered in this analysis. The fluid flows over a stretch sheet is considered. In addition, the energy equation also assumes the existence of a uniform heat source or sink and thermal radiation. Furthermore, the concentration equation emphasizes the chemical reaction. The current proposed model yields a set of nonlinear governing equations. The modeled formulation is transformed into a dimensionless system through the application of a suitable alteration. The complex nonlinear equation system was solved using the bvp4c through numerical methods. The main motive of this exploration is to emphasize the rate of heat and mass transfer in a flow of [Formula: see text], and [Formula: see text]-based hybrid nanofluid across a stretch sheet. The graphical study illustrates that Weissenberg number and magnetic field enhancement result in decreasing the velocity. But thermal layer, entropy production, and Bejan number are enhanced with larger values of Weissenberg number and magnetic field. This study focuses on different profiles with various flow parameters. Furthermore, we have compared the tri-hybrid nanofluid with the hybrid and mono nanofluid in all the figures and tabular format. Additionally, we have compared tri-hybrid, hybrid, and mono nanofluid using graphs for velocity, temperature, concentration, entropy production, and Bejan number.

A unified approach to the construction of higher-order derivative-free iterative methods for solving systems of nonlinear equations

In this article, we introduce a unified approach to constructing a higher-order derivative-free scheme based on the approximations of F'(zk)-1. A family of order p=6,7 derivative-free method is proposed and compared to some well-known methods. The necessary and sufficient condition for p-th order of convergence are given in terms of parameter matrices τ(k) and α(k) . Some good choices of and are offered. Numerical experiments were carried out to confirm the theoretical results.

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.

Exploring nanoparticle dynamics in binary chemical reactions within magnetized porous media: a computational analysis

Artificial Neural Networks are incredibly efficient at handling complicated and nonlinear mathematical problems, making them very useful for tackling these challenges. Artificial neural networks offer a special computational architecture that is extremely valuable in disciplines like biotechnology, biological computing, and computational fluid dynamics. The present work investigates the applicability of back-propagation artificial neural networks in conjunction with the Levenberg-Marquardt algorithm for evaluating heat transmission in hybrid nanofluids. This work focuses on the computational analysis of a MgO + GO/EG hybrid nanofluid’s steady mixed convection flow over an exponentially stretched sheet, considering multiple slip boundary conditions, thermal conductivity, heat generation, and thermal radiation. A nonlinear system of ordinary differential equations is produced from the basic associated partial differential system by performing the proper exponential similarities modifications. For generating benchmark datasets, the resulting ordinary differential equations are processed employing the bvp4c method. Considering benchmark datasets set aside for training (70%), testing (15%), and validation (15%), the Levenberg-Marquardt algorithm, which employs back-propagation in artificial neural networks, is implemented. The accuracy of the suggested strategy for handling nonlinear problems is verified utilizing mean squared error, error histograms, and regression analysis, which are all used to evaluate the methodology. Outstanding agreement is seen when ANN outputs are compared to numerical results. The flow properties, including temperature, velocity, and concentration profiles, are shown graphically and numerically. For practical purposes, it is therefore essential to analyze the flow and heat transfer in hybrid nanofluids over exponentially extending and shrinking surfaces under mixed convection and heat source scenarios. Hybrid nanofluid problems have a wide range of practical and industrial applications, such as medication delivery, manufacturing, microelectronics, nuclear plant cooling, and marine engineering.

Analytical simulation of Darcy-Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy-Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy-Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy's formula for porous media flow: two nanofluid particles, molybdenum disulfide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method (HAM) is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

Two-Step Fifth-Order Efficient Jacobian-Free Iterative Method for Solving Nonlinear Systems

This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments that confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

An elasto-visco-plastic constitutive model for snow: Theory and finite element implementation

Snow exhibits unique mechanical behaviour due to its evolving properties influenced by temperature, stress conditions, and viscous effects. This paper introduces a nonlinear constitutive model for snow, featuring new formulations for the yield function and strain rate potential, and incorporating viscosity, sintering, and degradation effects. The model is numerically integrated into the Abaqus/Standard FEM software using a fully implicit backward Euler method for time integration and Powell’s hybrid method for linearizing the nonlinear system of equations. The robustness and stability of the numerical scheme ensure accurate simulation of snow behaviour under various loading conditions. The model is finally validated against experimental data available in the literature, demonstrating its effectiveness and reliability in capturing the complex mechanical response of snow.

Bioconvective magnetized flow analysis of a tangent hyperbolic nanofluid with nonuniform source/sink over an exponentially porous stretching sheet

AbstractThis study use numerical method to investigate the transportation of a bioconvective magnetized tangent hyperbolic nanofluid across an exponentially porous stretched sheet. The flow model takes into account the important contributions of Joule heating, thermal radiation, activation energy, and nonuniform heat source/sink. Moreover, slip boundary conditions with gyrotactic microorganisms are taken into flow analysis. The flow model is converted into a system of nonlinear ordinary differential equations (ODEs) using suitable similarity variables. The bvp4c approach in MATLAB is used to get numerical solutions for this system. The study evaluates the influence of different parameters on the velocity, temperature, concentration, and microorganism profile via graphical representations. The results indicate that increasing the mixed convection parameters accelerates the velocity profile, whereas raising the magnetic parameter slows it down. The temperature profile increases with higher values of radiation parameter, however, it decreases as the Prandtl number and thermal slip parameter increase. Furthermore, the microorganism profile decreases as the bioconvection Lewis number and microorganism slip parameter increase.

Solving Nonlinear Difference Equations: Insights from Three-Dimensional Systems and Numerical Examples

This paper presents a study on nonlinear difference equation systems of 6k + 3 order. The equations are of the form pn+1 =pn−(6k+2) / (±1±qn−2k rn−(4k+1) pn−(6k+2)), qn+1 =qn−(6k+2) / (±1±rn−2kpn−(4k+1) qn−(6k+2)),rn+1 =rn−(6k+2) / (±1± pn−2kqn−(4k+1) rn−(6k+2)), k ≥ 0 where n is a non-negative integer (belonging to the set N0 = N ∪ {0}) and the starting values p−l , q−l , r−l , l ∈ {0, 1, . . . , 6k+2} are arbitrary nonzero real numbers. We propose a systematic approach to solve this system, introducing a novel technique to find explicit solutions. The main outcomes of our study are the explicit solutions derived from the considered system. The study examines four different cases of this system and provides numerical examples to illustrate the results. The numerical examples demonstrate the behavior of the system for various initial conditions. The study is concluded with graphical representations of the solutions for each case, providing insights into the behavior of the systems.

Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies

Investigation of nanofluid flows under various conditions is becoming fascinating research due to its numerous applications in many science and engineering fields such as solar energy, geothermal energy, water purification, glass blowing, enhanced oil recovery, petroleum production and food processing. For the present requirements in thermal energy applications, this mathematical model is considered to explore the diffusion and heat source effects on the flow of conductive Casson and Maxwell nanofluids in a porous medium on an elongating sheet with a chemical reaction. Further, the effects of viscous dissipation and thermal radiation are also examined. Moreover, the entropy generation is analyzed since this approach is employed in solar energy exchangers. The governing equations and the corresponding boundary conditions are constructed. This system of partial differential equations (PDEs) is changed into a system of nonlinear ordinary differential equations (ODEs) with the help of a suitable transformation and then solved numerically by the bvp5c MATLAB package. The influences of the physical parameters on the flow phenomena are discussed and presented through figures and tables. This analysis explains that the Casson fluid flow has a higher velocity field than the Maxwell fluid flow. The Nusselt number becomes higher as the Dufour numbers increase. It has also been noticed that increasing the Brinkmann number improves the entropy and, an increase in radiation enriches the temperature profile. The concentration of the fluid is decreased for the increasing values of the higher-order chemical reaction parameter. Furthermore, comparisons of the current results were conducted for limiting examples of the problem and found to be in good agreement with existing results in the literature.

Efficient Solutions for Stochastic Fractional Differential Equations with a Neutral Delay Using Jacobi Poly-Fractonomials

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method’s capability to produce accurate and efficient solutions.

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

Electroosmotically assisted peristaltic propulsion of blood-based hybrid nanofluid through an endoscope with activation energy

The main aim of this research work is to analyze the peristaltic pumping of hybrid nanofluid through an endoscope. Here, blood is considered as base fluid and iron-oxide and gold are considered as nanoparticles. The non-Newtonian behavior of blood is studied through the Casson fluid model. The effect of nanoparticles is included in the mathematical formulation by using the modified version of Buongiorno model. Moreover, the effects of motion due to electroosmosis, activation energy, and Joule heating are also considered in fluid flow. The obtained non-linear system of equations after imposing lubrication approach is solved using Mathematica. The flow properties subject to variation in different involved parameters is analyzed through graphical results. It is assessed from graphical results that for larger Casson fluid parameters, fluid flow is more accelerated. It is also observed that presence of hybrid nanoparticles improves the velocity profile as well as cooling tendency of the working fluid when compared with the case of mono nanofluid. The electroosmotic phenomenon works a key parameter that can control the fluid flow by either assisting and opposing the flow. It has observed that a larger Debye length parameter, assisting electric field, and larger activation energy parameter reduce the nanoparticle concentration.

Existence and forms of entire solutions to system of non-linear partial differential equations

Existence and forms of entire solutions to system of non-linear partial differential equations

Historical knowledge transfer driven self-adaptive evolutionary multitasking algorithm with hybrid resource release for solving nonlinear equation systems

In reality, there are many extremely complex nonlinear optimization problems. How to locate the roots of nonlinear equation systems (NESs) more accurately and efficiently has always been a major numerical challenge. Although there are many excellent algorithms to solve NESs, which are all limited by the fact that the algorithm can solve at most one NES in a single run. Therefore, this paper proposes a historical knowledge transfer driven self-adaptive evolutionary multitasking algorithm framework (EMSaRNES) with hybrid resource release to solve NESs. Its core is that in one run, EMSaRNES can efficiently and accurately locate the roots of multiple NESs. In EMSaRNES, self-adaptive parameter method is proposed to dynamically adjust parameters of the algorithm. Secondly, adaptive selection mutation mechanism with historical knowledge transfer is designed, which dynamically adjusts the evolution of populations with or without knowledge sharing according to changes in the current population diversity, thereby balancing population diversity and convergence. Finally, hybrid resource release strategy is developed, which archives the roots that meet the accuracy requirements, and then three distributions are selected to generate new populations, thus ensuring that the population diversity is maintained at high level. After a variety of experiments, it has been proven that compared to comparative algorithms EMSaRNES has superior performance on 30 general NESs test sets. In addition, the results on 18 extremely complex NESs test sets and two real-life application problems further prove that EMSaRNES finds more roots in the face of complex problems and real-life problems.

Hybridized Brazilian–Bowein type spectral gradient projection method for constrained nonlinear equations

This paper proposes a hybridized Brazilian and Bowein derivative-free spectral gradient projection method for solving systems of convex-constrained nonlinear equations. The method avoids solving any subproblems in each iteration. Global convergence is established under appropriate assumptions on the functions involved. Additionally, numerical experiments are conducted to evaluate the algorithm’s performance, providing evidence of its efficiency compared to similar algorithms from the existing literature. The results demonstrate that the method outperforms some existing approaches in terms of the number of iterations, function evaluations, and time required to obtain a solution based on the examples considered.