Solutions Of Nonlinear Equations Research Articles

Insight into Functional Boiti–Leon–Mana–Pempinelli Equation and Error Control: Approximate Similarity Solutions

The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal functions. In these cases, the F-BLMPE reduces to an explicit form in the dependent variable. In addition to this, it is worth deriving approximate similarity solutions of the F-BLMPE with constant coefficients using the extended unified method (EUM). In this method, nonlinear partial differential equation (NLPDE) solutions are expressed in polynomial and rational forms through an auxiliary function (AF) with adequate auxiliary equations. Exact solutions are estimated using formal solutions substituted into the NLPDEs, and the coefficients of the AF of all powers are set equal to zero. This approach is valid when the NLPDE is integrable. However, this technique is not valid for non-integrable equations, and only approximate solutions can be found. The maximum error can be controlled by an adequate choice of the parameters in the residue terms (RTs). Multiple similarity solutions are derived, and the ME is depicted in various examples within this work. The results found here confirm that the EUM is an efficient method for solving NLPDEs of the F-BLMPE type.

Lump wave solutions, lump-stripe soliton inelastic collision phenomena and rogue-type wave solutions for a generalized breaking soliton system in (3+1)-dimensions

Based on five types of structures of the functional multi-layer neural network model, some explicit exact solutions for a (3+1)-dimensional generalized breaking soliton system including lump waves, lump-stripe solitons and rogue-type waves are explored by using bilinear neural network method without the datasets training. The 3-dimensional plots, 2-dimensional curves and density plots of these results obtained are provided to display their dynamic characteristics. In particular, the complete processes of lump wave and stripe soliton fusion phenomena are presented. We hope the formulations established in this paper could be applied to study the exact traveling wave solutions of other integrable nonlinear evolution equations.

Data-driven fusion and fission solutions in the Hirota–Satsuma–Ito equation via the physics-informed neural networks method

Fusion and fission are two important phenomena that have been experimentally observed in many real physical models. In this paper, we investigate the two phenomena in the (2+1)-dimensional Hirota–Satsuma–Ito equation via the physics-informed neural networks (PINN) method. By choosing suitable physically constrained initial boundary conditions, the data-driven fusion and fission solutions are obtained for the first time. Dynamical behaviors and error analysis of these solutions are investigated via illustratively numerical figures, which show that good results are achieved. It is pointed out that the PINN method adopted here can be effectively used to construct the data-driven fusion and fission solutions for other nonlinear integrable equations. Based on the powerful predictive capability of the PINN method and wide applications of fusion and fission in many physical areas, it is hoped that the data-driven solutions obtained here will be helpful for experts to predict or explain related physical phenomena.

The dynamics of fractional KdV type equations occurring in magneto-acoustic waves through non-singular kernel derivatives

To study magneto-acoustic waves in plasma, we will use a numerical method based on the Natural Transform Decomposition Method (NTDM) to find the approximative solutions of nonlinear fifth-order KdV equations. The method combines the familiar Natural transform (NT) with the standard Adomian decomposition method. The fractional derivatives considered are the Caputo–Fabrizio and the Atangana–Baleanu derivatives in the sense of Caputo derivatives. Adomian polynomials may be employed to tackle nonlinear terms. In this method, the solution is calculated as a convergent series, and it is demonstrated that the NTDM solutions converge to the exact solutions. A range of two- and three-dimensional figures have been used to illustrate the dynamic behavior of the derived solutions. The tables provide a visual representation of numerical data. The physical behavior of the derived solutions about fractional order is further demonstrated by several simulations. When addressing nonlinear wave equations in science and engineering, the NTDM offers a broad range of applications. Several examples are given to highlight the importance of this work and to demonstrate the simplicity and trustworthiness of the method.

Basis Function Approaches for Numerical Solutions of Nonlinear Partial Differential Equations

Basis Function Approaches for Numerical Solutions of Nonlinear Partial Differential Equations

Novel Explosive and Super Fractional Nonlinear Schrödinger Structures

The fractional nonlinear Schrödinger equation solutions have been investigated via fraction space-time derivative sense. We applied the unified technique for this model to extract new structures of waves. The fractional property structures were obtained from the model in form of hyperbolic, soliton, shock, explosive, superperiodic, and trigonometric structures. It was found that increasing fractal factors produces a change in the phase and wave frequency of propagating nonlinear waves. The physical models that explain tidal energy generation are crucial to the development of contemporary green power systems. The parametric description for wave characteristics in this process is created by the solution of nonlinear equations. The obtained solutions are applicable in new communications, energy applications, fractional quantum modes, and in science and complex phenomena in astrophysics. Finally, the proposed technique can be implemented for further fractional physical models.

Solution of Multi-Dimensional Non-linear Fractional Differential Equations of Higher Orders

In our paper, we are used here two methods to solve non-linear differential equations from a higher order: the first-one is domain decomposition method is used to estimate the Maxi. Abso. Trunc. Error of Adomain series and the second-one proposed numerical (PNM), these types of equations are studied. When we use these methods, an exclusive solution will be provided, and the approximate analyses of this method applied to these types of equations will be overlooked, and the maximum error that has been informed to solve the ADOMIANS series will be classified. A digital example is prepared clarify the impact method provided and significant following of these equations in our paper is Bagley-Torvik equation.

Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow

The present work introduces a deep learning approach to describe the perturbations of the pressure and radius in arterial blood flow. A mathematical model for the simulation of viscoelastic arterial flow is developed based on the assumption of long wavelength and large Reynolds number. Then, the reductive perturbation method is used to derive nonlinear evolutionary equations describing the resistance of the medium, the elastic properties, and the viscous properties of the wall. Using automatic differentiation, the solutions of nonlinear evolutionary equations at different time scales are represented using state-of-the-art physics-informed deep neural networks that are trained on a limited number of data points. The optimal neural network architecture for solving the nonlinear partial differential equations is found by employing Bayesian Hyperparameter Optimization. The proposed technique provides an alternate approach to avoid time-consuming numerical discretization methods such as finite difference or finite element for solving higher order nonlinear partial differential equations. Additionally, the capability of the trained model is demonstrated through graphs, and the solutions are also validated numerically. The graphical illustrations of pulse wave propagation can provide the correct interpretation of cardiovascular parameters, leading to an accurate diagnosis and successful treatment. Thus, the findings of this study could pave the way for the rapid development of emerging medical machine learning applications.

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

The aim of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher-order dispersion term. For this purpose, the first and second simplest methods are used to build soliton solutions of travelling wave solutions. Furthermore, the bifurcation behavior of traveling waves including new types of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and beneficial tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher-order dispersion will add some value to the literature of mathematical and plasma physics.

Uncovering diverse soliton solutions in the modified Schrödinger’s equation via innovative approaches

The topic of soliton propagation in optical fibers is explored in our research paper, with a focus on the utilization of the modified nonlinear Schrödinger’s equation with perturbation terms. Two effective techniques have been employed in this model, resulting in the generation of wave structures in a closed-form manner and the production of a wide range of distinct solutions. These solutions hold potential applications in various fields such as optical fibers, plasma fluids, and biomolecular dynamics. The proposed approaches are characterized by their simplicity, robustness, and ability to generate new solutions for nonlinear partial differential equations in the field of mathematical physics. Captivating figures depicting the propagation of traveling wave solutions for carefully chosen parameter values are included in the paper. Overall, valuable insights into the behavior of soliton propagation in optical fibers are provided by this research, and new avenues for future research in this field are offered.

Solving PDEs arising in the formation of liquid drop pattern using Sumudu transform based technique

In this research work, we look into the solitary wave solutions of 2D and 3D nonlinear Rosenau Hyman type equations with time- fractional derivatives, arising during the formation of liquid drop pattern. We implement relatively new scheme, which is the combination of an integral transform such as Sumudu transform and the Homotopy perturbation method. Some theorems have been stated here to describe the convergence of the proposed technique. To demonstrate the ease of use and precision of the proposed methodology, four test examples with various parameter settings have been addressed.

The dynamical study and analysis of diverse bright-dark and breathers wave solutions of nonlinear evolution equations and their applications

In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, [Formula: see text]-dimensional generalized shallow water wave (SWW) and [Formula: see text]-dimensional Fokas dynamical models by applying two-variable [Formula: see text]- expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more.

Modeling of water table profile variations owing to stream-aquifer interaction.

Spatial and temporal variations of the water table could be explained by the one-dimensional Boussinesq equation by incorporating the variables of evapotranspiration and groundwater recharge with appropriate initial and boundary conditions. In this study, the stream-aquifer interaction has been investigated through a numerical example model with the implementations of Galerkin's method-based Finite Element Solution (FES), Hybrid Finite Analytic Solution (HFAS), Fully Implicit Finite Difference Solution (FIFDS) of one-dimensional nonlinear Boussinesq equation, and analytical solutions of the Boussinesq equation linearized by Baumann's transformation (AS I) as well as linearized by Werner's transformation (AS II). Considering HFAS as the benchmark solution, it was observed that in both recharging and discharging aquifers, water table profiles at 1 day and 5 days as obtained from FES followed by FIFDS were observed quite close to HFAS. Based on L2 and Tchebycheff norms, FES and FIFDS were ranked in first and second place, respectively. L2 and Tchebycheff norms could not consistently establish the performance ranking of analytical solutions but their performance ranking was certainly below the numerical solutions. The performance ranking of analytical solutions could not consistently be established using the L2 and Tchebycheff norms, but it was certainly below the numerical solutions.

Asymptotical Stability Criteria for Exact Solutions and Numerical Solutions of Nonlinear Impulsive Neutral Delay Differential Equations

In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results.

Solitary waves of the complex Ginzburg-Landau equation with anti-cubic nonlinearity

The complex Ginzburg-Landau equation with anti-cubic nonlinearity is considered. Using a modification of the simplest equation method for finding exact solutions of nonlinear partial differential equations we get new solitary waves with arbitrary amplitude and speed of this special case of the complex Ginzburg -Landau equation. The first integral of nonlinear ordinary differential equation is found. The general solution of equation is presented taking into account the first integral of equation. Conserved density of the solitary wave solution is calculated.

Complex solutions for nonlinear fractional partial differential equations via the fractional conformable residual power series technique and modified auxiliary equation method

The fractional-order nonlinear Gardner and Cahn–Hilliard equations are often used to model ultra-short burst beams of light, complex fields of optics, photonic transmission systems, ions, and other fields of mathematical physics and engineering. This study has two main objectives. First, the main objective of this investigation is to solve the fractional-order nonlinear Gardner and Cahn–Hilliard equations by using the modified auxiliary equation method, which is not found in the literature. Second, the exact and approximate solutions of these equations are obtained by utilizing the fractional conformable residual power series algorithm and the modified auxiliary equation method. For the analytical and numerical solutions to two equations, we employ two separate techniques and establish consistency between the precise answers that are derived and the compatible numerical solution. To the best of our knowledge, this method of solving equations has never been investigated in this manner. The 2D and 3D contours have been defined using appropriate parametric values to support the physical compatibility of the results. The assessed findings suggested that the approach used in this study to recover inclusive and standard solutions is approachable, efficient, and faster in computing and can be considered a useful tool in resolving more complex phenomena that arise in the field of engineering, mathematical physics, and optical fiber.

Diverse optical soliton solutions of two space-time fractional nonlinear evolution equations by the extended kudryashov method

This study investigates the inclusive optical soliton solutions to the (2+1)-dimensional nonlinear time-fractional Zoomeron equation and the space-time fractional nonlinear Chen-Lee-Liu equation using the extended Kudryashov technique. The beta derivative is used to conduct the fractional terms and investigate wide-spectral soliton solutions to the considered models. The obtained solutions yield a variety of typical soliton shapes, including ant-peakon soliton, V-shaped soliton, anti-bell-shaped soliton, kink soliton, periodic soliton, singular periodic soliton for the specific value of the parameters. The three-dimensional, contour, and two-dimensional graphs of the derived solitons have been plotted to illustrate the structure, propagation, and influence of the fractional parameter. It is observed that the fractional parameter affects the amplitudes and periods of certain solitons. The precision of the acquired solutions is confirmed by reintroducing them into the original equation using Mathematica. The findings of this study indicate that the employed method has the capability of yielding compatible, creative, and useful solutions for diverse nonlinear evolution equations with fractional derivatives. This approach could introduce novel ways for unraveling other nonlinear equations and have implications in diverse sectors of nonlinear science and engineering.

Bridging machine learning and weighted residual methods for delay differential equations of fractional order

In this research, an attempt has been made to build a bridge between a machine learning model called Least-Squares Support Vector Regression (LS-SVR) and Weighted Residual Methods (WRMs) to solve delay differential equations in fractional derivative order. Our machine learning algorithm, which is influenced by the collocation method, uses Legendre polynomials as a kernel function to approximate the solution of linear and nonlinear delay differential equations, including the Lane–Emden pantograph equation and some fractional cases. In the proposed algorithm, the estimated solution of delay differential equations is approximated as a linear combination of M degrees of Legendre polynomials and a set of weights that are learned during the fitting process. The roots of the Legendre functions are utilized as training data to develop the algorithm. The effectiveness of our method is demonstrated through the use of numerical graphs and tables. A comparative analysis of the numerical results has shown that the proposed method yields better accuracy and convergence compared to other numerical techniques.

Multi-soliton solutions of the Sawada-Kotera equation using the Hirota direct method: Novel insights into nonlinear evolution equations

Recently, mathematicians, engineers, and scientists have explored the unique characteristics and potential applications of multi-solitons, which is an expanding domain of study. There are various approaches to obtaining multi-soliton solutions of an integrable system, such as the Bäcklund transform, the nonlinear transform method, the Hirota direct method, etc. But each approach has some own characteristics and attributes; however, the Hirota direct method is dominant among them and provides further multi-soliton solutions of nonlinear evolution equations. In this article, we use the Hirota direct method to investigate the single-soliton, double-soliton, and triple-soliton solutions, known as multi-soliton solutions, of the integral Sawada-Kotera equation. We also demonstrate and discuss the effects of amplitude on the fluctuation of wave number in different ranges by comparing 2-D and 3-D plots.

Optimal mode movement of the robot manipulator on an elastic base according to the criterion of the mean square value of the acceleration of the drive torque

Previous studies of the optimization of the manipulator motion mode on an elastic base based on the criteria of the rms value of the drive torque and the rate of change of the rms value of the drive torque established that the use of optimal manipulator motion modes, which are created on the basis of higher derivatives of the characteristics of the change of the drive torque, allow to obtain more smooth movements of the boom system of the manipulator, thereby reducing its oscillations on the elastic support. At the same time, the obtained optimal modes of movement didn't completely eliminate the boom system oscillations during the change of departure. By observing the oscillations of the elastic link of the manipulator, the quality of the obtained optimal modes of movement was evaluate and the presence of residual oscillations was shown.
 This work considers the possibility of optimizing the manipulator's motion mode based on the criterion of the mean square value of the acceleration of the drive torque change. For this purpose, the second derivative of the driving torque function of the manipulator boom system drive was determined first. Such a criterion for optimizing the motion mode is an integral functional, the minimization of which is carry out by the methods of variational calculus. The regularity of the change of the driving torque it was find from the dynamic equation of motion.
 The search for the optimal motion mode of the manipulator realized through the solution of the Poisson boundary nonlinear differential equation of the fourth order.
 The results of the conducted research made it possible to clarify much better the nature of the vibrations of the manipulator boom system and, as a result, to develop a drive control system that allows the realization of the obtained optimal mode of movement.