Solutions Of Nonlinear Equations Research Articles

A deterministic approach to investigate nonlinear evolution equations for large balance numbers

When the balance number is greater than one, the modified simple equation (MSE) method typically fails to yield analytical wave solutions for nonlinear evolution equations (NLEEs) that appear in engineering and mathematical physics. We have addressed this shortcoming in this article and established a technique to implement the MSE approach to investigate NLEEs for balancing number two. Two NLEEs, namely, the regularized long wave and the Jimbo-Miwa equations, have been investigated in order to affirm the approach. Through this method, we found further generic wave solutions related to physical parameters, and when the parameters receive particular values, solitons emerge from the exact solutions. Graphs are used to investigate the solitary wave features of the attained solution functions, which illustrate the usefulness, validity, and compatibility of the scheme.

Neuro‐PINN: A hybrid framework for efficient nonlinear projection equation solutions

AbstractNonlinear projection equations (NPEs) provide a unified framework for solving various constrained nonlinear optimization and engineering problems. This paper presents a deep learning approach for solving NPEs by incorporating neurodynamic optimization and physics‐informed neural networks (PINNs). First, we model the NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization, and the objective becomes solving this ODE system. Second, we use a modified PINN to serve as the solution for the ODE system. Third, the neural network is trained using a dedicated algorithm to optimize both the ODE system and the NPE. Unlike conventional numerical integration methods, the proposed approach predicts the end state without computing all the intermediate states, resulting in a more efficient solution. The effectiveness of the proposed framework is demonstrated on a collection of classical problems, such as variational inequalities and complementarity problems.

On Data Dependency and Solutions of Nonlinear Fredholm Integral Equations with the Three-Step Iteration Method

In this study, the solution of the second type of homogeneous nonlinear Fredholm integral equations is investigated using a three-step iteration algorithm. In other words, it has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Also, data dependency is obtained for the second type of homogeneous nonlinear Fredholm integral equations and this result is supported by an example.

Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

Multiple cylindrically symmetric solutions of nonlinear Maxwell equations

In this paper, we study the following nonlinear time-harmonic Maxwell equations \begin{equation}\label{equation 0.1} \nabla\times(\nabla \times E)-\omega^2\varepsilon(x)E =P(x)|E|^{p-2}E+Q(x)|E|^{q-2}E, \end{equation} where $\varepsilon(x)$ is the permittivity of the material, $x\in\mathbb{R}^{3}$, $1< q< {p}/({p-1})< 2< p< 6$, $P(x),Q(x)\in C\left(\mathbb{R}^{3},\mathbb{R}\right)$. Under some special cylindrical symmetric conditions for $\varepsilon(x)$, $P(x)$ and $Q(x)$, we obtain infinite many cylindrically symmetric solutions of \eqref{equation 0.1} by using variational method and fountain theorems without $\tau$-upper semi-continuity.

New Numerical and Analytical Solutions for Nonlinear Evolution Equations Using Updated Mathematical Methods

This study explores adapted mathematical methods to solve the couple-breaking soliton (BS) equations in two-dimensional spatial domains. Using these methods, we obtained analytical soliton solutions for the equations involving free parameters such as the wave number, phase component, nonlinear coefficient, and dispersion coefficient. The solutions are expressed as hyperbolic, rational, and trigonometric functions. We also examined the impact of wave phenomenon on two-dimensional diagrams and used composite two-dimensional and three-dimensional graphs to represent the solutions. We used the finite difference method to transform the proposed system into a numerical system to obtain numerical simulations for the Black–Scholes equations. Additionally, we discuss the stability and error analysis of numerical schemes. We compare the validity and accuracy of the numerical results with the exact solutions through analytical and graphical comparisons. The methodologies presented in this research can be applied to various forms of nonlinear evolutionary systems because they are appropriate and acceptable.

Existence and asymptotic behavior of solutions for non-linear wave equations of Kirchhoff type with viscoelasticity

"In this paper we discuss the global existence and the asymptotic behavior of solutions of an initial boundary value problem of a non-linear wave equation of Kirchhoff type. "

Physical structure and multiple solitary wave solutions for the nonlinear Jaulent–Miodek hierarchy equation

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences.

Analytical study of nonlinear models using a modified Schrödinger’s equation and logarithmic transformation

Optical solitons are special waves that maintain their shape while traveling. Hence, solitary optical waves have great significance when it comes to representing pulse propagation through nonlinear partial differential equations. In our research, we have uncovered unique solutions for the formation of solitary wave patterns in the equation of the Heisenberg ferromagnetic spin chain (HFSC). This equation aptly describes the behavior of electromagnetic waves in contemporary magnetism. By employing innovative techniques based on logarithmic transformations and analytical methods, we have obtained various solution forms expressed in a concise manner using elementary functions. We have tested the correctness of these solutions by substituting them directly into the main equation. These recent discoveries provide novel perspectives on the complex realm of solitons as depicted by this model. Moreover, since the model finds applications in fiber optic communications, fluid dynamics, and other fields, the implications of these discoveries are broad. The utilized methods stand out for being simple, reliable, and capable of creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. The research findings presented here showcase the effectiveness of the applied methods in reliably studying nonlinear phenomena in the HFSC equation and other nonlinear problems in mathematical physics. By utilizing these tools, scholars have the opportunity to expand and enhance their understanding of the complex mathematical frameworks underlying real-world problems.

Extension of optimal auxiliary function method to non-linear fifth order lax and Swada-Kotera problem

In this article, a semi-analytical approach known as the optimal auxiliary function method is extended to the approximate solution of non-linear partial differential equations. The fifth order lax and swada-kotera equations are taken as test examples. Utilizing the well-known least squares method, the optimal convergence control parameter values in the auxiliary function have been determined. The outcomes of the proposed method are contrasted with those of a new iterative approach and a homotopy perturbation method. It has been demonstrated that the suggested method for solving non-linear partial differential equations is straightforward and rapidly convergent. The numerical outcomes demonstrate the effectiveness and reliability of the suggested approach. Additionally, using higher order approximations can increase the suggested method's accuracy.

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.

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.