Solutions Of Nonlinear Equations Research Articles

New exact soliton wave solutions appear in optical fibers with Sardar sub equation and new auxiliary equation techniques

This paper comprehensively analyzes exact solutions for the nonlinear long-short wave interaction system within the optical field. Consider two general techniques in this field, the Sardar sub-equation method, and a new auxiliary-equation technique. These methods are applied to derive a wide range of soliton solutions for nonlinear partial differential equations. By transforming the original partial differential equation into an ordinary differential equation using an appropriate transformation, various types of solitary wave solutions are obtained. The novelty of this work lies in the application of two powerful analytical methods. The study significantly broadens the scope of these techniques and their applications, providing a diverse set of exact solutions. To enhance comprehension, the obtained solutions are visualized through 3D, 2D, contour, and density plots, offering clear insights into the dynamics of solitary waves. Long-short-wave interaction model has many applications in different kinds of areas such as in optical fiber communication, to understand the interaction between different wave components that can influence the transmission of signals. This model is used to study the interaction between ion-acoustic waves and electron plasma waves. This helps in understanding energy transfer and wave stability in plasma, which is essential for applications like fusion energy research and space plasma. This is important in coastal engineering for predicting wave behaviors that affect coastal structures, sediment transport, and tsunami dynamics.

Some novel exact solutions for the generalized time-space fractional coupled Hirota-Satsuma KdV equation

In this paper, two efficient methods, namely the modified G′/G,1/G-expansion method and the G′/bG′+G+a-expansion method, are employed to obtain novel exact wave solutions for the generalized time-space fractional coupled Hirota-Satsuma KdV equation. Various types of analytical explicit solutions, including well-known bell-shape solitons, mixed solitary wave solutions, and periodic wave solutions are obtained. These solutions are of great significance for revealing the nonlinear interaction between two long waves with different dispersion effects. The two-dimensional and three-dimensional distribution maps and contour plots corresponding to partial solutions are simulated to visually display the evolution process of relevant physical quantities over time. Moreover, the potential applications of these solutions in nano/micro devices and systems, especially in MEMS (Micro-Electro-Mechanical Systems) are discussed. It is demonstrated that the methods and processes utilized have strong applicability for constructing analytical solutions of nonlinear evolution equations.

Evolution dynamics of fundamental mechanisms of wave propagation in gas bubble-liquid interactions and soliton solutions

Abstract This paper investigates the dynamics of solitary wave solutions based on the (3+1)-dimensional nonlinear wave equation with variable coefficients expressed to describe gas-bubble-liquid interactions. The generalised (1/G’)-expansion method, new solitary wave solutions for this equation have been successfully derived to better understand the underlying dynamics of wave phenomena, especially in gas-bubble-liquid systems. Thanks to this method, the results of the variations of physical parameters in the generated solutions are also emphasised. The physical dynamics of each dimension in the generated solutions allow both the dimensions to be compared with each other and the equation to be compared with the existing literature with a holistic understanding. With this application, we have also analyzed the direct effects of the viscosity of the fluid on the dispersion of the bubble. These solitary solutions have helped us to better explain the wave behavior in gas-bubble-liquid systems and provided a new perspective on the solution of nonlinear wave equations. The structure of the study contributes to a deeper understanding of wave phenomena by discussing the (3+1) dimensional variable coefficient nonlinear wave equation within the framework of both mathematical analysis and physical analysis.

Nonlinear oscillators at resonance with periodic forcing

In this note we unify the results of P. O. Frederickson and A. C. Lazer [J. Differential Equations 5 (1969), 26-270], A. C. Lazer [Electron. J. Differ. Equ. Conf., vol. 5, 2000, pp. 113–119], A. C. Lazer and D. E. Leach [Ann. Mat. Pura Appl. (4) 82 (1969), 49–68], J. M. Alonso and R. Ortega [Nonlinearity 9 (1996), no. 5, 1099–1111], and P. Korman and Y. Li [Acta Math. Sci. Ser. B (Engl. Ed.) 34 (2014), no. 4, 1025–1028] on periodic oscillations and unbounded solutions of nonlinear equations with linear part at resonance and periodic forcing. We give conditions for the existence and nonexistence of periodic solutions, and we obtain a rather detailed description of the dynamics for nonlinear oscillations at resonance, in case periodic solutions do not exist.

Existence and uniqueness results of nonlinear hybrid Caputo-Fabrizio fractional differential equations with periodic boundary conditions

In this manuscript , we establish the existence and uniqueness of solutions for nonlinear hybrid fractional differential equations involving Caputo-Fabrizio fractional derivatives of order $\varrho\in(0,1)$. The proofs are based on Banach fixed point theorem and some basic concept of Caputo-Fabrizio fractional analysis. As an application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.

A General Perturbed Newtonian Framework and Critical Solutions of Nonlinear Equations

A General Perturbed Newtonian Framework and Critical Solutions of Nonlinear Equations

On Lie symmetries, soliton interaction nature and conservation laws of Broer-Kaup-Kupershmidt system in shallow water of uniform depth

Abstract The exact solutions of nonlinear partial differential equations are significant to analyze the soliton nature of physical phenomena. Therefore, the Lie symmetry method is introduced to obtain exact solutions of the Broer-Kaup-Kupershmidt system. It represents two-dimensional propagation of nonlinear and dispersive long gravity waves in shallow water with uniform depth. The present work is devoted to derive symmetry reductions, soliton solutions, and conserved vectors under one-parameter Lie group transformations. Meanwhile, the invariance property of Lie groups validates the reducibility of the system. Therefore, a continual process of reductions transmutes the test system into a system of ordinary differential equations. Thus, this integrability property leads to generalized exact solutions with parametric constraints. These solutions entail all three arbitrary functions, f 1(y), f 2(t), f 3(t) existing in infinitesimals and various free parameters. On account of existing functions and free parameters, the obtained solutions are more admirable than those of preceding works (Kassem and Rasheed 2019 Chinese J. Phys. 57 90–104, Kumar et al 2022 Math. Comput. Simulat. 196 319–35, Tanwar and Kumar 2024 J. Ocean Eng. Sci. 9 199–206) and helpful to describe significant physical nature under wide parametric ranges. The deductions of previous results on imposing the particular values to arbitrary functions and constants ensure the generalness and novelty of derived results. Moreover, adjoint equations and conserved quantities associated with Lagrangian formulation have been evolved via Lie symmetries. We numerically simulate these solutions to analyze their physical nature. Consequently, soliton fusion and fission, line solitons, doubly solitons, multisolitons, and bright and dark soliton nature are discussed, which validate the physical relevance of the solutions.

Oscillation Solutions for Nonlinear Second-Order Neutral Differential Equations

This research investigates the oscillation criteria of nonlinear second-order neutral differential equations with multiple delays, focusing on their noncanonical forms. By leveraging an innovative iterative technique, new relationships are established to enhance the monotonic properties of positive solutions. These advancements lead to the derivation of novel oscillation criteria that significantly extend and refine the existing body of knowledge in this domain. The proposed criteria address gaps in the literature, providing a robust framework for analyzing such differential equations. To demonstrate their practical implications, three illustrative examples are presented, showcasing the applicability and effectiveness of the results in solving real-world problems involving delay differential equations.

Periodicity and positivity of solutions for first-order nonlinear neutral differential equations with iterative terms and impulsive effects

Periodicity and positivity of solutions for first-order nonlinear neutral differential equations with iterative terms and impulsive effects

Optical soliton solutions of the M-fractional paraxial wave equation

This research used a modified and extended auxiliary mapping method to examine the optical soliton solutions of the truncated time M-fractional paraxial wave equation. We employed the truncated time M-fractional derivative to eliminate the fractional order in the governing model. The few optical wave examples of the paraxial wave condition can assume an insignificant part in depicting the elements of optical soliton arrangements in optics and photonics for the investigation of different actual cycles, including the engendering of light through optical frameworks like focal points, mirrors, and fiber optics. We identified the solution using a few free parameters regarding hyperbolic function form. We discovered periodic wave, bright and dark kink wave, bell wave, and singular soliton solution for the numerical values of the free parameters. To explain the behavior of various solutions, we have spoken the obtained solutions graphically for a physical explanation using MATLAB. The strategy introduced is fundamental and robust as a smart soliton solution for nonlinear partial differential equations, and it may play a crucial role in nonlinear optics, fiber optics, and communication systems.

Noether and partial Noether approach for the nonlinear (3+1)-dimensional elastic wave equations

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

On k-fuzzy metric spaces with applications

With application point of view, Gopal et al. [D. Gopal, W. Sintunavarat, A.S. Ranadive, S. Shukla, The investigation of k-fuzzy metric spaces with the first contraction principle in such spaces, Soft Comput., 27:11081–11089, 2023] generalized the conceptions of a fuzzy metric space and introduced the definition of k-fuzzy metric space. Here a fuzzy set defined in k-fuzzy metric space is a membership function FY : X × X × (0, +∞)k -> [0; 1], that is, the fuzzy distance between two points of the set depends on more than one parameter, and then also introduced first contraction principle in this space. In this sequel, we extend the work on k-fuzzy metric spaces by generalizing Banach contraction principle by introducing various type of inequalities. Here we introduce Tirado-type k-fuzzy contraction condition and prove fixed point theorem for Tirado-type contractive mapping. We also discuss the k-fuzzy ψ-contractive mapping, where ψ ∈ Ψ, and Ψ is a class of mappings defined from ψ : [0; 1] -> [0; 1] that has certain properties, and also obtained fixed point for such class of mappings. Later, we define Ćirić-type contraction inequalities to prove fixed point results by restricting ourselves on l-natural property of the fuzzy space to ensure the existence of fixed point. Between all results, a set of supportive examples are also produced to validate the results. In application section, we discuss the solutions of Volterra-type integral equations and second-order nonlinear ordinary differential equation.

Toward Efficient Numerical Solutions of Nonlinear Integral Equations with (T.L.D) Algorithm

This study focuses on a specific type of nonlinear second-order integral equations defined over a considerable interval. We propose a new numerical method, (T.L.D), in three phases: transformation of the equation, linearization with the Newton-Kantorovich method, and descritization with the Nyström technique. We theoretically define the convergence conditions and illustrate the accuracy of our method on several practical examples.

Fixed Points Results in Algebra Fuzzy Metric Space with an Application to Integral Equations

This paper introduces a new class of mappings termed (α̂,β̂)−Ω-contraction mapping (briefly, (α̂,β̂)−Ω−CMt;) and establishes certain fixed-point (FP) results in the framework of Algebra fuzzy metric space. Additionally, we expanded our results to include the existence of a nonlinear integral equation solution. Results from this study improve, expand and generalization certain previously published results in the literature.

On solution of non-linear FDE under tempered Ψ−Caputo derivative for the first-order and three-point boundary conditions

In this article, the existence and uniqueness of solutions for non-linear fractional differential equation with Tempered Ψ−Caputo derivative with three-point boundary conditions were studied. The existence and uniqueness of the solution were proved by applying the Banach contraction mapping principle and Schaefer’s fixed point theorem.

Exact and soliton solutions of nonlinear evolution equations in mathematical physics using the generalized (G′/ G)-expansion approach

Abstract The nonlinear evolution equations play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Klein–Gordon equation, a second-order equation in both space and time, is closely related to the Schrödinger equation and describes the behavior of spinless particles in theoretical physics. The equation is utilized extensively in many areas related to modern physics, including astrophysics, cosmology, and the theory of classic mechanics. The uKdV equation is suitable for use in various fields, including fluid dynamics, oceanography, and optical solitons research. It is useful for simulating events with wave-like characteristics and soliton dynamics, which contributes to the theoretical development of mathematical physics. This article addresses the exact and soliton solutions of nonlinear evolution equations in mathematical physics and engineering using the new generalized ( G ′ / G ) -expansion approach. We examine a large number of soliton solutions, including kink-shaped, singular-kink, single soliton, singular-soliton, periodic, and others. To better understand the characteristics of the solutions, some three-dimensional plots are set up for visualization. It has been demonstrated that the suggested approaches are more attractive as well as effective for getting single solutions to nonlinear evolution equations.

ON THE ABILITY OF DIFFERENTIAL TRANSFORM METHOD AND ITS IMPROVEMENTS TO SOLVE PROPERLY THE SHIP’S ROLL EQUATION

A major concern regarding the ship safety is related to severe rolling oscillations which, in certain circumstances, can even lead to its capsizing. Assuming the rolling motion decoupled from its other five possible motions, one results a mathematical model associated to a second order differential equation where the main parameters (inertia, damping, stiffness and excitation) reveal a significant nonlinearity. In the absence of exact analytical solutions, the amplitude and period characteristics of the ship rolling can be evaluated by approximate numerical or analytical methods. In this work, we checked if the performant differential transform method (DTM) and its improvement with Pade approximants is able to provide approximate analytical solutions for nonlinear roll equation, valid for both the transitory and stationary states. The obtained results were verified against those produced by MatLab numerical generated simulations. We noticed that for the linearized roll equation, which describes quite correctly a significant part of the situations of practical interest, the DTM doubled by the Pade approximation [4/4] offers the exact solution. If the nonlinear terms from damping and restoring moments are included in the study but the sea is considered waveless, the investigated technique proves a good accuracy in describing the attenuation over time of the initial excitation. DTM and Pade [4/4] cease to offer reasonable solutions as soon as the exciting moment of the waves is included in the procedure. We gave clear explanations for this impasse and showed that the use of higher-order Pade approximations can solve (even if with additional efforts) totally or at least partially this problem.

BASINS OF ATTRACTIONS OF NEW ITERATIVE METHODS FOR FINDING SIMPLE ZEROS

In this paper we present two new variants of Homeier’s iterative method for finding simple, real or complex, solution of nonlinear equations. Increasing the order of convergence from three to four is achieved by one additional term. Through many numerical examples, by classical and criteria based on basins of attraction, it is shown that the new methods can be competitive to other fourth-order methods.

STUDY OF THE BIAS OF N-PARTICLE ESTIMATES OF THE MONTE CARLO METHOD IN PROBLEMS WITH PARTICLE INTERACTION

The paper gives a theoretical and numerical justification of the bias with the 𝑂(1/𝑁) order for the 𝑁-particle statistical estimates of the functionals of the solution of nonlinear kinetic equations for the model with interaction of particle trajectories. An estimate of the coefficient in the corresponding bias formula is obtained.

ON EXACT SOLUTIONS OF MULTIDIMENSIONAL GENERALIZED MONGE–AMPERE EQUATION

Exact solutions of some multidimensional generalized Monge–Ampere equations are found. These solutions are a superposition of a quadratic form of spatial variables and solutions of nonlinear ordinary differential equations generated by the Monge–Ampere equations.