Class Of Nonlinear Differential Equations Research Articles

Periodic Solutions for a Class of Nonlinear Differential Equations

Periodic Solutions for a Class of Nonlinear Differential Equations

Soliton structures for the (3 + 1)-dimensional Painlevé integrable equation in fluid mediums

The (3 + 1)-dimensional Painlevé integrable equation are a class of nonlinear differential equations with special properties, which play an important role in nonlinear science and are of great significance in solving various practical problems, such as many important models in fields such as quantum mechanics, statistical physics, nonlinear optics, and celestial mechanics. In this work, we utilize the Hirota bilinear form and Mathematica software to formally obtain the interaction solution among lump wave, solitary wave and periodic wave, which has not yet appeared in other literature. Additionally, using the (G′/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G)$$\\end{document}-expansion method, we provide a rich set of exact solutions for the (3 + 1)-dimensional Painlevé integrable equation, which includes two functions with arbitrary values. This method is the first to be applied to the (3 + 1)-dimensional Painlevé integrable equation. By giving some 3D graphics and density maps, the dynamic properties are analyzed and demonstrated, which is beneficial for promoting understanding and application of the (3 + 1)-dimensional Painlevé integrable equation.

Periodic solutions in reversible symmetric second order systems with multiple distributed delays

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result.

An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations

This article presents a novel approach using the Fibonacci wavelet collocation method (FWCM) for the numerical solution of Emden-Fowler-type equations. The Emden-Fowler equations are a class of nonlinear differential equations that arise in various fields of science and engineering, particularly in astrophysics and fluid dynamics. Due to their nonlinear and singular nature, the conventional approaches to solving these equations encounter difficulties. This method is particularly effective in handling problems with singularities, as it adapts naturally to the local behaviour of the solution. Here, we introduced the Fibonacci wavelet collocation method as a powerful numerical technique for tackling Emden-Fowler-type equations. The Fibonacci wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Emden-Fowler equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, several examples demonstrate the method's flexibility when dealing with different singularity levels. This paper contributes to numerical analysis by introducing the Fibonacci wavelet method as a robust tool for solving Emden-Fowler-type equations.

New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings.

Superconductors and the periodic penetration parameter: Defining and utilizing in diverse applications

There are various types of materials that have different levels of electrical conductivity, and one category is known as superconductors or superconducting materials. Superconducting materials are characterized by their complete lack of electrical resistivity. These materials are highly important due to their wide range of applications in electricity transmission, although they do have certain limitations. The Bardeen–Cooper–Schryver theory and the Ginzburg–Landau theory are two significant theories used to explain the nature of superconducting materials. Of particular interest in this study is the Ginzburg–Landau differential equation, which is considered a vital equation in this field. This equation belongs to a class of nonlinear differential equations. Our research focuses on simulating solutions to the Ginzburg–Landau equation under steady-state conditions. We conducted simulations for several superconducting materials, including aluminum, niobium, lead, tin, niobium germanide, niobium tin, vanadium silicate, lead hexa-molybdenum octa-sulfur, magnesium diboride, uranium triplatinum, potassium, barium copper oxide, yttrium, calcium copper oxide, and barium mercury. We define a new parameter of the superconductor conduction materials, which is the periodic parameter of the superconductor. By analyzing the periodic solutions obtained from the Ginzburg–Landau differential equation, we were able to determine the values of the periodic penetration parameters for each material. Notably, monatomic superconducting materials exhibited periodic penetration parameters in the range of tens of micrometers, while tetra- and penta-elements materials had values in the tens of nanometers. Superconducting materials of two or three different elements showed average values for these parameters. These findings provide valuable insights into the characteristics and behavior of various superconducting materials.

Theoretical analysis of a class of $ \varphi $-Caputo fractional differential equations in Banach space

<abstract><p>A study of a class of nonlinear differential equations involving the $ \varphi $-Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results.</p></abstract>

A RESULT OF MEAN SQUARE EXPONENTIAL STABILITY FOR DIFFERENTIAL DELAY EQUATIONS WITH STOCHASTIC NOISE.

In the present paper, we aim to study of a class of nonlinear differential equations with stochastic noise. Firstly, we introduce the condition of local Lipschitz and a new non-linear growth condition. Then by applying Lyapunov function and semi-martingale convergence theorem, we prove that the stochastic system under consideration has a unique global solution. Additionally, we also investigate the exponential stability of the mean square.

Almost Automorphic Solutions to Nonlinear Difference Equations

In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions.

New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

A novel approach for an approximate solution of a nonlinear equation of charged damped oscillator with one degree of freedom

A novel technique is proposed for finding an approximate solution of the strongly nonlinear ordinary differential equation for the charged damped pendulum with one degree of freedom. The method relies on a transformation of the governing nonlinear differential equation that keeps unchanged the order of the highest derivative, in conjunction with a modified homotopy perturbation technique (MHPM). Only quadratic damping is considered for the numerical computations. To validate the used technique, the obtained results are compared to those arising from a numerical solution by Runge–Kutta of the fourth order (RK4) and by finite differences (FD). Good agreement between the two solutions is reached when quadratic damping is suppressed. In the presence of damping, agreement takes place only for a rather limited range of times. Plots of the analytical solutions are provided for both cases. The proposed method may be used to analyze a wide class of nonlinear differential equations.

On the Qualitative Analysis of Solutions of a Class of Nonlinear Differential Equations of the Second Order with Constant Delay

On the Qualitative Analysis of Solutions of a Class of Nonlinear Differential Equations of the Second Order with Constant Delay

A Deterministic Setting for the Numerical Computation of the Stabilizing Solutions to Stochastic Game-Theoretic Riccati Equations

In this paper, we are interested in the numerical aspects of the class of generalized Riccati difference equations which are involved in linear quadratic (LQ) stochastic difference games. More specifically, we address the problem of the numerical computation of the stabilizing solutions for this class of nonlinear difference equations. We propose an iterative deterministic algorithm for the computation of such a global solution. The performances of the proposed algorithm are illustrated with some numerical examples.

On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$

In this paper, we aim to investigate the qualitative behavior of a general class of non-linear difference equations. That is, the prime period two solutions, the prime period three solutions and the stability character are examined. We also use a new technique introduced in [1] by E. M. Elsayed and later developed by O. Moaaz in [2] to examine the existence of periodic solutions of these general equations. Moreover, we use homogeneous functions for the investigation of the dynamics of the aforementioned equations.

Peridynamic differential operator-based nonlocal numerical paradigm for a class of nonlinear differential equations

Peridynamic differential operator-based nonlocal numerical paradigm for a class of nonlinear differential equations

Solvability and representations of the general solutions to some nonlinear difference equations of second order

<abstract><p>We give detailed theoretical explanations for getting the closed-form formulas and representations for the general solutions to four special cases of a class of nonlinear difference equations of second order considered in the literature, present an extension of the class of difference equations which is solvable in closed form, analyze some results on the long-term behavior of the solutions to the class of equations, and give some results on convergence.</p></abstract>

Влияние возмущения подвижной особой точки на структуру аналитического приближенного решения одного класса нелинейных дифференциальных уравнений третьего порядка в комплексной области

The authors prove the theorem of existence and uniqueness of the solution, construct an analytical approximate solution in the complex domain for one class of nonlinear differential equations of the third order, the solution of which are discontinuous functions. The solution of the listed mathematical problems is based on the classical approach. Since the existing methods allow obtaining moving singular points only approximately, it is necessary to investigate the effect of a perturbation of a moving singular point on the structure of an analytical approximate solution in the complex domain. Since the existing methods allow obtaining moving singular points only approximately, it is necessary to investigate the effect of a perturbation of a moving singular point on the structure of an analytical approximate solution in the complex domain. A theorem that allows us to determine a priori estimates of the error of the analytical approximate solution is proved. The study applies the classical approach to estimation and illustrates the application of series with fractional negative powers. The article presents the results of numerical experiment confirming the validity of the obtained theoretical position. The technique of optimization of a priori estimates of analytical approximate solution in the vicinity of perturbed value of moving singular point using a posteriori estimates is presented. The results allow expanding the classes of nonlinear differential equations used as a basis for mathematical models of processes and phenomena in various fields of human activity. In particular, the class of equations under consideration can be applied in the study of wave processes in elastic beams, which is confirmed by theoretical data

Fibonacci collocation method for solving a class of nonlinear differential equations

In this study, a collocation method based on Fibonacci polynomials is used for approximately solving a class of nonlinear differential equations with initial conditions. The problem is firstly reduced into a nonlinear algebraic system via collocation points, later the unknown coefficients of the approximate solution function are calculated. Also, some problems are presented to test the performance of the proposed method by using error functions. Additionally, the obtained numerical results are compared with exact solutions of the test problems and approximate ones obtained with other methods in literature.

Existence and uniqueness of solution for fractional differential equations with integral boundary conditions and the Adomian decomposition method

We propose an Adomian decomposition method to solve a class of nonlinear differential equations of fractional‐order with modified Caputo derivatives and integral boundary conditions. Our approach uses the integral boundary conditions to derive an equivalent nonlinear Volterra integral equation before establishing existence and uniqueness of solution and a recursion scheme for the solution. The convergence of the method is proved and an error analysis given. Two numerical examples are solved by obtaining a rapidly converging sequence of analytical functions to the solution.

Analytic Approximate Solution in the Neighborhood of a Moving Singular Point of a Class of Nonlinear Equations

The paper considers a class of nonlinear differential equations which are not solvable in quadratures in general case. The author’s technology for solving such equations contains six problems. In this article, the solution to one of these problems is given, a real area in which it is possible to calculate an analytical approximate solution in the case of an approximate value of a moving singular point is obtained. Obtained results are based on the application of elements of differential calculus in finding estimates for the approximate solution error. Theoretical provisions are confirmed by numerical calculations, which characterize their reliability.