Solutions Of Nonlinear Differential Equations Research Articles

METHOD FOR CONSTRUCTING PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

A justification is given for the analytical method for constructing periodic solutions for nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series, in which the coefficients are polynomials depending on the parameter without the assumption of its smallness. Two examples are considered — the van der Pol equation and the Lorentz system.

STUDY OF LOADER TRANSMISSION DYNAMICS WITH HIGH TORQUE STEP-ADJUSTABLE HYDROMOTOR-WHEELS

Goal. The purpose of the article is to analyze the dynamics of the hydraulic fluid power of the forklift transmission equipped with radial-piston high- torque multi-cycle hydraulic motors with a step change in working volume. Such a transmission is offered as a gearless one compared to the use of traditional transmissions with axial-piston hydraulic motors and gearboxes in front of the drive wheel hub. Research methodology. The traction-speed characteristic of a serial wheel loader is built taking into account the constant power of the drive internal combustion engine for rotation of the axial- piston reversible adjustable transmission pump. The dependence of the torque on the wheel hub of the loader depending on its speed was analyzed. The selection of a high-torque radial-piston hydraulic motor of multi-cycle action, which satisfies the parameters of the transmission in terms of torque and maximum rotation frequency and does not have an intermediate gearbox on the hub, has been made. To reduce the set power of the drive motor of the loader pump and fuel consumption, the hydraulic motors of the transmission are also equipped with a two-stage regulator for changing the working volume. Constructed block diagrams in the VisSim application program package for the solution of nonlinear differential equations for determining the speed of the loader, the pressure in the hydraulic drive and its useful power. The results. The analysis of oscillograms of dynamic processes in the hydraulic drive of the loader showed that by means of modeling the pump supply modes and step change of the working volume of the multi-cycle radial piston hydraulic motor, it is possible to detect extreme modes that lead to fluctuations in the speed and power of the loader, as well as pressure in the hydraulic drive.

Existence and Stability for Fractional Differential Equations with a ψ–Hilfer Fractional Derivative in the Caputo Sense

This article aims to explore the existence and stability of solutions to differential equations involving a ψ-Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Dynamics of the rational difference equations

Discrete-time equations are sometimes used to explain natural phenomena that happen in nonlinear sciences. We study the periodicity, boundedness, oscillation, stability, and certain exact solutions of nonlinear difference equations in this paper. Using the standard iteration method, exact solutions are obtained. Some well-known theorems are used to test the stability of the equilibrium points. Some numerical examples are also provided to confirm the theoretical work’s validity. The numerical component is implemented with Wolfram Mathematica. The method presented may be simply applied to other rational recursive issues. In this article, we have explored solutions for the following difference equations: \begin{equation*} x_{n+1}=\frac{x_{n}x_{n-8}}{ \pm x_{n-7} \pm x_{n}x_{n-7}x_{n-8}}, \quad n \in \mathbb{N}_{0}. \end{equation*} These equations are subject to initial conditions $x_{-8}$, $x_{-7}$, $x_{-6}$, $x_{-5}$, $x_{-4}$, $x_{-3}$, $x_{-2}$, $x_{-1}$ and $x_{0}$, which are arbitrary non-zero real numbers. Additionally, we have provided solutions for specific cases of these equations and conducted an analysis of their dynamic behavior. Lastly, we have derived estimates for the initial coefficients.

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

Existenceof Heteroclinic Solutions in Nonlinear Differential Equations of the Second-Order Incorporating Generalized Impulse Effects with the Possibility of Application to Bird Population Growth

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with ϕ-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that ϕ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of L1-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth.

Dynamical analysis and solutions of nonlinear difference equations of thirty order

Discrete-time systems are sometimes used to explain natural phenomena that happen in nonlinear sciences. We study the periodicity, boundedness, oscillation, stability, and certain exact solutions of nonlinear difference equations in this paper. Using the standard iteration method, exact solutions are obtained. Some well-known theorems are used to test the stability of the equilibrium points. Some numerical examples are also provided to confirm the theoretical work’s validity. The numerical component is implemented with Wolfram Mathematica. The method presented may be simply applied to other rational recursive issues. \par In this paper, we explore the dynamics of adhering to rational difference formula \begin{equation*} x_{n+1}=\frac{x_{n-29}}{\pm1\pm x_{n-5}x_{n-11}x_{n-17}x_{n-23}x_{n-29}}, \end{equation*} where the initials are arbitrary nonzero real numbers.

One class of solutions to the equations of ideal plasticity

Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Analytical solution to boundary layer flow and convective heat transfer for low Prandtl number fluids under the magnetic field effect over a flat plate

AbstractThe present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force.

Method for Constructing Periodic Solutions of Nonlinear Differential Equations

Method for Constructing Periodic Solutions of Nonlinear Differential Equations

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times

In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times.

The general Bernstein function: Application to χ‐fractional differential equations

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

Novel Θ‐Fuzzy‐Contraction Mappings and Existence of Solution of Nonlinear Differential Equations

This study pioneers the introduction of two novel concepts in the realm of double‐controlled metric spaces: the Θ‐fuzzy double‐controlled contraction mapping and the Θ‐fuzzy almost generalized double‐controlled contraction mapping. These concepts represent a significant expansion of the existing framework of generalized contractions. This research establishes the existence and uniqueness of fixed points for each of these contraction mappings and provides exemplary illustrations to clarify the results. Moreover, we demonstrate the practical applicability of our findings by showing the existence of a solution to a nonlinear differential equation. The established theorems also yield various corollaries, which confirm that our results generalize and extend previously established findings in the field.

Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The Stability and Catastrophic Behavior of Finite Periodic Solutions in Non-Linear Differential Equations

This study focuses on the stability and catastrophic behavior of finite periodic solutions in non-linear differential equations. The occurrence of folding surfaces and their relationship with saddle-node bifurcations are explored, being classified as fold and butterfly types of catastrophes. Additionally, the application of catastrophe theory is discussed to analyze the qualitative changes in solutions with the change in system parameters.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

A Chebyshev polynomial approach to approximate solution of differential equations using differential evolution

Most of the problems associated with physical, biological and engineering systems are complex and are usually modeled with the help of linear as well as non-linear differential equations. In general, the exact solutions of non-linear differential equations may not be obtained by analytical methods and thus, in this paper, a numerical method is proposed to find approximate solutions of differential equations using Chebyshev polynomials and metaheuristic optimization algorithms. In this paper, Ordinary differential equations (ODEs) are modeled as optimization problems to determine their approximate solutions by considering Chebyshev polynomials as the base approximation function, taking initial and/or boundary conditions as constraints. The unknown coefficients of the polynomials are calculated using Differential Evolution (DE) in order to get the approximate solution with minimal error. The proposed method is applied to find the approximate solution to both Initial and Boundary Value Problems. For all test problems considered in this paper, the algorithms have been programmed using MATLAB software to run on computer. The effectiveness of the method is demonstrated by graphical comparison of the computed approximate solutions with that of the exact ones. Further, the performance of the method proposed in this paper is shown by finding out the Root Mean Square Error (RMSE) between the exact and approximate solution, which is found to be better than that of the related existing papers.

Quasi-periodic development of the economy

The article considers the mathematical definition of the space of states of the macroeconomy, where each point represents a certain distribution of the growth of the main factors of production – labor, capital, and natural resources. We study the violation of the sustainable regime of economic development, described by the system of differential equations in partial derivatives. The phenomenon of stable development of the economy, as well as the presence of more than one stable regime of economic development, depending on political and international factors, is studied. The role of periodic development in the economy and the existence of multiple attractors that create various stable regimes are considered in detail. The various stable regimes that arise are solutions of non-linear differential equations. We study the process of the emergence of instability, when the economic number that defines a particular economy continues to increase after reaching its first critical value. A further increase in the economic number leads to the fact that periodic development becomes unsustainable. The trajectory analysis of production factors is performed by the method presented by L.D. Landau and E.M. Lifshitz. The emerging stabilization of the economy can be broken again as the economic number approaches the next critical value; instability reappears with the advent of a new frequency. After passing through several critical values of the economic number, a stable limit cycle can appear on the torus, which is a geometric interpretation of the trajectories of economic development. The appearance of a stable limit cycle means the manifestation of synchronization of oscillations the disappearance of the quasi-periodic and the establishment of a new periodic motion. The birth of a stable limit cycle prevents the emergence of a regime of superpositions of motions of trajectories with many incommensurable frequencies.