Solutions Of Nonlinear Differential Equations Research Articles

Strongly unpredictable solutions of difference equations

The theory of difference equations has wide application for approximation of differential operators, for constructing various discrete models, etc. Their usefulness is confirmed by numerous scientific publications, articles, and many international conferences. This article discusses new types of oscillations, unpredictable sequences, the presence of which shows the existence of chaos. The existence and uniqueness of strongly unpredictable solutions of non-linear difference equations are proved. An example with numerical simulations is presented to illustrate the theoretical results.

High accuracy least-squares solutions of nonlinear differential equations.

This study shows how to obtain least-squares solutions to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by any existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a constrained expression, derived from Theory of Connections. In this expression, the differential equation constraints are embedded and are always satisfied. The resulting constrained expression is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10-12. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.

Mathematical modeling of complex structures and nonlinear differential equations with movable points

Mathematical modeling as one of the strict methods of research justification is applied in different fields of human activity including structural mechanics. Differential equations represent one of the variants of the basics of a mathematical model. If linear differential equations do not cause difficulties, the nonlinear ones will present a serious problem for a researcher due to the appearance of movable singular points. Such differential equations are related to the class of equations generally not solvable in quadratures. This fact actualizes the problem of the approximate methods development to solve this category of equations, since the existing classical methods are very often turned out to be ineffective. This paper presents the development of an analytical method for approximate solutions of nonlinear differential equations with movable singular points. The formulation of the emerged tasks is given as well as the technique of their solution. The obtained theoretical material is tested by numerical experiments confirming the reliability of the results.

On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations

In this study, an optimal homotopy analysis approach will be described to deal with nonlinear fractional differential equations. The proposed approach presents a procedure to find an optimal auxiliary linear operator and the corresponding optimal initial approximation that will accelerate the convergence of series solutions for nonlinear differential equations with fractional derivatives. Then, a reliable modified version of the homotopy analysis method is presented to facilitate the calculations. Numerical comparison will be made to examine the computational efficiency and the pertinent features of the proposed algorithm. This algorithm is expected to be further employed to solve wide classes of nonlinear problems in fractional calculus.

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.

Exact solutions of the equation for surface waves in a convecting fluid

A method for finding exact solutions and the first integrals is presented. The basic idea of the method is to use the value of the Fuchs index that appears in the Painlevé test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations. It allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations. The main feature of the method is that we do not assign a solution function at the beginning, we find this function during calculations. This approach is conceptually equivalent to the third step of the Painlevé test and sometimes allows us to change this step. Our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations. We demonstrate a method for finding the traveling wave solutions and the first integrals of the well-known nonlinear evolution equation for description of surface waves in a convecting liquid. The general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth-order equation are found.

Green’s functions for higher order nonlinear equations

The well-known Green’s function method has been recently generalized to nonlinear second-order differential equations. In this paper, we study possibilities of exact Green’s function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green’s function can be represented in terms of the homogeneous solution. Specific examples and a numerical analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg–de Vries equations, we are forced to introduce higher order Green’s functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.

Msthematical Model of Operational Competition

They developed a mathematical model of operational competition on a linear range. The model is represented by an initial-boundary problem for the system of evolution equations. It is shown that the competition on the recovered trophic resource does not lead to the disappearance of one of the populations. The estimation of small population propagation rates in the range is given. They obtained the conditions for the existence of an autowave solution on an infinite straight line. The solution of nonlinear differential equations is based on numerical method use.

A New Approach to Approximate Solutions for Nonlinear Differential Equation

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

Normally generated line bundle and laurent series solutions of nonlinear differential equations

The aim of this paper is to demonstrate the rich interaction between the properties of dynamical systems, the geometry of its asymptotic solutions, and the theory of Abelian varieties. We are going to illustrate as well that the nature of many methods for finding solutions of some dynamical systems is determined by the Laurent series for solutions of nonlinear differential equations. We apply the methods to the Kowalewski’top a solid body rotating about a fixed point, the Kirchhoff’s equations of motion of a solid in an ideal fluid and the Ramani-Dorizzi-Grammaticos (RDG) series of integrable potentials.

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painleve test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.

Dhage iteration method for approximating solutions of nonlinear differential equations with maxima

Dhage iteration method for approximating solutions of nonlinear differential equations with maxima

Oscillation of Solutions of Nonlinear Difference Equation With a Super-linear Neutral Term

Abstract This paper deals with the oscillation of solutions of certain class of neutral difference equation ∆(an∆(χn + pnχαn−k)) + qnχβn+1−l = 0, where α and β are ratio of odd positive integers. New sufficient conditions are obtained for the oscillation of studied equation and examples illustrating the main results are provided.

Dynamic characteristic of spur gear with flexible support of gearbox

In this study, a nonlinear translation-torsion model of spur gear pair with flexible support of gearbox is proposed. The time-varying meshing stiffness, transmission error and backlash are considered in this model. Lagrange’s equations are used for establishing the mathematic model. The numerical method is presented for solutions of nonlinear differential equations. The effect of rotating speed and support stiffness of gearbox is analyzed. The numerical results show that the flexibility of the support of gearbox has a significant effect on the amplitude-frequency characteristic of the spur gear pair at low rotating speeds. The response shows flexibility while the support stiffness is smaller than the bearings and rigidity while the support stiffness is larger than the bearings. The maximum deformation of the driving gear bearings under the flexible support is generally greater than the one under rigid support.

Substrate mass transfer: analytical approach for immobilized enzyme reactions

In this paper, the boundary value problem in immobilized enzyme reactions is formulated and approximate expression for substrate concentration without external mass transfer resistance is presented. He’s variational iteration method is used to give approximate and analytical solutions of non-linear differential equation containing a non linear term related to enzymatic reaction. The relevant analytical solution for the dimensionless substrate concentration profile is discussed in terms of dimensionless reaction parameters α and β.

Analytic Solutions of Nonlinear Partial Differential Equations by the Power Index Method

An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or 180 ° rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations.

Modified harmonic balance method for the solution of nonlinear jerk equations

In this paper, a second approximate solution of nonlinear jerk equations (third order differential equation) can be obtained by using modified harmonic balance method. The method is simpler and easier to carry out the solution of nonlinear differential equations due to less number of nonlinear equations are required to solve than the classical harmonic balance method. The results obtained from this method are compared with those obtained from the other existing analytical methods that are available in the literature and the numerical method. The solution shows a good agreement with the numerical solution as well as the analytical methods of the available literature.

Differential quadrature method (DQM) for studying initial imperfection effects and pre- and post-buckling vibration of plates

The effects of initial geometric imperfection and pre- and post-buckling deformations on vibration of isotropic rectangular plates under uniaxial compressive in-plane load have been studied. The differential equations of plate motions, using the Mindlin theory and Von-Karman stress-strain relations for large deformations, were extracted. The solution of nonlinear differential equations was assumed as the summation of dynamic and static solutions. Due to a large static plate deflection as compared with its vibration amplitude, the differential equations were solved in two steps. First, the static equations were solved using the differential quadrature method and the arc-length strategy. Next, considering small vibration amplitude about the deformed shape and eliminating nonlinear terms, the natural frequencies were extracted using the differential quadrature method. The results for different initial geometric imperfection and different boundary conditions reflect the impact of the mentioned factors on the natural frequencies of plates.

NUMERICAL WITHOUT ITERATION METHOD OF MODELING OF ELECTROMECHANICAL PROCESSES IN ASYNCHRONOUS ENGINES

Purpose. Development of calculation of electromagnetic and electromechanic transients is in asynchronous engines without iterations.Methodology. Numeral methods of integration of usual differential equations, programming. Findings. As the system of equations, describing the dynamics of asynchronous engine, contents the products of rotor and stator currents and product of rotation frequency of rotor and currents, so this system is nonlinear one. The numeral solution of nonlinear differential equations supposes an iteration process on every step of integration. Time-continuing and badly converging iteration process may be the reason of calculation slowing. The improvement of numeral method by the way of an iteration process removing is offered. As result the modeling time is reduced. The improved numeral method is applied for integration of differential equations, describing the dynamics of asynchronous engine.Originality. The improvement of numeral method allowing to execute numeral integrations of differential equations containing product of functions is offered, that allows to avoid an iteration process on every step of integration and shorten modeling time.Practical value. On the basis of the offered methodology the universal program of modeling of electromechanics processes in asynchronous engines could be developed as taking advantage on fast-acting.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.