On bounded nonoscillatory solutions of third-order nonlinear differential equations

Seven common errors in finding exact solutions of nonlinear differential equations

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

On the solution of nonlinear fractional order differential equation

The continuation of solutions for certain nonlinear differential equations and its applications

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

In this paper, we study the existence of quasi-periodic solutions and the boundedness of solutions for a wide class nonlinear differential equations of second order. Using the KAM theorem of reversible systems and the theory of transformations we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

We prove that all the meromorphic solutions of the nonlinear differential equation c0u‴+6u4+c1u″+c2uu′+c3u3+c4u′+c5u2+c6u+c7=0 are elliptic or degenerate elliptic, and we build them explicitly.

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

Existence and stability of stationary solutions of nonlinear random difference equations are studied in this note. Firstly, we give the weak conditions that guarantee the continuity of Lypanunov exponents under small random perturbations. Secondly, we find out the conditions under which the ratio of the random norm and the standard Euclidean norm has deterministic bounds. Based on these new results, we provide easy-to-use conditions that guarantee the existence and almost sure stability of a stationary solution. In addition, we also prove that the stationary solution converges with probability one to the fixed point of the corresponding deterministic system as the noise intensity tends to zero.

Asymptotic to polynomials solutions for nonlinear differential equations

Realistic scenarios typically involve several interacting dependent variables, leading to nonlinearity in mathematical models. Nonlinear differential equations are a crucial mathematical phenomenon for describing real-world problems. On the other hand, fuzzy set theory is a valuable tool for modelling the uncertainty present in a system. So, nonlinear mathematical models under uncertainty need an explanatory study on a system of nonlinear fuzzy differential equations. In this paper, we analyse nonlinear fuzzy differential equations with possible applications. This paper advances the existing literature by introducing a definition of a fuzzy sequence and developing the existence and uniqueness theory for fuzzy solutions of nonlinear differential equations using the fixed-point theorem. The component-wise analysis of a fuzzy nonlinear system has been conducted. Furthermore, the analysed theory precedes an aptly fitted application of a fuzzy nonlinear dynamical system using a numerical scheme.

One of the major difficulties encountered in the general solutions of nonlinear differential equations, in contrast to the solutions of linear ones, is the manner in which the arbitrary constants enter these solutions. In this paper information is given and results are found in connection with this subject.

1.1. This paper is a theoretical investigation in the real domain of the existence of subharmonic solutions of non-linear differential equations of the formwhere F is analytic and of least period 2π/ω in t; ε = (ε1, …, εn) is small; and F(x, ẋ, 0, t) is not linear in x and ẋ.