Abstract
The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions. logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution
Highlights
Nonlinear differential equations and their solutions play an important role at description of physical and other processes and as a result we can observe a large number of publications in this area
In this paper we demonstrate the most simple method for finding solitary wave solutions of nonlinear differential equations with the application of the logistic function
In this paper we have considered the generalized Kuramoto – Sivashinsky equation (7) using the Painleve analysis for nonlinear ordinary differential equations
Summary
Nonlinear differential equations and their solutions play an important role at description of physical and other processes and as a result we can observe a large number of publications in this area. In this paper we demonstrate the most simple method for finding solitary wave solutions of nonlinear differential equations with the application of the logistic function. The aim of this paper is to to use the logistic function to look for the analytical solutions of the generalized nonlinear wave equation in the form: ut + α un ux − δ (um ux)x + uxx + σ uxxx + uxxxx = 0,. In the case when there are three arbitrary constants in the Laurent series we have the necessary condition for integrable nonlinear differential equation As this takes place we can obtain three arbitrary constants in the expansion: two arbitrary coefficients in (13) and constant z0 which can be added to the variable z. The general solution of equation (14) has the pole of the second order
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