Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity.

Nikolay K Vitanov,Zlatinka I Dimitrova

doi:10.3390/e23121624

Abstract

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

Highlights

Complex systems are numerous in human societies and in Nature
simple equations method (SEsM) is connected to the possibility of using more than one simple equation in order to construct the solution of the solved non-linear differential equation
We have demonstrated that there are cases when solutions of the solved nonlinear differential equation exist, but they cannot be constructed by elementary functions or by the known special functions

Summary

Introduction

Complex systems are numerous in human societies and in Nature. Several examples are research groups, communities, traffic networks, stock markets, etc. [1–8]. We discuss the SEsM (Simple Equations Method) for obtaining exact solutions of non-linear differential equations. SEsM is connected to the possibility of using more than one simple equation in order to construct the solution of the solved non-linear differential equation. An important part of SEsM is the construction of the solution of the solved equation This solution is a composite function of solutions of more simple differential equations. We will use the capacity of the SEsM to obtain exact analytical solutions of many equations containing polynomial non-linearity. We emphasize the usefulness of a special function, which is a solution of an ordinary differential equation containing polynomial non-linearity. The application of (2) to (1) leads to non-linear differential equations for the functions Fi. We do not know the general form for the transformation T. This case is reduced to the previous one and a nontrivial solution is possible

General Considerations

Example 1

Example 2

Concluding Remarks