On the Use of Composite Functions in the Simple Equations Method to Obtain Exact Solutions of Nonlinear Differential Equations

Nikolay K Vitanov,Kaloyan N Vitanov,Zlatinka I Dimitrova

doi:10.3390/computation9100104

Nikolay K Vitanov, Kaloyan N Vitanov + Show 1 more

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https://doi.org/10.3390/computation9100104

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Journal: Computation	Publication Date: Sep 27, 2021
Citations: 14	License type: CC BY 4.0

Affiliation: Institute of Mechanics, Bulgarian Academy of Sciences

Abstract

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of a class of nonlinear differential equations containing polynomial nonlinearities. We present an amended version of the methodology, which is based on the use of composite functions. The number of steps of the SEsM was reduced from seven to four in the amended version of the methodology. For the case of nonlinear differential equations with polynomial nonlinearities, SEsM can reduce the solved equations to a system of nonlinear algebraic equations. Each nontrivial solution of this algebraic system leads to an exact solution of the solved nonlinear differential equations. We prove the theorems and present examples for the use of composite functions in the methodology of the SEsM for the following three kinds of composite functions: (i) a composite function of one function of one independent variable; (ii) a composite function of two functions of two independent variables; (iii) a composite function of three functions of two independent variables.

Highlights

We discuss in this article the mathematical problem for obtaining exact analytical solutions of nonlinear differential equations
Theorem 1 is for the case of a differential equation containing polynomial nonlinearities where the unknown function h depends on two independent variables
The theorem states that for the case when the unknown function is a composite function, constructed by exponential functions, the solved equation can be reduced to a system of nonlinear algebraic equations

Summary

Introduction

We discuss in this article the mathematical problem for obtaining exact analytical solutions of nonlinear differential equations. The effects connected to the nonlinearity are studied by means of time series analysis or by means of models based on differential or difference equations [14,15,16,17,18,19]. The following points from the history of the methodology for obtaining exact solutions of nonlinear differential equations are relevant for our study: 1. It transforms the nonlinear Burgers equation to the linear heat equation; 2. Almost at the same time, Hirota developed a method for obtaining exact solutions of nonlinear partial differential equations [25,26]. The Hirota method is connected to an appropriate transformation of the nonlinearity of the equation. The truncated Painleve expansions may lead to many of these transformations [27,28,29,30,31]

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