Vitanov Nikolay K, Dimitrova Zlatinka I
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria.
Climate, Atmosphere and Water Research Institute, Bulgarian Academy of Sciences, Blvd. Tzarigradsko Chaussee 66, 1784 Sofia, Bulgaria.
Entropy (Basel). 2021 Dec 2;23(12):1624. doi: 10.3390/e23121624.
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.
我们讨论了简单方程法（SEsM）在求解含非多项式非线性的几类方程的非线性微分方程精确解中的应用。该研究的主要思路是在简单方程法的步骤（1.）中使用适当的变换。此变换必须将非多项式非线性转化为多项式非线性。然后，构建一个适当的解。该解是更简单方程解的复合函数。该解的应用将微分方程简化为一个非线性代数方程组。我们列出了10种可能的适当变换。给出了该方法应用的两个例子。在第一个例子中，我们得到了所求解方程的扭结解和反扭结解。第二个例子说明了该研究的另一个要点。要点如下。在某些情况下，简单方程法中使用的简单方程没有由初等函数或常用特殊函数表示的解。在这种情况下，我们可以使用一个特殊函数，它是一个含多项式非线性的适当常微分方程的解。第二个例子给出了使用此函数的具体情况。
