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

Area of Science:

• Applied Mathematics
• Non-linear Dynamics
• Mathematical Physics

Background:

• Non-linear differential equations are crucial in modeling complex phenomena.
• Obtaining exact solutions for equations with non-polynomial non-linearity remains a challenge.
• The Simple Equations Method (SEsM) offers a framework for solving such equations.

Purpose of the Study:

• To extend the applicability of the Simple Equations Method (SEsM) to non-polynomial non-linear differential equations.
• To introduce a transformation technique to convert non-polynomial non-linearity into polynomial non-linearity within SEsM.
• To demonstrate the method's effectiveness through illustrative examples, including the derivation of specific solution types.

Main Methods:

• Application of a novel transformation at Step 1 of the Simple Equations Method (SEsM).
• Conversion of non-polynomial non-linear terms into polynomial non-linear terms.
• Construction of composite solutions from solutions of simpler equations.
• Reduction of differential equations to systems of non-linear algebraic equations.
• Utilization of special functions as solutions when elementary functions are insufficient.

Main Results:

• Successfully applied SEsM to non-polynomial non-linear differential equations.
• Derived kink and anti-kink exact solutions for a specific equation.
• Demonstrated the use of special functions for equations lacking elementary solutions.
• Identified 10 potential transformations for converting non-polynomial to polynomial non-linearity.

Conclusions:

• The proposed transformation within SEsM effectively handles non-polynomial non-linear differential equations.
• The method provides a systematic way to obtain exact solutions, including kink/anti-kink types.
• The flexibility to incorporate special functions expands the scope of SEsM for complex problems.