Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

• Published in: Entropy (Basel, Switzerland) Switzerland), 2022-09, Vol.24 (9), p.1288
• Main Author: Nikolova, Elena V.
• Format: Article
• Language: English
• Subjects: Composite functions; Differential equations; exact travelling-wave solutions; extended fifth-order Korteweg-de Vries equation; Independent variables; Korteweg-Devries equation; Linear equations; Mathematical analysis; Methods; Partial differential equations; Polynomials; Riccati equation; Series (mathematics); simple equations method; Traveling waves
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e24091288

We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg-de Vries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation.