Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method

In this article, we apply the generalized Kudryashov method for finding exact solutions of three nonlinear partial differential equations (PDEs), namely: the Biswas-Milovic equation with dual-power law nonlinearity; the Zakharov--Kuznetsov equation (ZK(m,n,k)); and the K(m,n) equation with the generalized evolution term. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, and hyperbolic function solutions. Physical explanations for certain solutions of the three nonlinear PDEs are obtained.

On Fibonacci Functions with Period

A function is said to be a Fibonacci function if for all . In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period .

Exploration of Fibonacci Function

In this paper, I define Fibonacci function (probably unknown) on Real number field, for all x∈R, F:R → R,э F (x+n) = ∫nF(x+1)+∫n-1F(x). Also, I defined the limit value of Fibonacci function, which is closed to 1.618… where x tends to infinity. Including, Fibonacci sum as well.

THE QUASI-SINE FIBONACCI HYPERBOLIC DYNAMIC SYSTEM

This paper researches the dynamic behavior of a general form of the Fibonacci function, which is a quasi-sine Fibonacci function. It analyses the fixed points of the quasi-sine Fibonacci function on the real axis and the complex plane, and then constructs the Julia set of it using the escape-time method, discovering that the Julia set is fractal and it is on the x-axis symmetry. Using the conception of critical point, the quasi-sine Fibonacci function is generalized. Later the paper examines the dynamic behavior of the generalized quasi-sine Fibonacci function on critical points, and finds that the Mandelbrot set is also on the x-axis symmetry. Finally, it is discovered that there is a jumping phenomenon on the critical points.