This paper is devoted to improving the general Kudryashov method by a new and general auxiliary equation. So, a new method is introduced, which we call ‘the general improved Kudryashov method’, to produce exact solutions for nonlinear evolution equations arising in mathematical physics. As application examples, exact traveling wave solutions for the combined Korteweg–de Vries and modified Korteweg–de Vries (KdV–mKdV) equation and the (2+1)-dimensional Zakharov–Kuznetsov equation are obtained. These solutions can be classified as solitary and periodic wave solutions. Some of the obtained solutions are graphically sketched.

In this work, a new generalized concept of conformable derivative is given and named generalized θ –conformable derivative (GTCD). Several properties are studied such as the chain rule, the quotient rule, the product rule, the Rolle’s theorem, the mean value theorem and the fundamental theorems of calculus. Also, the geometrical and physical interpretations of the GTCD are presented. It is very easy to see that the GTCD is comprehensive and includes many past derivatives as special cases. An application of the generalized θ –conformable derivatives (GTCDs) is presented along with the G ′ / G , 1 / G –approach to construct a novel gathering of exact solutions for the nonlinear (2+1)-dimensional biological population model (BPM) involving GTCDs. Moreover, some comparisons and graphical interpretations are given to support our results related to the (2+1)-dimensional BPM. All acquired results regarding the GTCD show its validity to applying in many problems in mathematical physics, engineering, biology and others.

We deal with the Wick-type stochastic fractional Korteweg de–Vries (KdV) equation with conformable derivatives. With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.

In this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R) , $s\geq -1/4$s≥−1/4 . Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.