Fractional quasi AKNS‐technique for nonlinear space–time fractional evolution equations

Abstract

This paper aims to formulate the fractional quasi-inverse scattering method.Also, we give a positive answer to the following question: can the Ablowitz-Kaup-Newell-Segur (AKNS) method be applied to the space-time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi-conservation laws for the considered fractional equations from the defined fractional quasi AKNS-like system. The nonlinear fractional differential equations to be studied are the space-time fractional versions of the Kortweg-de Vries equation, modified Kortweg-de Vries equation, the sine-Gordon equation, the sinh-Gordon equation, the Liouville equation, the cosh-Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.

Approximation of functions by complex conformable derivative bases in Fréchet spaces

“…Remark The two definitions (10) and (12) are the extension of conformable fractional derivative and integral defined by Khalil et al, 27 which has many application in physics, mathematical problems, plasma, astronomy, and engineering; see, for example, previous works 36,38,55–58 and reference therein. Furthermore, there are several applications of the two definitions (10) and (12) in the theory of conformal complex analysis, see previous studies 30–33 …”

“…For more information about the study of fractional integral and differential operators, we refer to literature 34–44 …”

Stationary Response of a Kind of Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative Damping

“…A fractional derivative is a generalization of the integer-order derivatives, which have been widely applied in different fields, such as dynamics, engineering, computer science, etc. [22][23][24][25][26][27][28]. Fractional derivative damping is usually utilized to describe the viscoelastic damping model.…”

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

“…Fractional nonlinear models (FNMs) were also widely utilized in different domains of engineering, 4,5 physics, 6–9 biology, 10–14 and mathematics 15,16 . Because many systems exhibit post effects or memory, fractional derivatives are a better explanation, 17 and thus, FNMs can be extended to model a variety of complex phenomena 18–20 …”

“…Diverse methods including F‐expansion method, 29 extended sinh‐Gordon equation method, 27 and (G'/G)‐expansion method 28 have been put forward to study FNLSEs. These methods in the previous studies 19,20,27–29 simplified FPDEs into ordinary differential equations that are easier to solve based on Jumarie's basic formulae 30 . However, some researchers 31 have pointed that these transformations used in these methods 20,29,30 are not true because Jumarie's basic formulae used in the previous studies 20,27–29 are not correct.…”