This work explores Kink soliton solution, periodic and rational function solutions for the fractional generalized anti-cubic (FGAC) nonlinearity in fiber Bragg gratings (BGs). The ((D_ζ^α G)/G)-expansion method is employed conjunction with idea of a conformable derivative. Due to its nature, solution looks have some restrictions.

In this paper, we deal with the Duhamel formula, existence, uniqueness, and stability of mild solutions a class nonlocal impulsive differential equations conformable fractional derivative. The main results are based on semigroup theory combined some fixed point theorems. We also give an example to illustrate applicability our abstract results.

In this work, we study the following conformable fractional Sturm--Liouville
 problem
 \[
 l[y]=-T_{\alpha }(p(t)T_{\alpha }y(t))+q(t)y(t),
 \]
 where $t\in \lbrack 0,\infty ),$ real-valued functions $p$ and $q$
 satisfy conditions:
 \begin{array}{cc}
 (i) & q\in L_{\alpha }^{2}[0,\infty ), \\ 
 (ii) p\ \text{is\ absolutely\ continuous\ on}\ [0,\...

In this paper, the WKB method is extended to be applicable for conformable Hamiltonian systems where concept of operator with fractional order $\alpha$ used. The approximation $\alpha$-wavefunction derived when potential slowly varying in space. paper furnished some illustrative examples demonstrate method. quantities form are found inexact agreement traditional $\alpha=1$.

In this paper, a modified nonlinear Schrödinger equation with spatiotemporal dispersion is formulated in the senses of Caputo fractional derivative and conformable derivative. A new generalized double Laplace transform coupled Adomian decomposition method has been defined applied to solve newly dispersion. The approximate analytical solutions are obtained compared each other graphically.