We address the propagation of vortex beams with circular Airy–Gaussian shape in a ( 2 + 1 )-dimensional optical waveguide modeled by fractional nonlinear Schrödinger equation. Systematic analysis autofocusing rev...

In this paper we present a numerical method for fuzzy differential equation of fractional order under gH-fractional Caputo differentiability. The main idea of this method is to approximate the solution of fuzzy fractional differential equation (FFDE) by an implicit method as corrector and explicit method as predictor. This method is tested on numerical examples.

We consider the Hankel determinant representation for the rational solutions of the Painlevé II equation. We give an explicit formula for the generating function of the entries in terms of logarithmic derivative of the Airy function, which by itself is a particular solution of the Painlevé II equation.

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy (L-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately sat...

An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial boundary value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of Airy equation and KdV equation in a finite interval.

In this paper we provide numerical and analytical evidence that some degenerate dispersive partial differential equations are ill-posed. Specifically we study the K(2, 2) equation ut = (u)xxx + (u)x and the ‘degenerate Airy’ equation ut = 2uuxxx . For K(2, 2) our results are computational in nature: we conduct a series of numerical simulations which demonstrate that data which is very small in ...

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...

in this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (fpdes) in the sense of modified riemann-liouville derivative. with the aid of symbolic computation, we choose the space-time fractional zakharov-kuznetsov-benjamin-bona-mahony (zkbbm) equation in mathematical physics with a source to illustrate the validity a...

in this paper, a numerical efficient method is proposed for the solution of time fractionalmobile/immobile equation. the fractional derivative of equation is described in the caputosense. the proposed method is based on a finite difference scheme in time and legendrespectral method in space. in this approach the time fractional derivative of mentioned equationis approximated by a scheme of order o...

in this paper, we develop an efficient legendre wavelets collocation method for well known time-fractional heat equation. inthe proposed method, we apply operational matrix of fractionalintegration to obtain numerical solution of the inhomogeneoustime-fractional heat equation with lateral heat loss and dirichletboundary conditions. the power of this manageable method isconfirmed. moreover, the ...