It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article to use fractional KP not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, definitions conformable partial derivatives and integrals together with a physical interpretation are introduced then integrable consisting KPI KPII equa...

Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective, fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two quadratures. This theoretical result is corroborated by numerical simulations for different shapes of the periodic potential. Normal and fractional spreading process...

Fractional, anomalous diffusion in space-periodic potentials is investigated. The analytical solution for the effective fractional diffusion coefficient in an arbitrary periodic potential is obtained in closed form in terms of two quadratures. This theoretical result is corroborated by numerical simulations for different shapes of the periodic potential. Normal and fractional spreading processe...

in this paper, a new numerical method for solving time-fractional diffusion equations is introduced. for this purpose, finite difference scheme for discretization in time and chebyshev collocation method is applied. also, to simplify application of the method, the matrix form of the suggested method is obtained. illustrative examples show that the proposed method is very efficient and accurate.

We generalize a method introduced by Bourgain in [2] based on complex analysis to address two spatial dimensional models and prove that if a sufficiently smooth solution to the initial value problem associated with the Kadomtsev-Petviashvili (KP-II) equation (ut + uxxx + uux)x + uyy = 0, (x, y) ∈ R, t ∈ R, is supported compactly in a nontrivial time interval then it vanishes identically.

Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-diierential operators lead, in turn, to three diierent classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modiied Kadomtsev-Petviashvili and Dym hierarchies in 2+1 dimensions. Miura-and auto-BB acklund transformations are show...

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as wel...

Anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional Fokker-Planck dynamics via the underlying continuous-time random walk. An efficient numerical algorithm is developed which is applicable for an arbitrary potential. This algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in d...

We consider a one-dimensional fractional diffusion equation: ∂α t u(x, t) = ∂ ∂x ( p(x) ∂u ∂x (x, t) ) , 0 < x < `, where 0 < α < 1 and ∂α t denotes the Caputo derivative in time of order α. We attach the homogeneous Neumann boundary condition at x = 0, ` and the initial value given by the Dirac delta function. We prove that α and p(x), 0 < x < `, are uniquely determined by data u(0, t), 0 < t ...