Image inpainting is the process of repairing damaged or missing areas an image by utilizing information from its surrounding environment. The fractional order operator, which processes details like textures and edges with a finer scale, has demonstrated better outcomes success in processing. This study draws inspiration definition singular integral Laplacian presents two adaptive variational fu...

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (− ) as dispersive symbol. In particular, we obtain that fractional powers 1 2 < α < 1 arise from long-range lattice int...

Abstract. Using the method of sub-super-solution, we construct a solution of (−∆)su − cuz − f(u) = 0 on R of pyramidal shape. Here (−∆)s is the fractional Laplacian of sub-critical order 1/2 < s < 1 and f is a bistable nonlinearity. Hence, the existence of a traveling wave solution for the parabolic fractional Allen-Cahn equation with pyramidal front is asserted. The maximum of planar traveling...

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−∆)u = g in Ω, u ≡ 0 in R\Ω, for some s ∈ (0, 1) and g ∈ L∞(Ω), then u is C(R) and u/δ|Ω is C up to the boundary ∂Ω for some α ∈ (0, 1), where δ(x) = dist(x, ∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreov...

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.

In this article, we show the existence of ground state solutions for the nonlinear Schrödinger equation with fractional Laplacian (−∆)u+ V (x)u = λ|u|u in R for α ∈ (0, 1). We use the concentration compactness principle in fractional Sobolev spaces Hα for α ∈ (0, 1). Our results generalize the corresponding results in the case α = 1.

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.

Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier ...

In this paper, we give some properties of the left and right finite Caputo derivatives. Such derivatives lead to finite Riesz type fractional derivative, which could be considered as the fractional power of the Laplacian operator modelling the dynamics of many anomalous phenomena in super-diffusive processes. Finally, the exact solutions of certain fractional diffusion partial differential equa...