We analyze the functional equation F(x+F(x))=−F(x) and reveal its relationship with a system of partial differential equations arising as hydrodynamic limit pinned billiard balls on line. The must freeze at some time, i.e., no velocity may change after freezing time. terminal time profiles play role boundary conditions for PDEs (qua conditions, despite being “initial conditions” from wave persp...

We consider the focusing L2-critical half-wave equation in one space dimension i∂tu = Du− |u|u, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold M∗ > 0 such that all H1/2 solutions with ‖u‖L2 < M∗ extend globally in time, while solutions with ‖u‖L2 > M∗ may develop singularities in finite time. In this paper, we first prove the ex...

The aim of this paper is to study a conservative wave equation coupled to a diffusion equation : this coupled system naturally arises in musical acoustics when viscous and thermal effects at the wall of the duct of a wind instrument are taken into account. The resulting equation, known as Webster-Lokshin model, has variable coefficients in space, and a fractional derivative in time. The port-Ha...

This paper presents a new numerical method to solve time fractional Fokker-Planck equation. The space dimension is discretized to the Gauss-Lobatto points, then we apply pseudo-spectral successive integration matrix for this dimension. This approach shows that with less number of points, we can approximate the solution with more accuracy. The numerical results of the examples are displayed.

We study a linear fractional Fokker–Planck equation that models non-local diffusion in the presence of a potential field. The non-locality is due to the appearance of the 'fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular diffusion. We prove existence of weak solutions by combining a splitting technique together with a Wa...

Numerical method for Cahn-Hilliard equation has been well-studied, but few can be generalized to fractional Cahn-Hilliard equation. In this project to modified the numerical method proposed by Brian Wetton et al in the paper High accuracy solutions to energy gradient flows from material science models[1]. The method they described in the paper is a pseudo-spectral method suitable for considerin...