Stochastic Diffusion Equation with Fractional Laplacian on the First Quadrant

Abstract

In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

A Neumann problem for a diffusion equation with n-dimensional fractional Laplacian

“…For example, Balanzario and Kaikina [2] studied the stochastic nonlinear Landau-Ginzburg equations on the half-line with Dirichlet white-noise boundary conditions, Shi and Wang [11] studied the solution for a stochastic fractional partial differential equation driven by an additive fractional space-time white noise. In Sanchez et al [10], studied the stochastic version of (1) for the 2-dimensional case; however, the n-dimensional case on R n + := {x = (x 1 , . .…”

Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

“…However, in order to describe and forecast a real phenomenon, it is necessary to introduce a component that captures the random behavior caused by a major source of uncertainty, that usually propagates in time. When we add such a component, the model obtained is now governed by a stochastic fractional differential equation [6,7]. On the order hand, the Itô stochastic calculus has been applied in several fields of knowledge; such as, engineering, physics, biology, among others [8,9].…”