Abstract
We study an initial-boundary value problem for a n-dimensional stochastic diffusion equation with fractional Laplacian on mathbb{R}_{+}^{n}. In order to prove existence and uniqueness, we generalize the Fokas method to construct the Green function for the associated linear problem and then we apply a fixed point argument. Also, we present an example where the explicit solutions are given.
Highlights
1 Introduction The classical diffusion phenomenon is governed by a second order linear partial differential equation, whose Green function is given by a Gaussian probability density function and which describes the movement of energy through a medium in response to a gradient of energy
The diffusion processes in various systems with complex structure, such as liquid crystals, glasses, polymers, biopolymers, and proteins, usually do not follow a Gaussian density, as a consequence the phenomenon is described by a fractional partial differential equation [7]
There is some previous work for the initial-boundary value problem on the first quadrant R2+ for fractional diffusion equations, where the Green function has been constructed and an integral representation of the solution was found [3, 6]
Summary
The classical diffusion phenomenon is governed by a second order linear partial differential equation, whose Green function is given by a Gaussian probability density function and which describes the movement of energy through a medium in response to a gradient of energy. Dipierro et al, [4] have studied the asymptotic behavior of the solutions of the time-fractional diffusion equation. There is some previous work for the initial-boundary value problem on the first quadrant R2+ for fractional diffusion equations, where the Green function has been constructed and an integral representation of the solution was found [3, 6]. We consider the equation ut = αu, (1)
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