Fractional stochastic heat equation on the half-line

Abstract

In this paper, we consider an initial-boundary value problem for a stochastic non-linear heat equation with Riemann-Liouville spacefractional derivative and white noise on the half-line. We construct the integral representation of the solution and prove existence and uniqueness. Moreover, we adapt stochastic integration methods to approximate the solutions

Solving PDEs of fractional order using the unified transform method

“…There exist some earlier works analysing fractional PDEs on the half-line using methods similar to the unified transform method, e.g. the recent work of Arciga et al [2,3,28] and some papers of Kaikina [19,20]. However, some of these works have used other fractional models than the classical Riemann-Liouville one -such as the Riesz, Caputo, or Abel fractional derivatives [27] -or considered a narrower class of PDEs than that which we shall analyse here, while others have used more complicated methods than the unified transform method.…”

“…for all integers n, i.e. that |4n ± α| ≥ 1 for all integers n. In other words, α must lie in one of the intervals [1,3], [5,7], [9,11], etc. The domain D + can be described as follows, working from (31):…”

“…In particular, if 1 < α ≤ 3 2 , then the intervals on the right do not intersect and so D + is empty. For now, let us assume 3 2 < α < 5 2 , so that we have:…”