Stochastic evolution equation with Riesz-fractional derivative and white noise on the half-line

摘要

In this work, we consider an initial boundary-value problem for a stochastic evolution equation with Riesz-fractional spatial derivative and white noise on the half-line, {u_t(x,t) =D_x~α u(x,t) +Nu(x,t) + B(x,t), x＞0, t∈[0,T], u(x,0) = u_0(x), x＞0, u_x(0,t)= g_1(t), t∈[0,T], where D_x~α is the Riesz-fractional derivative, α ∈ (2,3), N is a Lipschitzian operator and B(x, t) is the white noise. To construct the integral representation of solutions we use the Fokas method and Picard scheme to prove existence and uniqueness. Moreover, Monte Carlo methods are implemented to approximate solutions.