Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data

In this study, we investigate a problem of finding the function u (x, y, t) for the fractional Rayleigh-Stokes equation with nonlinear source as follows {partial derivative(t)u - (1 + alpha partial derivative(beta)(t)) Delta u = f(x, y, t, u), (x, y, t) epsilon Omega x (0, T), u(x, y, t) = 0, (x, y, t) epsilon partial derivative Omega x (0, T), (1) u(x, y, t) = v(x, y), (x, y) epsilon partial derivative Omega, where Omega = (0, pi) x(0, pi). The values of the final data v at n xm points (x(p), y(q)) of Omega are contaminated by n xm observations V-pq (p = 1, 2,..., n, q = 1, 2,..., m). From the known data V-pq, we recover the initial data u (x, y, 0). We show that our backward problem is illposed in the sense of Hadamard. To regularize the instable solution, we use the trigonometric method in nonparametric regression associated with Fourier truncated expansion method. The numerical results show that our regularization method is flexible and stable. (C) 2019 Elsevier B.V. All rights reserved.