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

Abstract

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(1){∂tu−(1+α∂tβ)Δu=f(x,y,t,u),(x,y,t)∈Ω×(0,T),u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T),u(x,y,T)=v(x,y),(x,y)∈Ω,where Ω=(0,π)×(0,π). The values of the final data v at n × m points (xp, yq) of Ω are contaminated by n × m observations Vpq (p=1,2,⋯,n,q=1,2,⋯,m). From the known data Vpq, we recover the initial data u(x, y, 0). We show that our backward problem is ill-posed 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.