A fractional sideways problem in a one-dimensional finite-slab with deterministic and random interior perturbed data

In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using the Cauchy data u(x0,t),ux(x0,t) measured at one interior point x=x0 is an element of(0,L) or using an interior data u(x0,t) and the assumption limx ->infinity u(x,t)=0. However, the flux ux(x0,t) is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x=1 and x=2, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill-posed and further construct an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well.