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

Abstract

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 theor using an interior data u(x 0 , t) and the assumption lim x→∞ u(x, t) = 0. However, the flux u x (x 0 , 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. KEYWORDS fractional diffusion equation, ill-posed problem, inverse problem, nonparametric regression, sideways problem MSC CLASSIFICATION 35J92; 35R06; 46E30; 35R06; 46E30 INTRODUCTIONUntil recent years, the classical partial differential equations (PDEs) are a tool used to model many spatial-time systems in nature (see Kumar et al 1 ). However, the integer-order derivatives are defined locally so the mathematical model does not seem to be quite suitable for reflecting the mutual spatial-influence and time-memory in the systems. To improve this inadequacy, nonlocal (space-fractional or time-fractional) derivatives were used in modeling of natural and engineering systems. Fractional derivatives have been found to be quite flexible in describing viscoelastic, viscoplastic flow and anomalous diffusion in fluid mechanic, quantum mechanic, etc (superdiffusion, subdiffusion), which might be inconsistent with the classical Brownian motion model. [2][3][4][5] Hence, in the last decades, the investigation of fractional heat Math Meth Appl Sci. 2020;43:5314-5338. wileyonlinelibrary.com/journal/mma