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A fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data
Author(s) -
Duc Trong Dang,
Hong Nhung Nguyen Thi,
Dang Minh Nguyen,
Nhu Lan Nguyen
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6272
Subject(s) - mathematics , interior point method , bounded function , regularization (linguistics) , inverse problem , cauchy distribution , mathematical analysis , mathematical optimization , computer science , artificial intelligence
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 ( x 0 , t ) , u x ( x 0 , t ) measured at one interior point x = x 0 ∈ ( 0 , L ) or using an interior data u ( x 0 , t ) and the assumptionlim x → ∞ u ( x , t ) = 0 . However, the fluxu 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.