A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

IF 4.4 2区 数学 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Dang Duc Trong , Dinh Nguyen Duy Hai , Nguyen Dang Minh , Nguyen Nhu Lan
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引用次数: 0

Abstract

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x(0,L),0<x1<x2L. The problem is challenging since it is severely ill-posed for x[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.
具有两个内部温度测量值的分数侧向问题的双参数提霍诺夫正则化
本文讨论的是根据两次内部温度测量值确定热体表面温度的分数侧向问题。在数学上,它被表述为具有阶数 α∈(0,1] 的卡普托分数时间导数的一维热方程问题,其中数据是在两个内部点给出的,即 x=x1 和 x=x2,并确定 x∈(0,L),0<x1<x2≤L 的解。这个问题具有挑战性,因为对于 x∉[x1,x2]来说,它是一个严重的问题。对于这个问题,我们采用希尔伯特尺度下的 Tikhonov 正则化方法来构造稳定的近似问题。利用双参数 Tikhonov 正则化,我们通过先验和后验参数选择策略获得了希尔伯特尺度下的阶次最优收敛估计值。数值实验证明了所提方法的有效性。
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来源期刊
Mathematics and Computers in Simulation
Mathematics and Computers in Simulation 数学-计算机:跨学科应用
CiteScore
8.90
自引率
4.30%
发文量
335
审稿时长
54 days
期刊介绍: The aim of the journal is to provide an international forum for the dissemination of up-to-date information in the fields of the mathematics and computers, in particular (but not exclusively) as they apply to the dynamics of systems, their simulation and scientific computation in general. Published material ranges from short, concise research papers to more general tutorial articles. Mathematics and Computers in Simulation, published monthly, is the official organ of IMACS, the International Association for Mathematics and Computers in Simulation (Formerly AICA). This Association, founded in 1955 and legally incorporated in 1956 is a member of FIACC (the Five International Associations Coordinating Committee), together with IFIP, IFAV, IFORS and IMEKO. Topics covered by the journal include mathematical tools in: •The foundations of systems modelling •Numerical analysis and the development of algorithms for simulation They also include considerations about computer hardware for simulation and about special software and compilers. The journal also publishes articles concerned with specific applications of modelling and simulation in science and engineering, with relevant applied mathematics, the general philosophy of systems simulation, and their impact on disciplinary and interdisciplinary research. The journal includes a Book Review section -- and a "News on IMACS" section that contains a Calendar of future Conferences/Events and other information about the Association.
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