A new regularization for time-fractional backward heat conduction problem

IF 0.9 4区 数学 Q2 MATHEMATICS
M. Nair, P. Danumjaya
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引用次数: 0

Abstract

Abstract It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} at a time t ≥ 0 {t\geq 0} from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) {g:=u(\,\cdot\,,\tau)} for τ > t {\tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {t\in(0,\tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.
时间分数型逆向热传导问题的一种新的正则化方法
摘要众所周知,{利用}{τ >t }{\tau}{ >t的g:=}u≠({⋅,t) g:=u(\, }{\cdot}{ \,, }{}{\tau}{)还原t≥0 t }{\geq}{ 0时刻温度u(\, }{\cdot}{ \,,}t)的逆向热传导问题是不适定的,因为{g的小误差会导致u≠(⋅,t) u(\, \cdot \,,t)的大偏差}。然而,对于时间分数阶反向热传导问题(TFBHCP),上述问题在t=0 t=0 t=0时是适定性{的,在t=0 t=0 }t=0时是不适定性{的}。我们利用这一观察结果,以t为{正则化参数 t \in (0, \tau)的稳定近似解,用于近似t=0 t=0处的解,}并在合适的源条件下得到误差估计。我们还将提供一些数值例子来说明正则解的近似性质。{}
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来源期刊
Journal of Inverse and Ill-Posed Problems
Journal of Inverse and Ill-Posed Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.60
自引率
9.10%
发文量
48
审稿时长
>12 weeks
期刊介绍: This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published. Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest. The following topics are covered: Inverse problems existence and uniqueness theorems stability estimates optimization and identification problems numerical methods Ill-posed problems regularization theory operator equations integral geometry Applications inverse problems in geophysics, electrodynamics and acoustics inverse problems in ecology inverse and ill-posed problems in medicine mathematical problems of tomography
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