Startseite A two-dimensional backward heat problem with statistical discrete data
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A two-dimensional backward heat problem with statistical discrete data

  • Nguyen Dang Minh EMAIL logo , Khanh To Duc , Nguyen Huy Tuan und Dang Duc Trong
Veröffentlicht/Copyright: 5. Mai 2017

Abstract

We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ=θ(x,y)=u(x,y,0) such that

{ut-a(t)(uxx+uyy)=f(x,y,t),(x,y,t)Ω×(0,T),u(x,y,t)=0,(x,y)Ω×(0,T),u(x,y,T)=h(x,y),(x,y)Ω¯,

where Ω=(0,π)×(0,π). In the problem, the source f=f(x,y,t) and the final data h=h(x,y) are determined through random noise data gij(t) and dij satisfying the regression models

gij(t)=f(Xi,Yj,t)+ϑξij(t),
dij=h(Xi,Yj)+σijεij,

where (Xi,Yj) are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.

MSC 2010: 35K05; 47A52; 62G08

Funding statement: This research was supported by Vietnam National University-Ho Chi Minh City (VNU-HCM) under the Grant number B2017-18-03.

Acknowledgements

We would like to thank two anonymous referees, an Associate Editor, whose constructive comments helped to improve the presentation of the paper.

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Received: 2016-6-17
Revised: 2017-2-14
Accepted: 2017-2-26
Published Online: 2017-5-5
Published in Print: 2018-2-1

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