The truncation method for a two-dimensional nonhomogeneous backward heat problem

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

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The determination of two moving heat sources in two-dimensional inverse heat problem

Applied Mathematical Modelling ◽

10.1016/j.apm.2005.03.012 ◽

2006 ◽

Vol 30 (3) ◽

pp. 278-292 ◽

Cited By ~ 18

Author(s):

Ching-yu Yang

Keyword(s):

Heat Sources ◽

Two Dimensional ◽

Heat Problem ◽

Inverse Heat Problem

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On the convergence analysis and stability of the RBF-adaptive method for the forward-backward heat problem in 2D

Applied Numerical Mathematics ◽

10.1016/j.apnum.2020.08.015 ◽

2021 ◽

Vol 159 ◽

pp. 297-310

Author(s):

Siamak Banei ◽

Kamal Shanazari

Keyword(s):

Convergence Analysis ◽

Adaptive Method ◽

Heat Problem ◽

Backward Heat Problem

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On the backward heat problem: evaluation of the norm of∂u∂t

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171280000464 ◽

1980 ◽

Vol 3 (3) ◽

pp. 599-603

Author(s):

Yves Biollay

Keyword(s):

Heat Equation ◽

Heat Problem ◽

Backward Heat Problem

We show in this paper that‖Δu‖=‖ut‖is bounded∀t≤T(0)<Tif one imposes onu(solution of the backward heat equation) the condition‖u(x,t)‖≤M. A Hölder type of inequality is also given if one supposes‖ut(x,T)‖≤K.

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A two-dimensional backward heat problem with statistical discrete data

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2016-0038 ◽

2018 ◽

Vol 26 (1) ◽

pp. 13-31 ◽

Cited By ~ 9

Author(s):

Nguyen Dang Minh ◽

Khanh To Duc ◽

Nguyen Huy Tuan ◽

Dang Duc Trong

Keyword(s):

Regression Models ◽

Initial Temperature ◽

Least Squares Method ◽

Random Noise ◽

Heat Problem ◽

Formula Group ◽

Noise Data ◽

Ill Posed ◽

Grid Points ◽

Backward Heat Problem

AbstractWe focus on the nonhomogeneous backward heat problem of finding the initial temperature {\theta=\theta(x,y)=u(x,y,0)} such that\left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm}where {\Omega=(0,\pi)\times(0,\pi)}. In the problem, the source {f=f(x,y,t)} and the final data {h=h(x,y)} are determined through random noise data {g_{ij}(t)} and {d_{ij}} satisfying the regression models\displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t),\displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij},where {(X_{i},Y_{j})} 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.

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