Sequential Solution of the Sideways Heat Equation by Windowing of the Data

Berntsson, Fredrik

doi:10.1080/1068276021000048564

Cited by 11 publications

(4 citation statements)

References 13 publications

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“…In references, there are many types of internal measurements. In Berntsson [16] and Tuan et al [17], the temperature is recovered from the history interior temperature

u\left({x}_0,t\right)

and the interior flux

{u}_x\left({x}_0,t\right)

, where

{x}_0

is an interior point of the heat body. However, in practice, it is difficult to accurately measure the interior heat flux.…”

Section: Introductionmentioning

confidence: 99%

Corrigendum to a fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Duc Trong,

Thi Hong Nhung,

Dang Minh

et al. 2024

Math Methods in App Sciences

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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 measured at one interior point or using an interior data and the assumption . However, the flux 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, and , 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 constructs 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.

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“…In references, there are many types of internal measurements. In Berntsson [16] and Tuan et al [17], the temperature is recovered from the history interior temperature

u\left({x}_0,t\right)

and the interior flux

{u}_x\left({x}_0,t\right)

, where

{x}_0

is an interior point of the heat body. However, in practice, it is difficult to accurately measure the interior heat flux.…”

Section: Introductionmentioning

confidence: 99%

Corrigendum to a fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Duc Trong,

Thi Hong Nhung,

Dang Minh

et al. 2024

Math Methods in App Sciences

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“…They are referred to as inverse heat conduction problems. This area of research have been studied extensively, in 1D setting, by various authors, such as [3,4,11,14,15,21,39]. The Cauchy problem for the steady-state anisotropic heat conduction in 2D and 3D has also been considered in [37].…”

Section: Introductionmentioning

confidence: 99%

Solving stationary inverse heat conduction in a thin plate

Chepkorir,

Berntsson,

Kozlov

2023

Partial Differ. Equ. Appl.

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We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

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“…In 1995, Teresa Regiflska [3] solved a sideways heat problem which consists in applying the wavelet basis decomposition of measured data in the quarter plane arXiv:1910.03228v1 [math.AP] 8 Oct 2019 (x ≥ 0, t ≥ 0). In 1999, F. Berntsson [4] investigated the following sideways heat equation…”

Section: Introductionmentioning

confidence: 99%

Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

Ngoc

Tuan

Kirane

2019

Journal of Inverse and Ill-Posed Problems

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AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

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Sequential Solution of the Sideways Heat Equation by Windowing of the Data

Cited by 11 publications

References 13 publications

Corrigendum to a fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Corrigendum to a fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Solving stationary inverse heat conduction in a thin plate

Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

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