Identifying the heat sink

J. D. Audu; A. Boumenir; K. M. Furati; I. O. Sarumi

doi:10.3934/dcdss.2021164

Article Contents

2022, Volume 15, Issue 5: 1045-1059. Doi: 10.3934/dcdss.2021164

This issue Previous Article Approximate controllability of nonsimple elastic plate with memory Next Article Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

Identifying the heat sink

(a).
Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
(b).
Department of Mathematics, University of West Georgia, GA 30118, USA

^*Corresponding author: A. Boumenir
^*Corresponding author: A. Boumenir

Received: August 2021

Revised: November 2021

Early access: December 2021

Published: May 2022

This work is supported by KFUPM grant SB191022

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Abstract

In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.

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Mathematics Subject Classification: 35R30, 65M32.

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Figure 1. Plots of relative errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 1

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Figure 2. Plots of relative errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 2

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Figure 3. Plots of errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 3

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Figure 4. Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 0.001 with Algorithm 1

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Figure 5. Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 1 with Algorithm 2

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Figure 6. Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 1 with Algorithm 3

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Table 1. Errors in the recovery of the Fourier coefficients using Algorithm 1

$ T $	$ \|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\| $	$ \|\|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\|\|_2 $
0.0001	$ [1.56e-06, 3.15e-06, 3.20e-06,5.44e-08, 3.80e-07]^\top $	4.77e-06
0.001	$ [2.22e-08,\ 1.59e-05,\ 7.39e-06,\ 1.34e-05, \ 2.07e-05]^\top $	$ 3.03e-05 $
0.01	$ [ 2.65e-04,\ 2.10e-03,\ 2.89e-04, 1.46e-03,\ 1.61e-03]^\top $	$ 3.05e-03 $
0.1	$ [5.71e-02,\ 2.47e-03,\ 8.77e-02, 2.30e-03, \ 4.49e-02]^\top $	$ 1.14\text{e-01} $

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Table 2. Errors in the recovery of the Fourier coefficients using Algorithm 2

$ T $	$ \|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\| $	$ \|\|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\|\|_2 $
0.1	$ [6.01e-4,\ 8.21e-04,\ 6.34e-04, 4.71e-04, \ 2.61e-04]^\top $	$ 1.31e-03 $
1	$ [2.91e-06,\ 9.58e-06,\ 4.64e-06,\ 2.06e-06,\ 6.03e-08]^\top $	$ 1.12e-05 $
2	$ [ 4.93e-01,\ 2.49e-01, \ 1.39e-01,\ 4.95e-01,\ 2.46e-02]^\top $	$ 2.96e-01 $
3	$ [ 7.1e-01,\ 1.02e-0,\ 5.88e-01,\ 2.06e-01,\ 9.62e-02]^\top $	$ 1.39e-0 $
5	$ [3.99e-01,\ 7.48e-01, \ 4.98e-01,\ 1.99e-01,\ 9.97e-02]^\top $	$ 1.01e-0 $
6	$ [5.12e-0, 7.5 e-01, 5e-01, 2e-01, 1e-01]^\top $	$ 5.20e-0 $
7	$ [ 3.4e-01, 7.5e-01, 5e-01, 2e-01, 1e-01 ]^\top $	$ 34.87e-0 $

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Table 3. Errors in the recovery of the Fourier coefficients using Algorithm 3

$ T $	$ \|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\| $	$ \|\|\boldsymbol{q}_5 - \tilde{\boldsymbol{q}}_5\|\|_2 $
0.1	$ [2.65e-06, 3.39e-06, 4.50e-07, 4.83e-07, 5.05e-07]^\top $	4.38e-06
1	$ [4.70e-10, 9.36e-10, 8.53e-10, 4.74e-10, 2.57e-10]^\top $	1.46e-09
5	$ [6.21e-10, 1.19e-09, 1.07e-09, 5.86e-10, 3.14e-10]^\top $	1.84e-09
10	$ [5.78e-10, 1.11e-09, 9.67e-10, 5.28e-10, 2.82e-10]^\top $	1.69e-09

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