May  2022, 15(5): 1045-1059. doi: 10.3934/dcdss.2021164

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

Received  August 2021 Revised  November 2021 Published  May 2022 Early access  December 2021

Fund Project: This work is supported by KFUPM grant SB191022

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.

Citation: J. D. Audu, A. Boumenir, K. M. Furati, I. O. Sarumi. Identifying the heat sink. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1045-1059. doi: 10.3934/dcdss.2021164
References:
[1]

K. Ammari and M. Choulli, Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations, J. Differential Equations, 259 (2015), 3344-3365.  doi: 10.1016/j.jde.2015.04.023.

[2] K. Ammari and S. Gerbi, Evolution Equations: Long Time Behaviour and Control, London Mathematical Society Lecture Note Series, 439. Cambridge University Press, Cambridge, 2018. 
[3]

K. Atkinson and W. Han, Elementary Numerical Analysis, 3$^{rd}$ edition, Wiley, 2004.

[4]

M. Belishev, On approximating properties of solutions of the heat equation, Control Theory of Partial Differential Equations, Chapman and Hall/CRC, 242 (2005), 43–50.

[5]

A. Boumenir and V. Tuan, Recovery of heat coefficient by two measurements, Inverse Probl. Imaging, 5 (2011), 775-791.  doi: 10.3934/ipi.2011.5.775.

[6]

H. Fattorini and D. Russell, Uniform bounds on biothorgonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972.

[7]

A. I. Prilepko and A. B. Kostin, On some inverse problems for parabolic equations with final and integral observation, Mat. Sb., 183 (1992), 49-68. 

[8]

A. I. Prilekpo, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monographs and Textbooks in Pure and Applied Mathematics, 231. Marcel Dekker, Inc., New York, 2000.

[9]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: Some results and open questions, SIAM Rev, 20 (1978), 639-739.  doi: 10.1137/1020095.

[10]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, C. M. Dafermos and E. Feireisl eds., Elsevier Science, 3 (2007), 527–621. doi: 10.1016/S1874-5717(07)80010-7.

show all references

References:
[1]

K. Ammari and M. Choulli, Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations, J. Differential Equations, 259 (2015), 3344-3365.  doi: 10.1016/j.jde.2015.04.023.

[2] K. Ammari and S. Gerbi, Evolution Equations: Long Time Behaviour and Control, London Mathematical Society Lecture Note Series, 439. Cambridge University Press, Cambridge, 2018. 
[3]

K. Atkinson and W. Han, Elementary Numerical Analysis, 3$^{rd}$ edition, Wiley, 2004.

[4]

M. Belishev, On approximating properties of solutions of the heat equation, Control Theory of Partial Differential Equations, Chapman and Hall/CRC, 242 (2005), 43–50.

[5]

A. Boumenir and V. Tuan, Recovery of heat coefficient by two measurements, Inverse Probl. Imaging, 5 (2011), 775-791.  doi: 10.3934/ipi.2011.5.775.

[6]

H. Fattorini and D. Russell, Uniform bounds on biothorgonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972.

[7]

A. I. Prilepko and A. B. Kostin, On some inverse problems for parabolic equations with final and integral observation, Mat. Sb., 183 (1992), 49-68. 

[8]

A. I. Prilekpo, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monographs and Textbooks in Pure and Applied Mathematics, 231. Marcel Dekker, Inc., New York, 2000.

[9]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: Some results and open questions, SIAM Rev, 20 (1978), 639-739.  doi: 10.1137/1020095.

[10]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, C. M. Dafermos and E. Feireisl eds., Elsevier Science, 3 (2007), 527–621. doi: 10.1016/S1874-5717(07)80010-7.

Figure 1.  Plots of relative errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 1
Figure 2.  Plots of relative errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 2
Figure 3.  Plots of errors between the known heat sink q(x) and heat sink $ \tilde{q}(x) $ recovered with Algorithm 3
Figure 4.  Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 0.001 with Algorithm 1
Figure 5.  Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 1 with Algorithm 2
Figure 6.  Matching of q(x) and the $ \tilde{q}(x) $ recovered at T = 1 with Algorithm 3
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} $
$ 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} $
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 $
$ 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 $
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
$ 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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