| Citation: |
Lea Safetly, Toni Sayah. ON THE INVERSE PROBLEM FOR THE BURGERS' EQUATION WITH SMOOTH INITIAL DATA IN A BOUNDED DOMAIN[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1014-1037. doi: 10.11948/20260119
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ON THE INVERSE PROBLEM FOR THE BURGERS' EQUATION WITH SMOOTH INITIAL DATA IN A BOUNDED DOMAIN
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Abstract
In this paper, we study the inverse problem of recovering smooth initial data for the one-dimensional Burgers' equation on a bounded spatial domain. Specifically, this problem focuses on recovering the initial condition from measurements at a part of the boundary over a given time interval. We therefore prove the uniqueness of the solution to the direct problem and establish, under certain assumptions, a stability estimate for the inverse problem based on a backward method in time and space, showing that the initial data can be reconstructed in this smooth case. To validate our theoretical results, we perform several numerical simulations and demonstrate the effectiveness of the proposed approaches.
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References
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Lea Safetly, Toni Sayah. ON THE INVERSE PROBLEM FOR THE BURGERS' EQUATION WITH SMOOTH INITIAL DATA IN A BOUNDED DOMAIN[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1014-1037. doi: 10.11948/20260119
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Figure 1.
Comparison of the exact and numerical solution for
$ t=0.1 $ -
Figure 2.
Comparison of the exact and numerical solution for t = 0.5.
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Figure 3.
Comparison of the exact and numerical solution for t = 1.
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Figure 4.
Comparison of the exact and numerical solution for t = 1.5.
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Figure 5.
Comparison between exact and recovered initial data for b = 0.14.
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Figure 6.
Comparison between exact and recovered initial data for b = 10.
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Figure 7.
Comparison between reconstructed and exact initial condition for m = 1 and b = 1.5.
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Figure 8.
Comparison between reconstructed and exact initial condition for m = 1 and b = 1.75.
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Figure 9.
Comparison between reconstructed and exact initial condition for m = 1 and b = 2.
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Figure 10.
Comparison between reconstructed and exact initial condition for m = 1 and b = 3.
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Figure 11.
Comparison between reconstructed and exact initial condition for m = 3 and b = 0.01.
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Figure 12.
Comparison between reconstructed and exact initial condition for m = 3 and b = 0.07.
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Figure 13.
Comparison between reconstructed and exact initial condition for m = 3 and b = 0.15.
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Figure 14.
Comparison of the exact and reconstructed initial conditions u0 with white noise on u at x = 1, for α = 0.1, m = 1 and b = 1.
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Figure 15.
Comparison of the exact and reconstructed initial conditions u0 with white noise on u at x = 1, for α = 0.1, m = 3 and b = 0.04.
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Figure 16.
Variation of log10(e2j) with respect to log10(e1j), for m = 1 and b = 1.
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Figure 17.
Variation of log10(e2j) with respect to log10(e1j), for m = 3 and b = 0.04.
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Figure 18.
Comparison between the exact and numerical solution for m = 1 and b = 1.75.
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Figure 19.
Comparison between the exact and numerical solution for m = 1 and b = 1.75.
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Figure 20.
Comparison between the recovered and exact initial data for m = 1 and b = 1.
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Figure 21.
Comparison between the recovered and exact initial data for m = 1 and b = 1.75.
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Figure 22.
Comparison between the recovered and exact initial data for m = 1 and b = 2.
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Figure 23.
Comparison between the recovered and exact initial data for m = 1 and b = 3.
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Figure 24.
Variation of log10(e2j) with respect to log10(e1j), for m = 1 and b = 1.
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Figure 25.
Comparison between exact and numerical initial data for b = 1.75.
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Figure 26.
Comparison between exact and numerical initial data for b = 0.05.