Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry
Abstract
:1. Introduction
2. Direct and Inverse Problems
2.1. Direct (Forward) Problem
2.2. Inverse Problems
3. Well-Posedness of the Direct Problem
4. Numerical Solution of the Direct Problem
5. Numerical Solution of the Inverse Problems
6. Computational Examples
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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J | ||||||||
---|---|---|---|---|---|---|---|---|
30 | 5.1356 × 10−2 | 5.1356 × 10−2 | 4.3835 × 10−2 | 4.8458 × 10−2 | ||||
60 | 1.2944 × 10−2 | 1.9882 | 1.2944 × 10−2 | 1.9882 | 1.1176 × 10−2 | 1.9717 | 1.2016 × 10−2 | 2.0118 |
120 | 3.2557 × 10−3 | 1.9913 | 3.2556 × 10−3 | 1.9913 | 2.8523 × 10−3 | 1.9702 | 2.9957 × 10−3 | 2.0040 |
240 | 8.1772 × 10−4 | 1.9933 | 8.1772 × 10−4 | 1.9933 | 7.2696 × 10−4 | 1.9722 | 7.4882 × 10−4 | 2.0002 |
480 | 2.0520 × 10−4 | 1.9946 | 2.0520 × 10−4 | 1.9946 | 1.8488 × 10−4 | 1.9753 | 1.8739 × 10−4 | 1.9986 |
960 | 5.1458 × 10−5 | 1.9956 | 5.1457 × 10−5 | 1.9956 | 4.6914 × 10−5 | 1.9785 | 4.6913 × 10−5 | 1.9980 |
J | ||||||||
---|---|---|---|---|---|---|---|---|
30 | 6.2045 × 10−2 | 5.4563 × 10−2 | 6.0694 × 10−2 | 5.0790 × 10−2 | ||||
60 | 1.5682 × 10−2 | 1.9842 | 1.3650 × 10−2 | 1.9990 | 1.4837 × 10−2 | 2.0324 | 1.2518 × 10−2 | 2.0205 |
120 | 3.9612 × 10−3 | 1.9851 | 3.4131 × 10−3 | 1.9998 | 3.6648 × 10−3 | 2.0174 | 3.1063 × 10−3 | 2.0107 |
240 | 1.0004 × 10−3 | 1.9853 | 8.5331 × 10−4 | 1.9999 | 9.1049 × 10−4 | 2.0090 | 7.7366 × 10−4 | 2.0054 |
480 | 2.5264 × 10−4 | 1.9855 | 2.1333 × 10−4 | 2.0000 | 2.2689 × 10−4 | 2.0046 | 1.9305 × 10−4 | 2.0027 |
960 | 6.3794 × 10−5 | 1.9856 | 5.3333 × 10−5 | 2.0000 | 5.6630 × 10−5 | 2.0024 | 4.8216 × 10−5 | 2.0014 |
0.01 | 2.2141 × 10−3 | 1.0083 × 10−2 | 1.1401 × 10−3 | 7.7277 × 10−3 | 1.8319 × 10−1 | 4.3869 × 10−2 |
0.03 | 6.1225 × 10−3 | 2.6653 × 10−2 | 3.2143 × 10−3 | 2.0666 × 10−2 | 6.6437 × 10−1 | 1.6369 × 10−1 |
0.05 | 1.0031 × 10−2 | 4.3224 × 10−2 | 5.2888 × 10−3 | 3.3605 × 10−2 | 1.1456 | 2.8357 × 10−1 |
0.1 | 1.9802 × 10−2 | 8.4650 × 10−2 | 1.0475 × 10−2 | 6.5954 × 10−2 | 2.3486 | 5.8330 × 10−1 |
0.03 | 0.001 | 1.051 × 10−1 | 4.042 × 10−3 | 2.287 × 10−2 | 2.770 × 10−3 | 1.768 × 10−1 | 3.826 × 10−2 | 7.408 × 10−3 | 1.645 × 10−3 |
0.01 | 0.003 | 2.763 × 10−2 | 1.239 × 10−2 | 6.186 × 10−3 | 8.439 × 10−3 | 6.863 × 10−2 | 1.404 × 10−2 | 2.107 × 10−2 | 4.680 × 10−3 |
0.05 | 0.005 | 1.630 × 10−1 | 2.081 × 10−1 | 3.568 × 10−2 | 1.417 × 10−2 | 2.980 × 10−1 | 6.130 × 10−2 | 3.490 × 10−2 | 7.833 × 10−3 |
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Koleva, M.N.; Vulkov, L.G. Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry. Symmetry 2024, 16, 1065. https://doi.org/10.3390/sym16081065
Koleva MN, Vulkov LG. Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry. Symmetry. 2024; 16(8):1065. https://doi.org/10.3390/sym16081065
Chicago/Turabian StyleKoleva, Miglena N., and Lubin G. Vulkov. 2024. "Inverse Boundary Conditions Interface Problems for the Heat Equation with Cylindrical Symmetry" Symmetry 16, no. 8: 1065. https://doi.org/10.3390/sym16081065