Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation
Abstract
:1. Introduction
2. The Forward Problem
3. Basic Identities
4. Inverse Problems
4.1. Solution of
4.2. Solution of
5. Numerical Results
- TP1: ;
- TP2: .
6. Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, UK, 1973. [Google Scholar]
- Gröger, K. Initial boundary value problems from semiconductor device theory. Z. Angew. Math. Mech. 1978, 67, 345–355. [Google Scholar] [CrossRef]
- Langer, R.E. A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoka Math. J. 1972, 35, 260–275. [Google Scholar]
- Fila, M.; Quittner, P. Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Math. Appl. Sci. 1997, 20, 1325–1333. [Google Scholar] [CrossRef]
- Koleva, M.N.; Vulkov, L.G. Blow-up of continuous and semilinear solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type. J. Comp. Appl. Math. 2007, 202, 414–434. [Google Scholar] [CrossRef]
- Jovanovic, B.S.; Vulkov, L.G. Convergence of difference schemes for the Poisson equation dynamical interface conditions. Comput. Methods Appl. Math. 2003, 3, 177–188. [Google Scholar] [CrossRef]
- Jovanovic, B.S.; Vulkov, L.G. Convergence of finite difference schemes for the Poisson’s equation with a dynamic boundary condition. Comput. Methods Appl. Math. 2005, 45, 275–284. [Google Scholar]
- Vabishchevich, P.N. Numerical solution of a problem for the elliptic equation with unsteady boundary conditions. Matem Model. 1995, 7, 49–60. (In Russian) [Google Scholar]
- Isakov, V. Inverse Source Problems; AMS: Providence, RI, USA, 1989. [Google Scholar]
- Grysa, K.; Macia̧g, A. Identifying heat source intensity in treatment of cancerous tumor using therapy based on local hyperthermia—The Trefftz method approachs. J. Therm. Biol. 2019, 84, 16–25. [Google Scholar] [CrossRef]
- Hasanoglu, A.; Romanov, V.G. Introduction to Inverse Problems for Differential Equations, 1st ed.; Springer: Cham, Switzerland, 2017; 261p. [Google Scholar]
- Kabanikhin, S.I. Inverse and Ill-Posed Problems; DeGruyer: Berlin, Germany, 2011. [Google Scholar]
- Lesnic, D. Inverse Problems with Applications in Science and Engineering; CRC Pres: Abingdon, UK, 2021; p. 349. [Google Scholar]
- Samarskii, A.A.; Vabishchevich, P.N. Numerical Methods for Solving Inverse Problems in Mathematical Physics; de Gruyter: Berlin, Germany, 2007; 438p. [Google Scholar]
- Ait Ben Hassi, E.M.; Chorfi, S.-E.; Maniar, L. Identification of source terms in heat equation with dynamic boundary conditions. Math. Meth. Appl. Sci. 2022, 45, 2364–2379. [Google Scholar] [CrossRef]
- Ivanov, D.K.; Kolesov, A.E.; Vabischevich, P.N. Numerical method for recovering the piecewise constant right-hand side function of an alliptic equation from a partial boundary observation data. J. Phys. Conf. Ser. 2021, 2092, 012006. [Google Scholar] [CrossRef]
- Liu, C.-S. A BIEM using the Treftz test functions for solving the inverse Cauchy and source recovery problems. Engn. Anal. Bound. Elem. 2016, 62, 177–185. [Google Scholar] [CrossRef]
- Liu, C.-S.; Cheng, C.-W. A global boundary integral equation method for recovering space-time dependent heat source. Int. J. Heat Mass Transf. 2016, 92, 1034–1040. [Google Scholar] [CrossRef]
- Yu, W. Well-posednes of determining the source term of elliptic equation. Bull. Austral. Math. Soc. 1994, 50, 383–398. [Google Scholar] [CrossRef]
- Tikhonov, A.N.; Goncharsky, A.V.; Stepanov, V.V.; Yagola, A.G. Numerical Methods for the Solution of Ill-Posed Problems, 1st ed.; Springer: Dordrecht, The Netherlands, 1995; 253p. [Google Scholar]
- Tikhonov, A.N.; Arsenin, V.Y. Solutions of Ill-Posed Problems; V. H. Winston & Sons: Washington, DC, USA; John Wiley & Sons: New York, NY, USA, 1977; 258p, (Translated from Russian). [Google Scholar]
- Belgacem, F.B. Why is the Cauchy problem severely ill-posed? Inverse Probl. 2007, 23, 823–836. [Google Scholar] [CrossRef]
- Cheng, J.; Hon, Y.C.; Wei, T.; Yamamoto, M. Numerical computation of a Cauchy problem for Laplace’s equation. Z. Angew. Math. Mech. 2001, 81, 665–674. [Google Scholar] [CrossRef]
- Joachimiak, M.; Ciałkowski, M.; Fra̧ckowiak, A. Stable method for solving the Cauchy problem with the use of Chebyshev polynomials. Int. J. Numer. Methods Heat Fluid Flow 2019, 30, 1441–1456. [Google Scholar] [CrossRef]
- Joachimiak, M.; Joachimiak, D.; Ciałkowski, M. Investigation on thermal loads in steady-state conditions with the use of the solution to the inverse problem. Heat Transf. Eng. 2023, 44, 963–969. [Google Scholar] [CrossRef]
- El Hajji, M.; Jday, F. Boundary data completion for a diffusion-reaction equation based on the minimization of an energy error functional using conjugate gradient method. Punjab Univ. J. Math. 2019, 51, 25–43. [Google Scholar]
- Kirkeby, A. Feynman’s inverse problem. arXiv 2023. [Google Scholar] [CrossRef]
- Alessandrini, G. A small collection of open problems. Rend. Istit. Mat. Univ. Trieste 2020, 52, 591–600. [Google Scholar]
- Rundell, R. Some inverse problems for elliptic equations. Appl. Anal. Int. J. 1988, 28, 67–78. [Google Scholar] [CrossRef]
- Slodička, M.; Lesnic, D. Determination of the Robin coefficient in a nonlinear boundary condition for a steady-state problem. Math. Meth. Appl. Sci. 2009, 32, 1311–1324. [Google Scholar] [CrossRef]
- Engl, H.W.; Leitão, A. A Mann iterative regularization method for elliptic Cauchy problems. Numer. Funct. Anal. Optim. 2001, 22, 861–884. [Google Scholar] [CrossRef]
- Kozlov, V.A.E.; Maz’ya, V.G.; Fomin, A.V. An iterative method for solving the Cauchy problem for elliptic equation. Comput. Math. Phys. 1991, 31, 45–52. [Google Scholar]
- Gong, R.; Wang, M.; Huang, Q.; Zhang, Y. Inverse Cauchy problems: Revisit and a new approach. arXiv 2022. [Google Scholar] [CrossRef]
- Jaoua, M.; Chaabane, S.; Elhechmi, C.; Leblond, J.; Mahjoub, M.; Partington, J. On some robust algorithms for Robin inverse problem. Rev. Arima 2008, 9, 287–307. [Google Scholar]
- Shirzadi, A.; Takhtabnoos, F. A local meshless method for Cauchy problem of elliptic PDEs in annulus domains. Inverse Probl. Sci. Eng. 2016, 24, 729–743. [Google Scholar] [CrossRef]
- Liu, C.-S.; Wang, F. A meshless method for solving the nonlinear inverse Cauchy problem of elliptic type equation in a doubly-connected domain. Comput. Math. Appl. 2018, 76, 1837–1852. [Google Scholar] [CrossRef]
- Wang, L.; Qian, Z.; Wang, Z.; Gao, Y.; Peng, Y. An efficient radial basis collocation method for the boundary condition identification of the inverse wave problem. Int. J. Appl. Mech. 2018, 10, 1850010. [Google Scholar] [CrossRef]
- Wang, L.; Wang, Z.; Qian, Z. A meshfree method for inverse wave propagation using collocation and radial basis functions. Comput. Methods Appl. Mech. Engrg. 2017, 322, 311–350. [Google Scholar]
- Hu, M.; Wang, L.; Yang, F.; Zhou, Y. Weighted radial basis collocation method for the nonlinear inverse Helmholtz problems. Mathematics 2023, 11, 662. [Google Scholar] [CrossRef]
- Ciałkowski, M.; Olejnik, A.; Joachimiak, M.; Grysa, K.; Fra̧ckowiak, A. Cauchy type nonlinear inverse problem in a two-layer area. Int. J. Numer. Methods Heat Fluid Flow 2021, 32, 313–331. [Google Scholar] [CrossRef]
- Karageorghis, A.; Lesnic, D.; Marin, L. A survey of applications of the MFS to inverse problems. Inverse Probl. Sci. Eng. 2011, 19, 309–336. [Google Scholar] [CrossRef]
- Liu, C.-S.; Qu, W.; Zhang, Y. Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method. Numer. Heat Transf. Part B 2018, 73, 48–61. [Google Scholar] [CrossRef]
- Chorfi, S.E.; El Guermai, G.; Maniar, L.; Zouhair, W. Numerical identification of initial temperatures in heat equation with dynamic boundary conditions. Mediterr. J. Math. 2023, 20, 256. [Google Scholar] [CrossRef]
- Constantin, A.; Escher, J. Global solutions for quasilinear parabolic problems. J. Evol. Equations 2002, 2, 97–111. [Google Scholar] [CrossRef]
- Craig, W. A Course on Partial Differential Equations. Amer. Math. Soc. 2018, 197, 205. [Google Scholar]
- Esher, J. Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions. In: Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991). Lect. Notes Pure Appl. Math. 1994, 155, 173–183. [Google Scholar]
- Yin, Z. Global existence for elliptic equations with dynamic boundary conditions. Arch. Math. 2003, 81, 567–574. [Google Scholar] [CrossRef]
- Liu, C.-S. A highly accurate MCTM for inverse Cauchy problems of Laplace equation in arbitrary plane domains. Comput. Model. Eng. Sci. 2008, 35, 91–111. [Google Scholar]
- Liu, C.-S. A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Eng. Anal. Bound. Elem. 2008, 32, 778–785. [Google Scholar] [CrossRef]
m | Max. Error | Max. Error | |||
---|---|---|---|---|---|
0.01 | 1 | 6.8956 | 1.3360 | ||
0.005 | 1 | 4.3177 | 0.6754 | 2.4273 | 2.4606 |
0.0025 | 1 | 2.1914 | 0.9784 | 3.4441 | 2.8171 |
0.00125 | 1 | 1.1003 | 0.9940 | 5.0705 | 2.7639 |
0.000625 | 1 | 5.5081 | 0.9983 | 8.1055 | 2.6452 |
0.0003125 | 1 | 2.7551 | 0.9995 | 1.4465 | 2.4864 |
0.00015625 | 1 | 1.3777 | 0.9998 | 2.8862 | 2.3253 |
0.000078125 | 1 | 6.8888 | 0.9999 | 6.2998 | 2.1958 |
0.01 | 3 | 4.3634 | 3.4898 | ||
0.005 | 3 | 3.4898 | 0.3223 | 1.8095 | 0.9476 |
0.0025 | 3 | 1.9202 | 0.8619 | 1.5003 | 0.2703 |
0.00125 | 3 | 1.0576 | 0.8605 | 3.9079 | 1.9407 |
0.000625 | 3 | 5.4102 | 0.9670 | 9.9157 | 1.9786 |
0.0003125 | 3 | 2.7232 | 0.9904 | 2.5132 | 1.9802 |
0.00015625 | 3 | 1.3642 | 0.9972 | 6.2961 | 1.9970 |
0.000078125 | 3 | 6.8247 | 0.9992 | 1.5335 | 2.0376 |
m | Max. Error | Max. Error | Iter | |
---|---|---|---|---|
= 0.02 | ||||
0.01 | 1 | 6.3069 | 1.1115 | 4.842 |
2 | 2.0456 | 9.1980 | 9.149 | |
3 | 4.0123 | 2.3232 | 15.733 | |
0.005 | 1 | 1.6550 | 2.0769 | 3.821 |
2 | 3.2464 | 1.5258 | 8.940 | |
3 | 4.3991 | 7.9018 | 19.010 | |
0.01 | 1 | 4.4107 | 7.3444 | 6.594 |
2 | 1.1629 | 5.2861 | 17.574 | |
3 | 3.9689 | 2.1198 | 21.306 | |
0.005 | 1 | 4.4905 | 3.9252 | 7.139 |
2 | 1.3193 | 7.0087 | 17.920 | |
3 | 4.0288 | 6.0438 | 24.303 |
m | Max. Error | Max. Error | Iter | |
---|---|---|---|---|
0.01 | 10 | 4.3234 | 1.9103 | 40.763 |
15 | 3.0266 | 1.6096 | 47.208 | |
20 | 2.7671 | 1.4172 | 51.000 | |
0.005 | 10 | 4.2076 | 1.8547 | 44.582 |
15 | 2.9288 | 1.5733 | 51.553 | |
20 | 2.7508 | 1.3997 | 60.816 |
Max. error | Max. Error | Max. Error | |||
---|---|---|---|---|---|
Iter | |||||
0.02 | 1 | 1.8693 | 4.2198 | 1.9027 | 13.821 |
2 | 2.6226 | 3.7340 | 2.8543 | 15.772 | |
3 | 2.4577 | 8.0663 | 2.8665 | 19.841 | |
0.2 | 1 | 5.8913 | 6.2845 | 2.6709 | 13.862 |
2 | 2.3012 | 2.4847 | 1.5154 | 15.861 | |
3 | 2.1429 | 6.2469 | 2.1894 | 19.594 |
Max. Error | Max. Error | Max. Error | ||
---|---|---|---|---|
Iter | ||||
10 | 1.7842 | 1.6864 | 2.5069 | 39.267 |
15 | 1.4291 | 1.4003 | 1.1296 | 41.733 |
20 | 1.2674 | 1.4083 | 1.9546 | 48.831 |
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Koleva, M.N.; Vulkov, L.G. Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation. Axioms 2024, 13, 64. https://doi.org/10.3390/axioms13010064
Koleva MN, Vulkov LG. Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation. Axioms. 2024; 13(1):64. https://doi.org/10.3390/axioms13010064
Chicago/Turabian StyleKoleva, Miglena N., and Lubin G. Vulkov. 2024. "Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation" Axioms 13, no. 1: 64. https://doi.org/10.3390/axioms13010064
APA StyleKoleva, M. N., & Vulkov, L. G. (2024). Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation. Axioms, 13(1), 64. https://doi.org/10.3390/axioms13010064