The time-dependent diffusion equation: An inverse diffusivity problem

Document Type : Research Article

Authors

• S.H. Tabasi
• H.D. Mazraeh
• A.A. Irani
• R. Pourgholi
• A. Esfahani

School of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364, Damghan, Iran.

10.22067/ijnao.2021.11309.0

Abstract

We find a solution of an unknown time-dependent diffusivity a(t) in a linear inverse parabolic problem by a modified genetic algorithm. At first, it is shown that under certain conditions of data, there exists at least one solution for unknown a(t) in (a(t), T (x, t)), which is a solution to the corresponding problem. Then, an optimal estimation for unknown a(t) is found by applying the least-squares method and a modified genetic algo rithm. Results show that an excellent estimation can be obtained by the implementation of a modified real-valued genetic algorithm within an Intel Pentium (R) dual-core CPU with a clock speed of 2.4 GHz.

Keywords

• Inverse parabolic problem
• Existence
• Uniqueness
• Green’s function
• Fixed point
• Contraction mapping
• Genetic algorithm

20.1001.1.24236977.2021.11.1.2.9

---

References

1. Baiocchi, C. Sui problemi ai limiti per le equazioni paraboliche del tipodel calore, (Italian) Boll. Un. Mat. Ital. 19(3) (1964), 407–422.
2. Bao, G., Ehlers T., and Li, P. Radiogenic source identification for the he lium production-diffusion equation, Commun. Comput. Phys. 14 (2013), 1–20.
3. Bao, G., Ehlers T., Li, P., and Zheng, G. A nonuniqueness result for an inverse diffusivity problem in thermochronometry, Submitted.
4. Bao, G. and Xu, X. An inverse diffusivity problem for the helium production-diffusion equation, Inverse Problems, 28(8) (2012), 085002, 15 pp.
5. Beck, J.V., and Murio, D.C. Combined function specification regularization procedure for solution of inverse heat condition problem, AIAA J., 24 (1986), 180–185.
6. Cabeza, J.M.G., Garcia J.A.M., and Rodriguez, A.C. A sequential algorithm of inverse heat conduction problems using singular value decomposition, J. Therm. Sci. 44 (2005), 235–244.
7. Cannon, J.R. The one-dimensional heat equation, With a foreword by Felix E. Browder. Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Read ing, MA, 1984.
8. Demir, A., and Bayrak, M. A. Inverse problem for determination of an unknown coefficient in the time fractional diffusion equation, Communications in Mathematics and Applications, 9(2) (2018), 229–237.
9. Foadian, S., Pourgholi, R., Tabasi, S. H., and Zeidabadi H. Solving an inverse problem for a generalized time-delayed Burgers-fisher equation by haar wavelet method, J. Appl. Anal. Comput. 10(2) (2020) 391–410.

10. Friedman, A. Partial differential equations parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
11. Holland, J.H. Adaptation in natural and artificial system: An introductory analysis with applications to biology, control, and artificial intelli gence, University of Michigan Press, Ann Arbor, Mich., 1975.
12. John, F. Partial differential equations, Springer-Verlag, New York, 1982.
13. Kythe, P.K., Puri, P., and Schaferkotter, M.R. Partial differential equations and mathematica, CRC Press, Boca Raton, FL, 1997.
14. Lesnic, D., Yousefi, S.A., and Ivanchov, M. Determination of a time dependent diffusivity from nonlocal conditions, J. Appl. Math. Comput. 41 (2013), 301–320.
15. Lions, J.C. and Magenes, E. Remarques sur les problèmes aux limites pour opérateurs paraboliques. Compt. Rend. Acad. Sci. (Paris) 251 (1960), 2118–2120.
16. Lions, J.C. and Magenes, E. Problèmes aux limites non homogènes. II, (French) Ann. Inst. Fourier (Grenoble) 11 (1961), 137–178.
17. Lions, J.C. and Magenes, E. Problèmes aux limites non homogènes. IV. (French) Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15(3) (1961), 311–326.
18. Lions, J.C., and Magenes, E. Problèmes aux limites non homogènes elapplications, Dunod, Paris,1968.
19. Ozisik, M.N. Heat conduction, John Wiley & Sons, Inc., New York, 1993.
20. Pourgholi, R., Azizi, N., Gasimov, Y.S., Aliev, F., and Khalafi, H.K. Removal of numerical instability in the solution of an inverse heat conduction problem, Commun. Nonlinear Sci. Numer. Simul. 14 (6) (2009), 2664–2669
21. Pourgholi, R., Foadian, S.A., and Esfahani, A. Haar basis method to solve some inverse problems for two-dimensional parabolic and hyperbolic equations, TWMS J. App. Eng. Math. 3 (2013), 10–32.
22. Zeidabadi, H., Pourgholi, R., and Tabasi, S.H. Solving a nonlinear inverse system of Burgers equations, Int. J. Nonlinear Anal. Appl. 10(1) (2019), 35–54.
23. Zeidabadi, H., Pourgholi R., and Tabasi S.H. A hybrid scheme for time fractional inverse parabolic problem, Waves Random Complex Media, 30(2) (2020), 354–368.