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Parameters estimation for a new anomalous thermal diffusion model in layered media. (English) Zbl 1409.35235

Summary: In this paper, we study an inverse problem of parameters estimation for a new time-fractional heat conduction model in multilayered medium. In the anomalous thermal diffusion model, we consider the fractional derivative boundary conditions and the conduction obeys modified Fourier law with Riemann-Liouville fractional operator of different order in each layer. For the direct problem, we construct an effective finite difference scheme by using the balance method to deal with the discontinuity interface. For the inverse problem, we apply the nonlinear conjugate gradient (NCG) method with different conjugated coefficients to simultaneously identify the fractional exponent in each layer. Finally, we use experimental data to verify the effectiveness of the proposed technique, in which the Jacobian matrix is achieved by a derivative-free approach. We analyze the sensitivity coefficients and the convergence behaviors of the NCG algorithm. The simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate fitting than the classical counterpart and the NCG method is a feasible and effective technique for the inverse problem of parameters estimation in fractional model.

MSC:

35R30 Inverse problems for PDEs
35R11 Fractional partial differential equations
35Q79 PDEs in connection with classical thermodynamics and heat transfer
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[1] Samko, S.; Kilbas, A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon Breach: Gordon Breach New York · Zbl 0818.26003
[2] Podlubny, I., Fractional Differential Equations (1993), Academic Press: Academic Press New York · Zbl 0918.34010
[3] Meerschaert, M. M.; Sikorskii, A., Stochastic Models for Fractional Calculus (2010), De Gruyter: De Gruyter Berlin · Zbl 1490.60004
[4] Green, A. E.; Naghdi, P. M., Thermoelasticity without energy dissipation, J. Elaticity, 31, 189-208 (1993) · Zbl 0784.73009
[5] Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31, 113-126 (1968) · Zbl 0164.12901
[6] Jiang, X.; Xu, M., The time fractional conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems, Physica A, 38, 3368-3374 (2010)
[7] Povstenko, Y., Non-axisymmetric solutions to time-fractional diffusion-wave equation in an inifite cylinder, Fract. Calc. Appl. Anal., 14, 418-438 (2011)
[8] Ilic, M.; Turner, I. W.; Liu, F.; Anh, V., Analytical and numerical solutions of a one-dimensional fractional-in-space diffusion equation in a composite medium, Appl. Math. Comput., 216, 2248-2262 (2010) · Zbl 1193.65168
[9] Chen, S.; Jiang, X., Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus, Physica A, 391, 3865-3874 (2012)
[10] Jiang, X.; Chen, S., Analytical andnumerical solutions of time fractional anomalous thermal diffusion equation in composite medium, ZAMM Z. Angew. Math. Mech., 95, 156-164 (2015) · Zbl 1322.80004
[11] Zhuang, Q.; Yu, B.; Jiang, X., An inverse problem of parameter estimation for time fractional heat conduction in a composite medium using carbon-carbon experimental data, Physica B, 456, 9-15 (2015)
[12] Yu, B.; Jiang, X.; Wang, C., Numerical algorithms to estimate relaxation parameters and caputo fractional derivative forafractional thermal wave model in spherical composite medium, Appl. Math. Comput., 274, 106-118 (2016) · Zbl 1410.80021
[13] Fan, W.; Jiang, X.; Qi, H., Parameter estimation for the generalied fractional element network zener model based on the Bayesian method, Physica A, 427, 40-49 (2015)
[14] Liu, F.; Burrage, K., Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62, 822-833 (2011) · Zbl 1228.93114
[15] Wei, H.; Chen, W.; Sun, H.; Li, X., A coupled method for inverse source problem of spatial fractional anomalous diffsuion eqautions, Inverse Probl. Sci. Eng., 118, 945-956 (2010) · Zbl 1204.65116
[16] Ghazizadeh, H. R.; Azimi, A.; Maerefat, M., An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation, Int. J. Heat Mass Transfer, 55, 2095-2101 (2012)
[17] Wang, L.; Liu, J., Total variation regulariztion for a backward time-fractional diffusion probelm, Inverse Problems, 29, Article 115013 pp. (2013) · Zbl 1297.65116
[18] Murio, D., Stable numercial solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 53, 1492-1501 (2007) · Zbl 1152.65463
[19] Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25, Article 115002 pp. (2009) · Zbl 1181.35322
[20] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447 (2011) · Zbl 1219.35367
[21] Zheng, G.; Wei, T., A new regulariztion method for the time-fractional inverse advection-dispersion problem, SIAM J. Numer. Anal., 49, 1972-1990 (2011) · Zbl 1239.65060
[22] Li, G.; Zhang, D.; Jia, X.; Yamamoto, M., Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29, Article 065014 pp. (2013) · Zbl 1281.65125
[23] Li, J.; Guo, B., Parameter identification in fractional differential equations, Acta Math. Sci., 33B, 855-864 (2013) · Zbl 1299.35316
[24] Tian, W.; Li, C.; Deng, W.; Wu, Y., Regularization methods for unknown source in space fractional diffusion equation, Math. Comput. Simulation, 85, 45-56 (2012) · Zbl 1260.35246
[25] Yu, B.; Jiang, X. Y., Numerical identification of the fractional derivatives in the two-dimensional fractional cable equation, J. Sci. Comput., 68, 252-272 (2016) · Zbl 1348.65135
[26] Chen, S.; Liu, F.; Jiang, X.; Turner, I.; Burrage, K., Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients, SIAM J. Numer. Anal., 54, 606-624 (2016) · Zbl 1382.65290
[27] Fan, W.; Jiang, X.; Chen, S., Parameter estimation for the fractional fractal diffusion model based on its numerical solution, Comput. Math. Appl., 71, 642-651 (2016) · Zbl 1443.65175
[28] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346
[29] Tadjeran, C.; Meerschaert, M. M.; Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 205-213 (2006) · Zbl 1089.65089
[30] Chen, C.; Liu, F.; Turner, I.; Anh, V., A fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 886-897 (2007) · Zbl 1165.65053
[31] Ervin, V. J.; Heuer, N.; Roop, J. P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45, 572-591 (2007) · Zbl 1141.65089
[32] Zhang, N.; Deng, W.; Wu, Y., Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4, 496-518 (2012) · Zbl 1262.65108
[33] Zeng, F.; Li, C.; Liu, F.; Turner, I., The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35, 2976-3000 (2013)
[34] Li, X. J.; Xu, C. J., A sapce-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 2108-2131 (2009) · Zbl 1193.35243
[35] Bueno-Orovio, A.; Kay, D.; Burrage, K., Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54, 937-954 (2014) · Zbl 1306.65265
[36] Wang, H.; Basu, T. S., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34, 2444-2458 (2012)
[37] Moroney, T.; Yang, Q., Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioner, J. Comput. Phys., 246, 304-317 (2013) · Zbl 1349.65398
[38] Burrage, K.; Hale, N.; Kay, D., An efficient implementation of an implicit fem scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34, A2145-A2172 (2012) · Zbl 1253.65146
[39] Yu, B.; Jiang, X.; Xu, H., A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation, Numer. Algorithms, 68, 923-950 (2015) · Zbl 1314.65114
[40] Sun, W.; Yuan, Y., Optimization Theory and Methods: Nonlinear Programming (2006), Springer: Springer New York
[41] Özisik, M. N.; Orlande, H. R.B., Inverse Heat Transfer: Fundamentals and Applications (2000), Taylor & Francis: Taylor & Francis New York
[42] Nocedal, J.; Wright, S. J., Numerical Optimization (2006), Springer: Springer New York · Zbl 1104.65059
[43] Samarskii, A. A., The Theory of Difference Schemes (2001), Marcel Dekker: Marcel Dekker New York · Zbl 0971.65076
[44] Dowding, K.; Beck, J.; Ulbrich, A.; Blackwell, B.; Hayes, J., Estimation of thermal properties and surface heat flux in carbon-carbon composite, J. Thermophys. Heat Transfer, 9, 345-351 (1995)
[45] Beck, J. V.; Blackwell, B.; Haji-sheikh, A., Comparision of some inverse heat conduction methods using experimental data, Int. J. Heat Mass Transfer, 39, 3649-3657 (1996)
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