Two-dimensional parabolic inverse source problem with final overdetermination in reproducing kernel space

Abstract

A new method of the reproducing kernel Hilbert space is applied to a two-dimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.

Keywords

Inverse source problem / Final overdetermination / Parabolic equation / Reproducing kernel

References

[1] Ebel A, Davitashvili T. Air, water and soil quality modelling for risk and impact assessment, 2007, Dordrecht: Springer

[2] Ling L, Yamamoto M, Hon Y C, Takeuchi T. Identification of source locations in two-dimensional heat equations. Inverse Probl., 2006, 22: 1289-1305

[3] Cheng W, Zhao L, Fu C. Source term identification for an axisymmetric inverse heat conduction problem. Comput. Math. Appl., 2010, 59: 142-148

[4] Badia A E, Ha-Duong T. On an inverse source problem for the heat equation, application to a pollution detection problem. J. Inv. Ill-Posed Problems, 2002, 10: 585-599

[5] Choulli M, Yamamoto M. Conditional stability in determining a heat source. J. Inv. Ill-Posed Problems, 2004, 12: 233-243

[6] Dehghan M, Tatari M. Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math. Comput. Model., 2006, 44: 1160-1168

[7] Sakamoto K, Yamamoto M. Inverse heat source problem from time distributing overdetermination. Appl. Anal., 2009, 88: 735-748

[8] Cannon J R, Pérez-Esteva S. Uniqueness and stability of 3D heat sources. Inverse Probl., 1991, 7: 57-62

[9] Choulli M, Yamamoto M. Generic well-posedness of an inverse parabolic problem-the Hölder-space approach. Inverse Probl., 1996, 12: 195-205

[10] Fan J, Nakamura G. Well-posedness of an inverse problem of Navier-Stokes equations with the final overdetermination. J. Inv. Ill-Posed Problems, 2009, 17: 565-584

[11] Choulli M, Yamamoto M. Generic well-posedness of a linear inverse parabolic problem with diffusion parameters. J. Inv. Ill-Posed Problems, 1999, 7: 241-254

[12] Kamynin V L. On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition. Math. Notes, 2003, 73: 202-211

[13] Isakov V. Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math., 1991, 44: 185-209

[14] Isakov V. Inverse problems for partial differential equations. Applied Mathematical Sciences, 2006, New York: Springer-Verlag

[15] Yi Z, Murio D A. Source term identification in 1-D IHCP. Comput. Math. Appl., 2004, 47: 1921-1933

[16] Trong D D, Long N T, Alain P N D. Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term. J. Math. Anal. Appl., 2005, 312: 93-104

[17] Trong D D, Quan P H, Alain P N D. Determination of a two-dimensional heat source: uniqueness, regularization and error estimate. J. Comput. Appl. Math., 2006, 191: 50-67

[18] Chen Q, Liu J. Solving an inverse parabolic problem by optimization from final measurement data. J. Comput. Appl. Math., 2006, 193: 183-203

[19] Choulli M. An inverse problem for a semilinear parabolic equation. Inverse Probl., 1994, 10: 1123-1132

[20] Yang L, Yu J, Deng Z. An inverse problem of identifying the coefficient of parabolic equation. Appl. Math. Model., 2008, 32: 1984-1995

[21] Takeuchi T, Yamamoto M. Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation. SIAM J. Sci. Comput., 2008, 31: 112-142

[22] Hon Y C, Takeuchi T. Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem. Adv. Comput. Math., 2011, 34: 167-183

[23] Cui M, Lin Y. Nonlinear numerical analysis in the reproducing kernel space, 2009, New York: Nova Science Publishers, Inc.

[24] Cui M, Geng F. A computational method for solving one-dimensional variable-coefficient Burgers equation. Appl. Math. Comput., 2007, 188: 1389-1401

[25] Zhou Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions. J. Comput. Appl. Math., 2009, 230: 770-780

[26] Wang W, Han B, Yamamoto M. Inverse heat problem of determining time-dependent source parameter in reproducing kernel space. Nonlinear Anal. Real, 2013, 14: 875-887

[27] Wang W, Yamamoto M, Han B. Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation. Inverse Probl., 2013, 29: 1-15