In this paper, we consider the problem for identifying the unknown source in the Poisson equation. A modified Tikhonov regularization method is presented to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable.<\/p>\r\n","references":"[1] A. Burykin, A. Denisov, Determination of the unknown sources in the heat-conduction equation, Computational Mathematics and Modeling 8 (4) (1997) 309-313.\r\n[2] J. Cannon, P. Duchateau, Structural identification of an unknown source\r\nterm in a heat equation, Inverse Problems 14 (3) (1998) 535-551.\r\n[3] J. Cannon, S. Perez-Esteva, Uniqueness and stability of 3d heat sources,\r\nInverse problems 7 (1991) 57.\r\n[4] M. Choulli, M. Yamamoto, Conditional stability in determining a heat\r\nsource, Journal of Inverse and Ill-posed Problems 12 (3) (2004) 233-243.\r\n[5] F. Dou, C. Fu, F. Yang, Optimal error bound and Fourier regularization\r\nfor identifying an unknown source in the heat equation, Journal of Computational and Applied Mathematics 230 (2) (2009) 728-737.\r\n[6] A. El Badia, T. Ha-Duong, On an inverse source problem for the heat equation. Application to a pollution detection problem, Journal of Inverse and Ill-posed Problems 10 (6) (2002) 585-600.\r\n[7] A. Farcas, D. Lesnic, The boundary-element method for the determination\r\nof a heat source dependent on one variable, Journal of Engineering Mathematics 54 (4) (2006) 375-388.\r\n[8] A. Kirsch, An introduction to the mathematical theory of inverse\r\nproblems, Springer-Verlag,New York, 1996.\r\n[9] G. Li, Data compatibility and conditional stability for an inverse source\r\nproblem in the heat equation, Applied Mathematics and Computation\r\n173 (1) (2006) 566-581.\r\n[10] L. Ling, M. Yamamoto, Y. Hon, T. Takeuchi, Identification of source\r\nlocations in two-dimensional heat equations, Inverse Problems 22 (4)\r\n(2006) 1289-1305.\r\n[11] H. Park, J. Chung, A sequential method of solving inverse natural\r\nconvection problems, Inverse Problems 18 (3) (2002) 529-546.\r\n[12] V. Ryaben'kii, S. Tsynkov, S. Utyuzhnikov, Inverse source problem and\r\nactive shielding for composite domains, Applied Mathematics Letters\r\n20 (5) (2007) 511-515.\r\n[13] M. Yamamoto, Conditional stability in determination of force terms of heat equations in a rectangle, Mathematical and Computer Modelling\r\n18 (1) (1993) 79-88.\r\n[14] L. Yan, C. Fu, F. Yang, The method of fundamental solutions for\r\nthe inverse heat source problem, Engineering Analysis with Boundary\r\nElements 32 (3) (2008) 216\u2013222.\r\n[15] L. Yan, F. Yang, C. Fu, A meshless method for solving an inverse\r\nspacewise-dependent heat source problem, Journal of Computational\r\nPhysics 228 (1) (2009) 123\u2013136.\r\n[16] F. Yang, The truncation method for identifying an unknown source in\r\nthe poisson equation, Applied Mathematics and Computation 22 (2011)\r\n9334\u20139339.\r\n[17] F. Yang, C. Fu, The modified regularization method for identifying the\r\nunknown source on poisson equation, Applied Mathematical Modelling\r\n2 (2012) 756\u2013763.\r\n[18] Z. Yi, D. Murio, Source term identification in 1-D IHCP, Computers &\r\nMathematics with Applications 47 (12) (2004) 1921\u20131933.\r\n[19] Z. Zhao, Z. Meng, A modified tikhonov regularization method for a\r\nbackward heat equation, Inverse Problems in Science and Engineering\r\n19 (8) (2011) 1175\u20131182.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 68, 2012"}