Numerical investigation of an inverse problem based on regularization method

In this study, a numerical approach of the spectral collocation method coupled with a regularization technique is applied for solving an inverse parabolic problem of the heat equation in a quarter plane. The problem includes the estimation of an unknown boundary condition from an overspecified condition. The stable solution of the problem exists and is proved by Tikhonov regularization technique. The algorithm works without any mesh points or elements, and accurate results can be obtained efficiently. By employing the numerical algorithm on the problem, the resultant matrix equation is ill-condition. To regularize this matrix equation, we apply regularization technique, with the L-curve and general cross-validation criteria for choosing the regularization parameter. For demonstrating the performance and ability of the proposed algorithm, a test example is presented. The numerical results showed that the solution obtained with the algorithm designed in this paper is stable with the noisy data and the unknown boundary condition was recovered very well.