An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Abstract

In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed (very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit (ADE) method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $lsqnonlin$ tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.

Keywords

• Existence and uniqueness
• Contraction mapping
• 2D parabolic equation
• Inverse identification problem
• Nonlinear optimization
• Tikhonov regularization

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