A Lie-Group Adaptive Method to Identify the Radiative Coefficients in Parabolic Partial Differential Equations

摘要

We consider two inverse problems for estimating radiative coefficients a(x) and a(x, y), respectively, in T,(x, t) =T_(xx)(x, t)-a(x)T(x, t), and T_t(x, y, t) = T_(xx)(x, y, t) +T_(yy)(x, y, t)-α(x, y)T(x, y, t), where α are assumed to be continuous functions of space variables. A Lie-group adaptive method is developed, which can be used to find α at the spatially discretized points, where we only utilize the initial condition and boundary conditions, such as those for a typical direct problem. This point is quite different from other methods, which need the overspecified final time data. Three-fold advantages can be gained by the present Lie-group adaptive method (LGAM): (i) no a priori information of radiative coefficients is required, (ii) no extra data are measured, and (iii) no complicated procedure is involved. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions for 1-D and 2-D cases.