Abstract:

The parabolic equation，a classic developing equation，is widely applied in the scientific and engineering fields，such as predicting the solute transportation in groundwater，simulating temperature of the thermal conductive materials and analyzing the population mutual effect.Generally，different practical problems are summarized into different models.However，there always exist some unknown conditions or parameters in the models when the parabolic models are applied to address some practical problems.The unknown conditions usually need reconstruction by some other additional data in an indirect way.Mathematically，these identification problems are called as parabolic equation inverse problems.

In application，the initial state or the inner source term in some diffusion system always needs identification.The identification problems are generally modeled as backward problem and inverse source problem for the parabolic equation，two classic inverse problems extensively studied by engineers and mathematicians.In theoretical aspect，the existence and uniqueness of the two inverse problems are proven by integral equation theories，Laplace transformation，Kalerman estimate and fixed point theory.In algorithm aspect，quasi-boundary regularization method，quasi-reversibility method，Tikhonov regularization method and projection method are usually adopted to solve the above inverse problems.According to published works，the additional data for the backward problem or the inverse source problem are required in the whole spatial domain on some terminal time.Studies of inverse problems for parabolic equation are scarce when the observation data comes from local observation，while the simultaneous identification problems for the parabolic equation are even scarcer when the additional data are taken from the local measurement.

Generally，the observation in the whole spatial domain is difficult.Obtaining the observation data in a local spatially domain is more practical.Driven by the real application，simultaneous identification of initial value and source term for a kind of parabolic equation is studied in this paper based on local measurements.First，the formal series solution to the direct problem is obtained by the eigenfunction expansion method，and the non-uniqueness of the simultaneous identification is proven when the additional data are given in a spatial sub-domain at two observation times.Then，the uniqueness of the simultaneous inversion problem is proven based on the local measurements at three observation times and the result of analytic continuation for the parabolic equation.Next，an easily paralleled inversion algorithm is proposed based on the technique of finite element interpolation and the principle of superposition.Last，several numerical examples including the cases of existing and non-existing analytical solutions are tested to demonstrate the efficiency of the inversion algorithm.