A Mathematica Program for heat source function of 1D heat equation reconstruction by three types of data
This work provides a flexible Mathematica program for heat source reconstruction, but it is an incremental application of existing methods to a specific problem.
The paper solves an inverse problem for the 1D heat equation to reconstruct the heat source function from three types of data, using simple inversion and Tikhonov regularization with two parameter estimation methods. The reconstruction accuracy is measured by comparison with the true source function.
We solve an inverse problem for the one-dimensional heat diffusion equation. We reconstruct the heat source function for the three types of data: 1) single position point and different times, 2) constant time and uniformly distributed positions, 3) random position points and different times. First we demonstrate reconstruction using simple inversion of discretized Kernel matrix. Then we apply Tikhonov regularization for two types of the parameter of regularization estimation. The first one, which is in fact exemplary simulation, is based on minimization of the distance in C space of reconstructed function to the initial source function. Second rule is known as Discrepancy principle. We generate the data from the chosen source function. In order to get some measure of accuracy of reconstruction we compare the result with the function from which data was generated. We also deliver corresponding application in symbolic computation environment of Mathematica. The program has a lot of flexibility, it can perform reconstruction for much more general input then one considered in the paper.