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.Comment: 10 pages, 7 figure