On several ill-posed and ill-conditioned mathematical problems of soil physics

Several well-known mathematical models of concentration fields in the soil (both at the single aggregate and the profile scales) are considered. It is shown that the respective boundary value problems for steady-state profiles belong to the class of ill-posed problems, since their solution does not exist. It occurs because a certain set of processes (for example, diffusion transport + first-order kinetic of the consumption) restricts possible boundary conditions, which, therefore, can no longer be arbitrary. Ill-posed inverse problems are also briefly described as well as one ill-conditioned inverse problem of parameters identification for mathematical model of the soil organic matter concentration profile. Exact solution for this model is the sum of two exponents. For a certain input data it was shown that this problem belongs to the class of ill-conditioned, since a small bias in the input data causes a significantly larger error in the solution (i.e. in calculated parameters).