Propagation dynamics of a diffusing substance on the surface and in the bulk of water

Two- and three-dimensional problems of propagation of a diffusing substance on the surface and in the bulk of water are considered. An analytic solution to boundary value problems for the diffusion equation is proposed in unbounded domains for the initial condition of special form. The above-threshold range of concentrations of the diffusing substance is analyzed. Propagation of the diffusing substance along the free surface and at the bottom of a basin is considered. Analytic solutions to the problems are obtained by the Fourier method followed by the expansion of an arbitrary function in terms of Bessel functions and Legendre polynomials. The analytic solutions constructed are compared with the numerical solutions of a boundary value problem obtained by the software package Mathematica. The size of the pollution spot as a function of time, as well as the effect of geometric and physical parameters used on the spot size, is analyzed. The mathematical models considered have an important applied value in the problem of environment protection in case of emergency situations on ships.