Let there be a function f(x), which is a realization of some stationary random function. It is required to find an approximate value of the integral ...

The creation of highly sensitive measuring equipment made it possible to determine small concentrations and fluxes of radioactive impurities in the atmosphere. Due to this, an important from a practical point of view, the task of determining the position and power of sources of impurity located underground, based on observations in the surface layer of the atmosphere has arisen. The solution to this problem must be based on a theory that explains the propagation of radioactive impurities in the two-layer earth-atmosphere medium.

Recently, the formal apparatus of delta functions, i.e., discontinuous functions defined by equalities, has been increasingly applied to the solution of various problems of mathematical physics ...

Let’s consider the following problem in the theory of heat conduction. An initial temperature distribution is given in a space consisting of two media separated by a flat interface. It is required to find the temperature at any point in the space, at any moment of time. The thermal characteristics of each of the two media are assumed to be constant. The formulated problem has been considered by a number of authors for the one-dimensional case. The possibility of solving the multidimensional case using integral equations was pointed out by Münz [4]. In [5], a two-dimensional problem was solved by the method of successive approximations. In the present paper, a closed-form solution of the problem under consideration is given for the two- and three-dimensional case. The solution method can be applied to a number of similar problems.