Abstract
This article is a review, which provides a solution to the one-dimensional and three-dimensional problem of the wave process. The equation in question is a hyperbolic equation that can be derived only for the one-dimensional case and describes the process of river flow, matter transfer, diffusive mass transfer, and flow through a porous medium. The solution of the equation is sought in the form of integro-differential operators, represented as a special series with constant coefficients determined from the recurrence relation. With this approach, general solutions are expressed in terms of arbitrary functions and are not related to solving another equation. The resulting form of general solutions makes it possible to apply the method of initial functions to solve boundary value problems, since arbitrary analytical functions included in general solutions can be expressed in terms of initial functions specified by the problem condition. From these formulas of general solutions, it is easy to find all particular solutions in various classes of analytical functions. Next, the solution of a special case of the Lame equation describing the wave process is considered. The solution of a special case of the Lame equation is also sought in the form of a series, using the recurrent operator method. The algorithm for solving the three-dimensional problem of the wave process is similar to that for the one-dimensional problem. This approach is very effective for constructing solutions to the equations of the theory of thermal conductivity and the theory of elasticity. The results obtained are illustrated graphically and shown in the tables.