Simulation of oscillatory processes using differential equations of Liouville type

The problems of mathematical modeling lead to the necessity to create computational algorithms directly related to finding solutions of differential equations with partial derivatives in explicit form. In this study, explicit solutions are original tests for approximate methods that reflect the essence of the general solution. Each explicit solution of the differential equation has great importance as an accurate representation of the physical phenomenon under study within the framework of this model, as an analysis of the verification of numerical methods, as a theoretical basis for further modeling of the researched process. There have been considered aspects of the application of mathematical modeling to the study of oscillatory processes. Methods of reducing the solution of differential equations to an explicit form are proposed. Solution is given through functions of real arguments. The possible field of application is the study of wave processes. There is being considered the problem of building a variety of explicit solutions of the nonlinear third-order differential equation with partial derivatives with two boundary singular planes in space and second-order equation of general form with hyper-singular lines in the plane. On the basis of the developed method there has been proved the uniqueness of the obtained integral representations, and the boundary value problem of Cauchy type is posed and solved. The results are formulated in the form of theorems.