The total first boundary value problem for equation of hyperbolic type with piecewise constant coefficients and $\delta$-singularities

Abstract

For the first time a new formal solving scheme of the general first boundary value problem for a hyperbolic type equation with piecewise constant coefficients and-singularities was proposed and justified. In the basis of the solving scheme is a concept of quasi-derivatives, a modern theory of systems of linear differential equations, the classical Fourier method and a reduction method. The advantage of this method is a possibility to examine a problem on each breakdown segment and then to combine obtained solutions on the basis of matrix calculation. Such an approach allows the use of software tools for solving the problem. MSC: 34B05.

General Scheme of Modeling of Longitudinal Oscillations in Horizontal Rods

“…The solution to this problem can be achieved to some extent through the use of modern calculation technologies using software tools [5]. The authors proposed to use the matrix form of the second-order differential equation and modern methods of calculations using mathematical software to model the oscillatory processes in the works [14]. The objective of the research is to find solutions of the equation of rod oscillations with different load and conditions using the concept of a quasi-derivative.…”