HYBRID ASYMPTOTIC APPROACH TO SOLVING NONLINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS IN THE PRESENCE OF THE δ-FUNCTION

When design engineers are creating new design methods, it is important for them to have analytical dependencies to assess the impact of the parameters and external loads of the system under study on its stability and dynamic behaviour. This is used to create heterogeneous structures for aerospace engineering, the construction industry, and mechanical engineering. The solution of a large number of mathematical physics problems that boil down to the necessity of solving singular differential equations with variable discontinuity coefficients is often based on the use of numerical methods that do not allow for a qualitative analysis of the dependencies obtained. A feature of this work is the development of a hybrid WKB-Galyorkin asymptotic approach to the solution of nonlinear singular (with a “small” parameter at the older derivative) differential equations with variable discontinuous coefficients in the presence of a δ-function in the right-hand side. An approximate algorithm of analytical solution suitable for solving of mathematical physics applied problems using the perturbation method, which allows estimating the influence of the nonlinear component of the equation and computer algebra. is proposed. As an example, a nonlinear Duffing type differential equation with variable coefficients is considered. Particular attention is paid to the influence of the nature of the change in the coefficients of the main singular differential equation on the effect of the presence of a δ-function in the first derivative. Numerical results of analytical solutions (depending on the value of the parameters of the asymptotic development in two approximations) and comparison of the approximate analytical solution with the direct numerical solution of the problem under study are provided. Using computer algebra software package “Mathematica”, graphs of the results of calculations of the main equation of the problem are constructed by the direct numerical integration and the hybrid asymptotic method.