Asymptotic simulation of the spectral problem

A mathematical model of a spectral problem containing a small parameter at the higher derivatives is investigated by an asymptotic method. The expected solution and its geometric parameters are presented as formal asymptotic decompositions. As a result, the original high-order differential task is reduced to a sequence of lower order tasks. Next, the zero approximation problem under the main boundary conditions is solved, the execution of which leads to a system of linear algebraic equations containing a spectral parameter. Equating to the zero of the determinant of the resulting system gives transcendent equations for eigenvalues. The asymptotic genesis of emerging transcendental equations allows asymptotic analysis to be applied to these equations. It helps to reveal the components of the equation that make the major contribution to the spectrum formation. Graphical solutions are used as leading considerations in this analysis. As a result of the asymptotic simulation, approximate formulas for the eigenvalues of the spectral problem have been obtained.