Development of the analytical method of the general mathieu equation solution
Journal Title: Восточно-Европейский журнал передовых технологий - Year 2018, Vol 4, Issue 4
Abstract
<p>An analytical method for solving the general Mathieu differential equation in the initial form is proposed. The method is based on the corresponding exact solution, which is found for arbitrary numerical parameters of the initial equation a and q. In turn, the exact solution is expressed through the fundamental functions, which are represented by series in powers of the parameters a and q with variable coefficients.</p><p>Along with Mathieu equation, the system of equivalent differential equations is also considered. It is shown that the Wronskian matrix, which is formed of the fundamental functions of the equation, is the transition matrix of the system. Thus, it is proved that the fundamental functions of the Mathieu equation satisfy the given conditions at the zero point.</p><p>In order to solve the problem of numerical realization of the exact formulas found, the fundamental functions are represented by power series. To calculate the coefficients of the power series, the corresponding recurrent relations are derived.</p><p>As a result of the research, finite analytical formulas for calculating the characteristic exponent v, the determination of which is the central part of any problem, the mathematical model of which is the Mathieu equation are obtained. In fact, a direct analytical dependence of v on the initial parameters of the equation a, q is established. This is especially important, since the parameter v plays the role of an indicator of such properties of solutions of the Mathieu equation as boundedness and periodicity.</p><p>The proposed analytical method is a real alternative to the application of approximate methods in solving any problems that are reduced to the Mathieu equation. The presence of finite analytic formulas will allow avoiding the procedure of finding the solutions of the equation in the future. Instead, to solve the problem in each specific case, it is enough to implement the obtained analytical formulas numerically.</p>
Authors and Affiliations
Yurii Krutii, Alexander Vasiliev
Keywords
Related Articles
- EP ID EP528068
- DOI 10.15587/1729-4061.2018.140634
- Views 56
- Downloads 0
Adaptive control over nonlinear objects using the robust neural network FCMAC
The paper explores issues related to the application of artificial neural networks (ANN) when solving the problems on identification and control of nonlinear dynamic systems. We have investigated characteristics of the n...
Aerodynamics of the turbulent flow around a multielement airfoil in cruse configuration and in takeoff and landing configuration
<p>Numerical modeling of multi-element airfoil's aerodynamics employs the Reynolds averaged Navier-Stokes equations of incompressible environment, which are related via a single-parametric differential turbulence model b...
Evaluation of dynamic properties of gas pumping units according to the results of experimental researches
Experimental studies of the dynamic properties of gas pumping units (GPU) of various types, which allowed us to obtain GPU acceleration curves and determine the parameters of the transfer function through various transmi...
Identification of distribution features of the instantaneous power components of the electric energy of the circuit with polyharmonic current
Many researchers pay attention to the problem of the distortion electric power. The known theoretical results that are reflected in the acting standards are reasonably criticized in the part of the distortion power gener...
Research of a microwave radiometer for monitoring of internal temperature of biological tissues
<p>Currently, there is growing interest among specialists in the use of non-invasive dose-free technologies for diagnosis and monitoring treatment of various diseases. Microwave radiometry enables non-invasive detection...