Approximate Solution of Mathieu's Differential Equation

Abstract

This paper presents a method for the approximate solution of the differential equations of the Mathieu-Hill type. This method is based on the analytical method of the Periodically Interrupted Electric Circuits. The second-order differential equations with periodic coefficients, considered in this paper, are represented by the general form : d²y/dz²+f(z)y=0, where f(z) is a single-valued periodic function of fundamental period Zᵀ, when f(z)=a+16q cos 2z, it is known as Mathieu's differential equation. Based on the procedure in this paper, the periodic function f(z) is subdivided into m functions, f₁(z), ··· , fᵣ(z), ··· , fₘ(z), each of which has a different interval zᵣ, (r=l, 2, ···, m) for one period zᴛ of f(z). Namely the function fᵣ(z) represents the linear approximation of f(z) in each interval, that is, fᵣ(z)=2cz+d, 0≦z≦zᵣ, (r=1, 2, ···, m) where the values of c and d are constant. From this practical linear approximation, the present method is adequate for the determination of the approximate solution of the differential equations of the Mathieu-Hill type and this method has certain advantages, especially for the stability of the solution and also the transient solution. The stability chart for Mathieu's differential equation is obtained and plotted for the ranges of -3≦a≦34 and 0≦q≦2. This result is very well coincident with Ince's numerical one computed for the range of q=0 to 5.0. The obtained solutions and their numerical results may be extensively accurate. And the procedure considered in this paper is useful for the mathematical analysis of a large class of physical problems.