On the convergence of WKB approximations of the damped Mathieu equation

Abstract

The form of the fundamental set of solutions of the damped Mathieu equation is determined by Floquet theory. In the limit as m → 0, we can apply WKB theory to get first order approximations of the fundamental set. WKB theory states that this approximation gets better as m → 0 and T is fixed. However, convergence of the periodic part and characteristic exponent is not addressed. We show that they converge to those predicted by WKB theory. We also provide a rate of convergence that is not dependent on T.

Differential equations of motion of a material point in the perpendicular plane to the plane of the gravitating disk

“…Nwaigwe [11] describe in detail the relevance of the investigated problem of the resonant relationship, which are perpendicular to the main plane and have an almost periodic force of action directed to the center of symmetry of the Galaxy, which can lead to a star moving away from the plane. Cases in which the stable position of the studied movement is violated is the main problem.…”

“…The differential equation of system (6) after substitution into the right side of the expression for u and θ0 from (7), expressed through θ0, can be transformed with the right side in the form of harmonics with coefficients 𝑐 0 , 𝑐 𝑖 , 𝑑 𝑖 , 𝑖 = 1,7 and can be determined by the following expressions depending on Simplifications of the first differential equation of the system (6) taking into account the Gilden choice (7) lead it to an inhomogeneous linear differential equation of the second order (9): under these assumptions, the solution can be represented (11): According to Gilden's interpretation of the application of the method from [10], it is noted that in subsequent approximations outside the signs of trigonometric functions even higher degrees will be encountered θ0, that the use of a variable θ0 does not significantly change the nature of the old methods. When strengthening the requirements of researchers for a variable θ0, as soon as in the form of an argument of trigonometric functions, it is necessary to resort to other artificial methods.…”