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Mathieu function. In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation. where a, q are real -valued parameters. Since we may add π/2 to x to change the sign of q, it is a usual convention to set q ≥ 0.
The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency. The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions. They have long been applied on a broad scope of wave ...
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The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained.
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form. with and being a piecewise continuous periodic function with period and defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due ...
The general solution of the above differential equation for a given value of a and q is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively. In general, the Mathieu functions are aperiodic; however, for characteristic values of a n ( q ) , b n ( q ) {\displaystyle a_{n}(q),b_{n}(q ...
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation. where is a periodic function with minimal period and average zero. By these we mean that for all. and. and if is a number with , the equation must fail for some . [1] It is named after George William Hill, who introduced it ...
Mathematically this can be modeled with the help of the Mathieu differential equation. [3] Ion path through a quadrupole. Ideally, the rods are hyperbolic, however cylindrical rods with a specific ratio of rod diameter-to-spacing provide an easier-to-manufacture adequate approximation to hyperbolas. Small variations in the ratio have large ...
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