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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 ...
Poincaré–Lindstedt method. In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms —terms growing without bound—arising in the ...
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 ...
The same illustration for The midpoint method converges faster than the Euler method, as . Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to ...
Differential equations. In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential ...
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 ...