Meissner Equation

	Meissner Equation The Meissner equation is a linear ordinary differential equation that is a special case of Hill's equation with the periodic function given as a square wave. There are many ways to write the Meissner equation. One is as : \frac + (\alpha^2 + \omega^2 \sgn \cos(t))y = 0 or : \frac + ( 1 + r f(t;a,b) ) y = 0 where : f(t;a,b) = -1 + 2 H_a( t \mod (a+b) ) and H_c(t) is the Heaviside function shifted to c. Another version is : \frac + \left( 1 + r \frac \right) y = 0. The Meissner equation was first studied as a toy problem for certain resonance problems. It is also useful for understand resonance problems in evolutionary biology. Because the time-dependence is piecewise linear, many calculations can be performed exactly, unlike for the Mathieu equation In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinary Differential Equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hill Differential Equation In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function by minimal period \pi . By these we mean that for all t :f(t+\pi)=f(t), and :\int_0^\pi f(t) \,dt=0, and if p is a number with 0 < p < \pi , the equation $f(t+p) = f(t)$ must fail for some $t$ . It is named after George William Hill, who introduced it in 1886. Because $f(t)$ has period $\pi$ , the Hill equation can be rewritten using the Fourier series of $f(t)$ : : $\frac+\left(\theta_0+2\sum_^\infty \theta_n \cos(2nt)+\sum_^\infty \phi_m \sin(2mt) \right ) y=0.$ Important special cases of Hill's equation include the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathieu Equation In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.Morse and Feshbach (1953).Brimacombe, Corless and Zamir (2021) They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.Gutiérrez-Vega (2015). Definition Mathieu functions In some usages, ''Mathieu function'' refers to solutions of the Mathieu differential equation for arbitrary values of a and q. When no confusion can arise, other authors use the term to refer specifically to \pi- or 2\pi-periodic solutions, which exist only for special valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Floquet Theory 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 :\dot = A(t) x, with \displaystyle A(t) a Piecewise#Continuity, piecewise continuous periodic function with period T and defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due to , gives a canonical form for each Fundamental solution, fundamental matrix solution of this common linear system. It gives a Change of coordinates, coordinate change \displaystyle y=Q^(t)x with \displaystyle Q(t+2T)=Q(t) that transforms the periodic system to a traditional linear system with constant, real coefficients. When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem. Note that the solutions of the linear differential equation form a vector space. A matrix \phi\,(t) is called a ''Fundamental ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]