# What physical phenomena are modelled by Chebyshev equation? [closed]

What physical phenomena are modeled by Chebyshev equation? The equation is below

$$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y ~=~ 0 .$$

I could not find it in Wikipedia or in Google (at least quickly). The answer should be simple for anyone who knows it. I think the answer to such a question should be quickly available but, it's not.

The Legendre polynomials arise naturally when solving the Poisson equation for a system with spherical symmetry (such as the hydrogen atom).

The Bessel functions arise naturally when solving the Poisson equation for a system with cylindrical symmetry.

In essentially the same way, the Chebyshev equation and its solutions arise when you consider a problem using an elliptical coordinate system.

I remember a side comment (perhaps in Arfken & Weber?) that all the named "special functions" arise from solving the Poisson equation in different coordinate systems. (I forget which one is toroidal coordinates.)

Despite the similarity of Chebyshev's equation with Legendre's equation, it does not occur often in physics or engineering, however, solutions of Chebyshev's equation are of much importance in topics of numerical analysis such as solution to partial differential equations, smoothing of data and others. While, on the other hand, its close associate Legendre's equation occurs quite often in areas such as electrodynamics and quantum mechanics, among others.

However, if you consider Chebyshev polynomials, they are of tremendous use in case of physical sciences.The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials.Then again,these polynomials find extensive use in the optimal control of time-varying linear systems.These are some major applications. Further more, I am linking the Google search page for your interest, if you like.

Chebyshev polynomials are also used in observations of oscillation phenomena, a notable example being the systems of two independent oscillators which plot the Lissajous curves. Chebyshev polynomials arise naturally when considering Lissajous curves with $a=1$ and $b=N$, which turn out to be Chebyshev polynomials of the first kind of degree N.

Anyone who's ever played with an oscilloscope knows that if the ratio of frequencies of some alternating currents passing through the oscilloscope is a natural number (i.e. ${\omega}_{2}=N{\omega}_{1}$) is aware that the oscilloscope plots a line, a parabola etc. (if phases are alligned, or else the curve becomes a general Lissajous curve), those being the first two polynomials.