Chebyshev polynomials, discovered by Pafnuty Chebyshev, are a set of orthogonal polynomials that create the basis of a chosen space. The Chebyshev polynomials have a recursive formula that is:

Chebyshev polynomials have numerous approximations such as with least squares analysis, in Clenshaw-Curtus method for numerical integration and even trigonometric multiple-angle formulas.

One of the interesting applications of Chebyshev polynomials is in the use of pseudo-spectral Chebyshev-Fourier method to solve for the three-dimensional Navier-Stokes equations. The Navier-Stokes equations describe fluid motion in liquids and in gases. A journal article published by E. Serre and P. Bontoux utilizes Chebyshev polynomials to find a solution to describe the vortex created by three dimensional swirling fluids.

The article presents the problem as trying to mathematically describe the “motion of a viscous fluid contained in a closed cylinder, with a rotating disk lid.” The boundary conditions are set out such as aspect ratio of the cylinder dimension and angular velocity of the top end. The article claims there isn’t an accepted mathematical theory that describes how the vortex created in the fluid. In their mathematical model, they use cylindrical polar coordinates and normalize the radial coordinates of the system to [-1, 1] in order to use Chebyshev polynomials. Chebyshev polynomials are used because they create an orthogonal basis and have an exponential convergence. Below is an equation for the flow variables (r is the radius, z is the axial component, t is time)

The terms T_n and T­_m are the Chebyshev Polynomials, which corresponds to the equations outlined earlier. Thus, Chebyshev Polynomials were used to fulfill Navier-Stokes equations as a least-squares spectral method.

http://journals.cambridge.org/download.php?file=%2FFLM%2FFLM459%2FS0022112002007875a.pdf&code=6083e86b3c3175fa7005a5003ed21b36

http://en.wikipedia.org/wiki/Chebyshev_polynomials