In this paper some useful properties of the ultraspherical polynomials are derived. An ultraspherical approximation of any continuous function is presented. An ultraspherical approximation of finite integrals is introduced. The error estimation for the ultraspherical approximation is derived. An algorithm that gives an optimal approximation of the integrals is introduced. Numerical results are included to confirm the efficiency and accuracy of the method.

Rayleigh-Ritz method is modified to use an unconstrained optimization method in its minimization process. Chebyshev spectral technique based on El-Gendi method is used to approximate the solution of the problem and its derivatives. The Falkner-Skan equation has been solved through the use of Chebyshev Spectral-Ritz method for different values of its parameters. Comparisons are made between the proposed method and the classical solution.