Numerical approximations using Chebyshev polynomial expansions: El-gendi's method revisited

The aim of this work is to nd numerical solutions for dierential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N. The solutions are exact at these points, apart from round-o com- puter errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial-value problems in time-dependent quantum eld theory, and second order boundary-value problems in fluid dynamics are pre- sented. of the functions rather than the Chebyshev coecients. The two approaches are formally equivalent in the sense that if we have the values of the function, the Chebyshev coecients can be calculated. In this paper we use the discrete orthogonality relation- ships of the Chebyshev polynomials to exactly discretize various continuous equations by reducing the study of the solutions to the Hilbert space of functions dened on the set of (N+1) extrema of TN(x), spanned by a dis- crete (N+1) term Chebyshev polynomial basis. In our approach we follow closely the procedures outlined by El-gendy (6) for the calculation of integrals, but extend his work to the calculation of derivatives. We also show that similar procedures can be applied for a second grid given by the zeros of TN(x). The paper is organized as follows: In Section II we re- view the basic properties of the Chebyshev polynomial and derive the general theoretical ingredients that allow us to discretize the various equations. The key element is the calculation of derivatives and integrals without ex- plicitly calculating the Chebyshev expansion coecients. In Sections III and IV we apply the formalism to obtain numerical solutions of initial-value and boundary-value problems, respectively. We accompany the general pre- sentation with examples, and compare the solution ob- tained using the proposed Chebyshev method with the numerical solution obtained using the nite-dierences method. Our conclusions are presented in Section V.