A Chebyshev expansion method for solving the time‐dependent partial differential equations

A Chebyshev expansion method for the parabolic and Burgers equations is developed. The spatial derivatives are approximated by the Chebyshev polynomials and the time derivative is treated by a finite-difference scheme. The accuracy of the resultant is modified by using suitable extrapolation scheme in the time direction. The Chebyshev expansion method is based on using an explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives. In addition, this formula is used in equating the order of derivatives appearing in the differential equation to the order of differential equation. The successive integration is used to obtain an algebraic system in the expansion coefficients. Finally, numerical examples are studied to demonstrate the viability of this method. Copyright © 2007 John Wiley & Sons, Ltd.