Abstract: In the present investigation,the unsteady flow of upper-convected Maxwell fluid in a horizontal circular pipe is studied by spectral method.The unste
ady problem is mathematically reduced to a partial differential equation of second order.By using spectral method the partial differential equation can be reduced to a system of ordinary differential equations for different terms of Chebyshev polynomials approximations.The ordinary differential equations are solved by the method of Laplace transform and the eigenvalue method that led to an analytical form of the solutions.

Key words: spectral method, unsteady flow, Chebyshev polynomial