A FAMILY OF METHODS EQUIVALENT TO THE TAU METHOD AND THEIR ERROR STRUCTURE

Mohamad  K. DAOU

The Tau Method as a method for solving ordinary and partial differential equations (ODE's and PDE's), is analyzed in its various forms, and used to develop new techniques for the collocation method and for Chebyshev series expansions methods on the basis of it. It is shown that through Ortiz's algebraic theory of the Tau Method it is possible to specify the Tau perturbation term (TPT) which produces a collocation approximant. As a result of that, the collocation method is represented in terms of the so-called shifted canonical polynomials, based on generalized Lanczos canonical polynomials. This approach has some computational advantages over collocation. Through this approach problems are made dependent, not on the collocation points, but on the coefficients of orthogonal polynomials. Subsequently we find an approximate solution solving an algebraic system whose dimension is independent of the degree of approximation.

The Operational approach to the Tau Method is also used to formulate a unified approach to Chebyshev and Taylor's truncated expansions techniques. The latter are then classified according to their respective TPTs. The Chebyshev collocation method is also discussed in terms of a projective method and a new projective space related to it is introduced. The implicit Runge Kutta methods (RKM), which are special cases of the collocation method, are examined and classified here in terms of their TPTs.

A comparison of Tau and collocation methods leads to an interesting analogy with the interpolation formula of Newton and Lagrange. The duality between 'Method' and 'Formula' is fully investigated and as a result a new procedure to generate the generalized canonical polynomials, based on Newton's formula, is developed. The results established above were extended to the step‑by‑step Tau Method and to piecewise collocation. This turned our attention to the question of local truncation errors. We discuss the following aspects of this problem:

Quantitative: High order estimates of the error for both function and derivative are given for the discrete and continuous cases when the coefficients of a second order ODE are perturbed; these estimates are applied to piecewise collocation and an analytic investigation is carried out to discuss the superconvergence phenomenon which occurs at the division points when Legendre polynomial are in use. In addition, those estimates are applied to nonlinear ODE’s and a new approximation method is introduced.

Structural: The behavior of the error curve is studied. It is found that the (known) TPT plays a leading role, giving valuable information on the unknown error function. As a consequence, a machinery to estimate the number of oscillations of the error curve is obtained. Following that, the discussion is extended to cover the case of the best polynomial approximation error. We give an upper bound bracketing up the lower bound already given by Chebyshev’s equioscillation theorem. Ortiz's algebraic theory is generalized to second order PDE's associated with initial, boundary or mixed conditions.

Some equivalence results between the global Tau Method and the traditional Galerkin method are deduced. It is found that, in some cases, the bipolynomial approximant obtained by The Tau Method is much more accurate than those obtained by Galerkin's. Finally, we propose a new purely recursive formulation of Galerkin's method and give numerical examples to test the various estimates given throughout this Thesis.