Let D y(x) = f(x) be a differential equation defined by a given differential operator D. For simplicity, let us assume that the equation is given by
with the initial condition y(0) = 1, for x in the interval [0, 1]. The solution of this equation is y(x) = e^{–x}, which has the infinite series expansion
Let us assume that we use only the first three terms of it and let y_{2}(x) stand for this polynomial, of degree 2. If we take for y(x) the approximate expression y_{2}(x) and replace it into the differential equation (1) we obtain one with a nonzero right hand side:
The higher the order of approximation n, the smaller the right hand side shall be (in our example it will be x^{n}/n! for x in the interval 0,1). It is well known that for any given n the monomials Kx^{n} with a nonzero coefficient K, are not the closest (or best uniform) approximation of zero (the original right hand side of Equation (1) in the interval 0,1. In fact for every value of n, the Chebyshev Polynomials T_{n}(x) have that singular property.
Therefore, it seems appropriate to reconsider the problem in a completely new way: instead of finding the error term (x^{n}/n! in our concrete example) after fixing the approximation Y_{n}(x), we should fix the error term to be minimal (i.e., a Chebyshev polynomial) and then determine the approximation Y_{n}(x) which satisfies it. This original idea was proposed by the eminent mathematician Cornelius Lanczos in his formulation of the Tau Method, first conceived while working under Albert Einstein in problems of relativity theory [see Lanczos (1956) for an account of the original formulation of this method].
Assume that for a given differential operator D we can determine a sequence of polynomials Q_{n}(x), n, = 0, 1, 2,..., called canonical polynomials, associated with D, and such that for any value of n,
(i.e., they generate a basis in the image space). The Chebyshev polynomials T_{n}(x) have the form
where the coefficients c_{k} are known for all k and n. To find a Tau approximation Y_{n}(x) we can start using τT_{n}(x) as the minimal right hand side. (This term is usually referred to as the Tau Method perturbation term.) The free parameter τ is a multiplier introduced for us to be able to satisfy the initial condition (y(0) = 1 in our example, more generally, initial, boundary or multipoint boundary conditions). Then, if we set
the Tau approximate solution Y_{n}(x) is immediately given in our concrete example by
where the parameter τ is such that
The main problem then is to find the sequence Q_{n}(x) associated with D for all n. It has been proved that a sequence of canonical polynomials is uniquely associated to every differential operator D and that it can be generated recursively through a simple expression (iteratively for nonlinear operators), given exclusively in terms of the coefficients of the differential equation [see Oritz (1968) for further details]. For example, for Equation (1) the sequence of canonical polynomials is recursively generated by
that is:
Suitable generalizations of the Tau Method have been applied successfully to the numerical approximation of complex systems of nonlinear partial differential equations involving large gradients in the function and or derivatives, such as in the case of soliton interactions and also in problems of fracture mechanics, fluid mechanics, mathematical physics and molecular biology. It has also been used in the approximation of differential eigenvalue problems where the eigenvalue parameter appears nonlinearly; such questions arise in difficult fluid dynamics problems see Oritz, (1994), for an overview of recent research into the Tau Method.
One advantage of the Tau Method, which accounts for the high precision of its approximations is that it does not require a discretization of the given differential operator, a process which often alters the behavior of solutions.
Several numerical approximation techniques can be interpreted as special cases of the Tau Method, corresponding to different choices of perturbation terms, and can then be treated in a more unified and systematic way. Among these are the Spectral Method, Collocation and several types of projection and finite difference and finite element methods. (See for example Spectral Analysis and Numerical Methods.)
REFERENCES
Lanczos, C. (1956) Applied Analysis, Prentice-Hall, New Jersey.
Ortiz, E. L. (1968) The Tau Method, SIAM J. Numer. Anal., 6, 480–492.
Ortiz, E. L. (1994) The Tau method and the numerical solution of differential equations: past and recent research, Proc. Cornelius Lanczos International Centenary Conference, 77–81, J. David Brown, Ed., SIAM, Philadelphia.
References
- Lanczos, C. (1956) Applied Analysis, Prentice-Hall, New Jersey.
- Ortiz, E. L. (1968) The Tau Method, SIAM J. Numer. Anal., 6, 480â€“492.
- Ortiz, E. L. (1994) The Tau method and the numerical solution of differential equations: past and recent research, Proc. Cornelius Lanczos International Centenary Conference, 77â€“81, J. David Brown, Ed., SIAM, Philadelphia.
Heat & Mass Transfer, and Fluids Engineering