An operational approach to the Tau method for the numerical solution of non-linear differential equations

A Convergence Result For The Tau Method

“…Looking at these papers it seems that except of some simpler cases there is no general result concerning the convergence of the constructed Tau approximants. starting from the operational approach given in [1], we show most interesting cases, using Chebyshev polynomials the Tau method is in fact a method Petrov type, thus convergence results follow theorems given in [2]. To determine y R we take from G only the first n+m+1 rows and columns, i.e., we solve the linear system < 6 > ân G n = y n j^v is a s-called Tau approximant, because The solution it generates the polynomial defect (7) (Dy n )(x) = a n ny = (a^Jv,, (a n n n+n+h ) v n+m+h , where h=max(a^ii) is the so-called height of the differential operator D. He notice that in view of (4) and (6) the components of the defect vanish with respect to the first n+1 elements of the base v.…”

Matrix Based Operatorial Approach to Differential and Integral Problems

“…The proposed package Chebpack, which is described in this chapter, is based on the representation (3) of the unknown functions and uses the tau method for linear operators (such as differentiation, integration, product with the independent variable,...) and the pseudospectral method for nonlinear operators -nonlinear part of the equations. The tau method was invented by Lanczos (1938Lanczos ( , 1956 and later developed in an operatorial approach by Ortiz and Samara (Ortiz & Samara, 1981). In the past years considerable work has been done both in the theoretical analysis and numerical applications.…”

Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems

“…So it is required to obtain an efficient approximate solution. There are different methods and approaches for ap-proximate numerical solution such as difference and compact finite difference method [1]- [3], Tau method [4], an extrapolation method [5], Taylor series method [6], method of regularization [7] [8], variational method [9], adomian decomposition method [10], variational iterations method [11] and references therein.…”