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Upper and lower error estimation for the Tau method and related polynomial techniques. (English) Zbl 0745.65048

A differential equation of the form \(A(x)y'+B(x)y+C(x)=0\) is considered whose solution \(y(x)\) is approximated by a polynomial \(y_ n(x)\) satisfying the perturbed equation \(A(x)y_ n'+B(x)y_ n+C(x)=\tau T^*_ n(x)\). The shifted Chebyshev polynomials \(T^*_ n(x)\) are used as the basis of expansions in the space \(\mathbb{C} [0,1]\) of continuous functions endowed with the uniform norm \(\| f\| = \max| f(x)|\). The approximation error is \(e_ n=y-y_ n\).
A recent technique by R. B. Adeniyi and P. Onumany [ibid. 21, No. 9, 19-27 (1991; reviewed above)] for an error estimation of the Tau method is discussed along with new approach for obtaining upper and lower bounds for the maximum Tau method error. In the latter the maximum deviation \(\| y_ n-y_{n+1}\|=\| e_ n-e_{n+1}\|\) is introduced between two successive approximate expansions as a means of estimating the error when the convergence is relatively fast.
This approach improves on the estimates determined by the technique mentioned above and that of the Lanczos method. An example of a nonlinear ordinary differential equation is given where this technique is applied and shown to be implemented for Chebyshev series expansions, collocation, and spectral methods.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0745.65046
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References:

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