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A unified approach to the tau method and Chebyshev series expansion techniques. (English) Zbl 0777.65051
The authors consider problems with differential equations of the following kind: (1) $$Dy(x):=\sum^ \nu_{r=0} p_ r(x)y^{(r)}(x)=f(x)$$, $$x\in[-1,1]$$ and (2) $$B_ i(y)=\gamma_ i$$, $$\gamma_ i=\text{constant}$$, $$i=1,\dots,\nu$$, where $$f$$ is a polynomial in $$x$$ of some given degree, the $$p_ r$$ are polynomial coefficients and $$B_ i$$, $$i=1,\dots,\nu$$, are linear functionals defined on $$C^ \nu[- 1,1]$$: $$B_ i(y)=\sum^{\nu-1}_{k=0}\sum^ \mu_{j=1} \beta^{(k)}_{ij} y^{(k)}(a_ j)$$, $$i=1,\dots,\nu$$, where $$a_ 1,\dots,a_ \mu$$ are fixed real numbers in $$[-1,1]$$ and $$\beta^{(k)}_{ij}$$ are some given constants.
The essential idea of the Lanczos tau method is to perturb problem (1), (2) in such a way that its exact solution becomes a polynomial, i.e. solving exactly the perturbed problem (3) $$Dy_ n(x)=f(x)+H_ n(x)$$, $$x\in[-1,1]$$, $$n\in\mathbb{N}$$, $$B_ i(y_ n)=\gamma_ i$$, $$i=1,\dots,\nu$$, where $$H_ n$$ is a polynomial perturbation term which reduces the exact solution of (3) to a polynomial $$y_ n$$ which will be called the $$n$$th tau approximation of $$y$$.
Two main choices of $$H_ n(x)$$ are considered in this paper: $$H_ n(V):=\sum^{r-1}_{i=1} \tau_ i V_{n+1}(x)$$ and $$H_ n(x,V):=\bigl(\sum^{r-1}_{i=0} \tau_ i x^ i\bigr) V_ n(x)$$, where $$V_ k$$ are polynomials of degree $$k$$ in $$x$$.
The authors show the equivalence between the above recursive formulation of the tau method and an alternative technique called the operational approach because it reduces differential problems to linear algebraic problems. The authors then use such methods as analytic tools in the simulation of two classical numerical techniques based on Chebyshev series expansions. The results make possible the recursive formulation of a variety of series expansion methods.