an:05543917 Zbl 1171.33006 Guo, Ben-Yu; Shen, Jie; Wang, Li-Lian Generalized Jacobi polynomials/functions and their applications EN Appl. Numer. Math. 59, No. 5, 1011-1028 (2009). 00248401 2009
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33C45 35J05 42C05 65N22 Jacobi polynomials; spectral approximation; error estimate; high-order differential equations For arbitrary real numbers $$\alpha,\,\beta$$ the authors define the generalized Jacobi polynomials/functions $$j_n^{\alpha,\beta}$$ by $j_n^{\alpha,\beta}(x)=\omega^{\hat{\alpha},\hat{\beta}}(x)\,J_{n_1}^{\tilde{\alpha},\tilde{\beta}}(x), \quad n\geq n_0^{\alpha,\beta},\;-1<x<1,$ where $\hat{\alpha}=\begin{cases} -\alpha,&\alpha\leq -1\\ 0,&\alpha>-1\end{cases}; \quad \tilde{\alpha}=\begin{cases}-\alpha,&\alpha\leq -1\\ \alpha &\alpha>-1\end{cases},\quad \omega^{a,b}(x)=(1-x)^a(1+x)^b$ and $n_0=n_0^{\alpha,\beta}:=[\hat{\alpha}]+[\hat{\beta}],\;n_1=n_1^{\alpha,\beta}:=n-n_0^{\alpha,\beta}.$ ($$[\cdot]$$ the greatest integer function) For $$\alpha,\,\beta>-1$$ and for $$\alpha,\,\beta$$ negative integers they are polynomials of degree $$n$$, coinciding---up to a multiplicative constant---with the definition in \textit{G. Szeg??} [Orthogonal Polynomials, AMS (1975; Zbl 0305.42011)]. Other values of the parameters lead to $j_n^{\alpha,\beta}(x)=\begin{cases}(1-x)^{-\alpha}\,(1+x)^{-\beta}\,J_{n_1}^{-\alpha,-\beta}(x), & \alpha,\beta\leq -1;\;n_1=n-[-\alpha]-[-\beta] \\ (1-x)^{-\alpha}\,J_{n_1}^{-\alpha,\beta}(x),&\alpha\leq -1,\beta > -1;\;n_1=n-[-\alpha]\\ (1+x)^{-\beta}\,J_{n_1}^{\alpha,-\beta}(x),&\alpha>-1,\beta\leq -1;\;n_1=n-[-\beta].\end{cases}$ The GJP/Fs satisfy orthogonality properties, a Sturm-Liouville equation, derivative recurrence relations, approximation properties on the underlying Sobolev space, etc. As an application spectral Galerkin methods for higher order differential equations are studied, including error estimates and, furthermore, some numerical results are given for the equation $u^{(6)}(x)-u(x)=f(x)$ on $$(-1,1)$$ with boundary conditions $$u(\pm 1),u'(\pm 1),u''(\pm 1)$$ and driving force $$f(x)$$ such that the exact solution is $u(x)=(1-x)e^x$ and $u(x)=(1+x)^pe^x.$ Marcel G. de Bruin (Haarlem) Zbl 0305.42011