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An efficient spectral method for ordinary differential equations with rational function coefficients. (English) Zbl 0846.65037
The authors show how to exploit the properties of the operator of integration for arbitrary families of classical orthogonal polynomials to arrive at efficient spectral algorithms for the approximate solution of a large class of ordinary differential equations of the form \[ Lu= \sum^n_{k= 0} (m_{n- k}(x) D^k) u= f(x),\quad x\in (a, b) \] subject to the constraints \(Tu= c\), where \(m_k\) are rational functions of \(x\), \(D^k\) denotes \(k\)th order differentiation with respect to \(x\), \(T\) is a linear functional of rank \(n\), and \(c\in \mathbb{R}^n\). Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65Z05 Applications to the sciences
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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