Barycentric rational interpolation offers an elegant approach to avoid a common problem of rational interpolation, namely the occurrence of poles in the interpolation interval, which is undesirable in many situations. More recently, Cirillo and Hormann [1] introduced an iterative approach to the Hermite rational interpolation problem. The main theme of this talk is to introduced quadrature rules based on barycentric rational Hermite interpolation. To this end, a barycentric rational Hermite quadrature, and a composite version of that will be introduced. Then, the proposed composite quadrature formula will be utilized to construct a direct method for solving Volterra integral equations (VIEs) \[\label{eq:VIE} y(t) = g(t) + \int_{t_0}^t k(t,s,y(s))\,\mathrm{d}s, \qquad t\in I=[t_0,T],\] where \(g\colon I\to\mathbb{R}^D\) and \(k\colon S\times\mathbb{R}^D\to\mathbb{R}^D\) are given functions, \(D\) stands for the dimension of the system, and \(S=\{(t,s):t_0\leq s\leq t\leq T\}\). To show the efficiency of the proposed method in solving VIEs and to validate the theoretical results, some numerical verifications will be presented.

Keywords: Linear barycentric rational interpolation, Hermite interpolation, Quadrature, Volterra integral equations.

References

1. Cirillo, E., Hormann, K.: An iterative approach to barycentric rational Hermite interpolation. Numer. Math. 140, 939–962 (2018)

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