SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Collocation for Singular BVPs in ODEs with Unsmooth Data

E.B. Weinmueller, I. Rachunkova and J. Vampolova

We deal with BVPs for systems of ODEs with singularities. Typically, such problems have the form \begin{eqnarray*} &&z'(t)=\frac{M(t)}{t}z(t)+f(t,z(t)), \enspace t\in (0,1], B_0z(0)+B_1z(1)=\beta,\end{eqnarray*} where $B_0$ and $B_1$ are constant matrices which are subject to certain restrictions for a well-posed problem. Here, we focus on the linear case where the function $f$ is unsmooth, $f(t)=g(t)/t$. We first deal with the analytical properties of the problem - existence and uniqueness of smooth solutions. To solve the problem numerically, we apply polynomial collocation and for the linear IVPs, we are able to provide the convergence analysis. It turns out that the collocation retains its high order even in case of singularities, provided that the analytical solution is sufficiently smooth. We illustrate the theory by numerical experiments; the related tests were carried out using the {Matlab} code sbvp [].

[1] W. Auzinger, G. Kneisl, O. Koch and E.B. Weinmüller, A Collocation Code for Boundary Value Problems in Ordinary Differential Equations, Numer. Algorithms, 33(2003), pp. 27-39.

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