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F. Auer, W. Auzinger, J. Burkotova, I. Rachunkova, E. Weinmüller:
"On nonlinear singular BVPs with nonsmooth data. Part 2: Convergence of collocation methods";
Applied Numerical Mathematics, 171 (2022), 149 - 175.

We discuss numerical solution of boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,
\[
x'(t) = \frac{M(t)}{t}x(t)+\frac{f(t,x(t))}{t}, \quad t \in (0,1], \quad b(x(0),x(1)) = 0,
\]
where \( M\colon [0,1] \to {\mathbb R}^{n \times n} \) and
\( f\colon [0,1] \times {\mathbb R}^n \to {\mathbb R}^n \) are continuous
matrix-valued and vector-valued functions, respectively. Moreover,
\( b\colon {\mathbb R}^n \times {\mathbb R}^n \to {\mathbb R}^n \) is a continuous nonlinear mapping
which is specified according to a spectrum of the matrix \( M(0)\) to guarantee the BVP to be well-posed.
For the case where \( M(0) \) has eigenvalues with nonzero real parts,
we prove new convergence results for the collocation method and analytical
results about the necessary smoothness of the solution to the problem required in the numerical analysis.

We illustrate the theory by means of numerical examples.

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BVPs, ODEs, time singularity, global existence and uniqueness, collocation, convergence

http://dx.doi.org/10.1016/j.apnum.2021.08.016