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A Riccati transformation method for solving linear BVPs. II: Computational aspects. (English) Zbl 0664.65075
The implementation aspects of the method described in part I [ibid. 25, No. 5, 1055-1073 (1988; reviewed above)] are discussed. It attempts to split the integration of the rapidly increasing (decreasing) solution components, typical for singularly perturbed two-point boundary value problems, into different initial value problems using the Riccati transformation \((1)\quad y_ 2(t)=R(t)y_ 1(t)+V(t).\)
The implementation of the method involves the following. An a priori upper bound or roof value, \(\rho\geq 1\), is assigned for \(\| R\| (:=\| R\|_{\infty})\). The initial value problems for \(R\) and \(V\) are integrated, and these solution values are monitored as they are returned by the integrator. At each such integration step, one checks if \(\| R\| >\rho\). If this is the case, say at \(t_ h\), then a reimbedding is performed, namely one does a decomposition with partial pivoting, such that \(\left( \begin{matrix} I_ q\\ R(t_ h)\end{matrix} \right)\to P^ T\left( \begin{matrix} L_{11}\\ L_{21}\end{matrix} \right)^ U_{-1},\) and set \(R(t_ h)=L_{21}L_{11}\) as the Riccati initial conditions for the new embedding corresponding to the permutation \(P\).
Upon reaching \(t=1\), the backward sweep for \(y_ 1\) is performed, representing \(R\) and \(V\) via cubic Hermite interpolants at certain previously computed values. The final solution \(y\) is then given using (1). The efficacy of the method is demonstrated on a number of examples.
Reviewer: L.M.Berkovich

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
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