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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

##### MSC:
 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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