# zbMATH — the first resource for mathematics

Numerical treatment of singular initial value problems. (English) Zbl 0659.65071
A modification of predictor-corrector schemes developed by Y. L. Luke, W. Fair and J. Wimp [ibid. 1, 3-12 (1975; Zbl 0329.65049)] and the author [ibid. 8, 231-239 (1982; Zbl 0483.65047)] is proposed. Numerical experiments on the problem $$y'=1+y^ 2$$, $$0\leq x\leq 1$$; $$y(0)=1$$ give efficient results in the neighborhood of the singularity of the solution as $$x=\pi /4$$.
Reviewer: L.J.Grimm

##### MSC:
 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory
##### References:
 [1] Aitken, A. C.: On interpolation by iteration of proportional parts. Proc. Edinburgh math. Soc. 2, 56-76 (1932) · Zbl 0005.02001 [2] Bader, G.; Deufhard, P.: A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. math. 41, 373-398 (1983) · Zbl 0522.65050 [3] Bulirsch, R.; Stoer, J.: Numerical treatment of ODES by extrapolation methods. Numer. math. 8, 1-13 (1966) · Zbl 0135.37901 [4] Deuflhard, P.: Order and stepsize control in extrapolation methods. Numer. math. 41, 399-422 (1983) · Zbl 0543.65049 [5] Deuflhard, P.: Recent progress in extrapolation methods for odes. SIAM rev. 27, 505-536 (1985) · Zbl 0602.65047 [6] Fair, W.; Luke, Y. L.: Rational approximations to the solution of the second order Riccati equation. Math. comp. 20, 602-606 (1966) · Zbl 0196.50003 [7] Fatunla, S. O.: Nonlinear multistep methods for ivps. Comp. maths applic. 8, 231-239 (1982) · Zbl 0483.65047 [8] Gragg, W. B.: On extrapolation algorithms for ordinary initial value problems. SIAM J. Numer. anal. 2, 384-404 (1965) · Zbl 0135.37803 [9] Hull, T. E.; Enright, W. H.; Fellen, B. M.; Sedgwick, A. E.: Comparing numerical methods for ODES. SIAM J. Numer. anal. 9, 603-637 (1973) · Zbl 0221.65115 [10] Lambert, J. D.; Shaw, B.: On the numerical solution of y’ = $$f(x, y)$$ by a class of formulae based on rational approximation. Maths. comp. 19, 456-462 (1965) · Zbl 0131.14402 [11] Lambert, J. D.; Shaw, B.: A method for the numerical solution of y’ = $$f(x, y)$$ based on a self-adjusting non-polynomial interpolant. Maths comp. 20, 11-20 (1966) · Zbl 0133.38205 [12] Luke, Y. L.; Fair, W.; Wimp, J.: Predictor-corrector formulas based on rational interpolants. Comp. maths applic 1, 3-12 (1975) · Zbl 0329.65049 [13] Neville, E. H.: Iterative interpolation. J. ind. Math. soc. 20, 87-120 (1934) · Zbl 0010.03102 [14] Shaw, B.: Modified multistep methods based on a non-polynomial interpolant. J. assoc. Comput. Mach 14, 143-154 (1967) · Zbl 0178.18405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.