Approximate solution of the differential equation with spline functions

Author:
Gh. Micula

Journal:
Math. Comp. **27** (1973), 807-816

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1973-0331789-X

Corrigendum:
Math. Comp. **29** (1975), 673-674.

Corrigendum:
Math. Comp. **29** (1975), 673-674.

MathSciNet review:
0331789

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Abstract | References | Similar Articles | Additional Information

Abstract: An approximate spline is constructed for the solution of Cauchy's problem regarding a second-order differential equation. The existence, uniqueness and convergence of the approximate spline solution are investigated.

**[1]**H. B. Curry & I. J. Schoenberg, "On Pólya frequency functions. IV. The fundamental spline functions and their limits,"*J. Analyse Math.*, v. 17, 1966, pp. 71-107. MR**36**#1884. MR**0218800 (36:1884)****[2]**P. Henrici,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley New York, 1962. MR**24**#B1772. MR**0135729 (24:B1772)****[3]**F. R. Loscalzo,*On the Use of Spline Functions for the Numerical Solution of Ordinary Differential Equations*, Doctoral Thesis, University of Wisconsin, Madison, Wis., 1968.**[4]**F. R. Loscalzo & T. D. Talbot, "Spline function approximations for solutions of ordinary differential equations,"*SIAM J. Numer. Anal.*, v. 4, 1967, pp. 433-445. MR**36**#4808. MR**0221756 (36:4808)****[5]**G. Micula, "Fonctions spline d'approximation pour les solutions des systèmes d'équations differentielles,"*An. Şti. Univ. "Al. I. Cuza" Iaşi*, v. 17, 1971, pp. 139-155. MR**0309315 (46:8425)****[6]**M. Sakai, "Spline interpolation and two point boundary value problems,"*Mem. Fac. Sci. Kyushu Univ. Ser. A*, v. 24, 1970, pp. 17-34. MR**42**#8702. MR**0273826 (42:8702)****[7]**E. Schechter, "Error bounds in the numerical integration of differential equations,"*Studio Univ. Babeş-Bolyai Ser. Math. Mech.*, v. 15, 1970, no. 1, pp. 43-53. MR**42**#1343. MR**0266437 (42:1343)****[8]**I. J. Schoenberg,*On Spline Functions*, Proc. Sympos. Inequalities (Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, New York, 1967, pp. 255-291. MR**36**#6848. MR**0223801 (36:6848)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0331789-X

Keywords:
Differential equation,
Cauchy problem,
spline function,
consistency relations,
fixed point,
discrete multistep method,
stable method,
convergence

Article copyright:
© Copyright 1973
American Mathematical Society