Global solution of the generalized Abel integral equation by implicit interpolation

Author:
H. Brunner

Journal:
Math. Comp. **28** (1974), 61-67

MSC:
Primary 65R05

MathSciNet review:
0331830

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Abstract: The construction of a (global) approximate solution for a given generalized Abel integral equation may be viewed as a problem of (implicit) interpolation in a prescribed linear space. In this paper, piecewise polynomials (extended spline functions) of a given degree and of class *C* are used to generate such an approximating function. Results on convergence and error bounds are given, and the practical application of this method is illustrated by a numerical example.

**[1]**M. Bôcher,*An Introduction to the Study of Integral Equations*, 2nd ed., Cambridge Univ. Press, London, 1914.**[2]**H. Brunner, "The numerical solution of the generalized Abel integral equation by piecewise polynomials,"*Notices Amer. Math. Soc.*, v. 19, 1972, p. A-662. (Abstract)**[3]**M. G. Cox,*Curve fitting with piecewise polynomials*, J. Inst. Math. Appl.**8**(1971), 36–52. MR**0287667****[4]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[5]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[6]**A. Huber,*Eine Näherungsmethode zur Auflösung Volterrascher Integralgleichungen*, Monatsh. Math. Phys.**47**(1939), no. 1, 240–246 (German). MR**1550815**, 10.1007/BF01695499**[7]**Richard Weiss,*Product integration for the generalized Abel equation*, Math. Comp.**26**(1972), 177–190. MR**0299001**, 10.1090/S0025-5718-1972-0299001-7**[8]**R. Weiss and R. S. Anderssen,*A product integration method for a class of singular first kind Volterra equations*, Numer. Math.**18**(1971/72), 442–456. MR**0312759**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0331830-5

Keywords:
Implicit interpolation,
generalized Abel integral equation,
continuous approximate solution,
piecewise polynomials

Article copyright:
© Copyright 1974
American Mathematical Society