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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Product integration for the generalized Abel equation
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by Richard Weiss PDF
Math. Comp. 26 (1972), 177-190 Request permission


The solution of the generalized Abel integral equation \[ g(t) = \int _0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 < \alpha < 1,\] where $k(t,s)$ is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for $\alpha \in [{\alpha _1},1)$ with ${\alpha _1} \doteqdot 0.2117$. This interval contains the important case $\alpha = \tfrac {1}{2}$. Convergence of order two for $\alpha \in (0,{\alpha _1})$ is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.
  • Gerald N. Minerbo and Maurice E. Levy, Inversion of Abel’s integral equation by means of orthogonal polynomials, SIAM J. Numer. Anal. 6 (1969), 598–616. MR 261814, DOI 10.1137/0706055
  • Robert Fortet, Les fonctions aléatoires du type de Markoff associées à certaines équations linéaires aux dérivées partielles du type parabolique, J. Math. Pures Appl. (9) 22 (1943), 177–243 (French). MR 12392
  • J. Durbin, Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test, J. Appl. Probability 8 (1971), 431–453. MR 292161, DOI 10.2307/3212169
  • Andrew Young, The application of approximate product integration to the numerical solution of integral equations, Proc. Roy. Soc. London Ser. A 224 (1954), 561–573. MR 63779, DOI 10.1098/rspa.1954.0180
  • P. Linz, Applications of Abel Transforms to the Numerical Solution of Problems in Electrostatics and Elasticity, MRC Technical Summary Report #826, University of Wisconsin, Madison, Wis., 1967. B. Noble, Lecture Notes, 1970.
  • 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 312759, DOI 10.1007/BF01406681
  • G. Kowalewski, Integralgleichungen, de Gruyter, Berlin und Leipzig, 1930.
  • L. V. Kantorovich and V. I. Krylov, Approximate methods of higher analysis, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen 1958. Translated from the 3rd Russian edition by C. D. Benster. MR 0106537
  • P. Linz, The numerical solution of Volterra integral equations by finite difference methods, MRC Technical Summary Report #825, University of Wisconsin, Madison, Wis., 1967.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Math. Comp. 26 (1972), 177-190
  • MSC: Primary 65P05
  • DOI:
  • MathSciNet review: 0299001