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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Product integration for the generalized Abel equation


Author: Richard Weiss
Journal: Math. Comp. 26 (1972), 177-190
MSC: Primary 65P05
MathSciNet review: 0299001
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Abstract: The solution of the generalized Abel integral equation

$\displaystyle g(t) = \int_0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 < \alpha < 1,$

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.

References [Enhancements On Off] (What's this?)

  • [1] 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 0261814 (41 #6426)
  • [2] 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 0012392 (7,19h)
  • [3] 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 0292161 (45 #1248)
  • [4] 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 0063779 (16,179b)
  • [5] 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.
  • [6] B. Noble, Lecture Notes, 1970.
  • [7] 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 (47 #1314)
  • [8] G. Kowalewski, Integralgleichungen, de Gruyter, Berlin und Leipzig, 1930.
  • [9] L. V. Kantorovich and V. I. Krylov, Approximate methods of higher analysis, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen, 1958. MR 0106537 (21 #5268)
  • [10] P. Linz, The numerical solution of Volterra integral equations by finite difference methods, MRC Technical Summary Report #825, University of Wisconsin, Madison, Wis., 1967.
  • [11] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685 (16,468c)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1972-0299001-7
PII: S 0025-5718(1972)0299001-7
Keywords: Generalized Abel equation, numerical solution, product integration methods
Article copyright: © Copyright 1972 American Mathematical Society