Product integration for the generalized Abel equation
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- by Richard Weiss PDF
- Math. Comp. 26 (1972), 177-190 Request permission
Abstract:
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.References
- 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
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 177-190
- MSC: Primary 65P05
- DOI: https://doi.org/10.1090/S0025-5718-1972-0299001-7
- MathSciNet review: 0299001