On solving weakly singular Volterra equations of the first kind with Galerkin approximations
HTML articles powered by AMS MathViewer
- by John M. Bownds PDF
- Math. Comp. 30 (1976), 747-757 Request permission
Corrigendum: Math. Comp. 31 (1977), 808.
Corrigendum: Math. Comp. 31 (1977), 808.
Abstract:
The basic linear, Volterra integral equation of the first kind with a weakly singular kernel is solved via a Galerkin approximation. It is shown that the approximate solution is a sum with the first term being the solution of Abel’s equation and the remaining terms computable as components of the solution of an initial-value problem. The method represents a significant decrease in the normal number of computations required to solve the integral equation.References
- Richard Weiss, Product integration for the generalized Abel equation, Math. Comp. 26 (1972), 177–190. MR 299001, DOI 10.1090/S0025-5718-1972-0299001-7 G. KOWALEWSKI, Integralgleichungen, de Gruyter, Berlin, 1930.
- Henry E. Fettis, On the numerical solution of equations of the Abel type, Math. Comp. 18 (1964), 491–496. MR 172477, DOI 10.1090/S0025-5718-1964-0172477-7
- 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 312759, DOI 10.1007/BF01406681
- R. S. Anderssen, F. R. de Hoog, and R. Weiss, On the numerical solution of Brownian motion processes, J. Appl. Probability 10 (1973), 409–418. MR 350882, DOI 10.2307/3212357
- John M. Bownds and Bruce Wood, On numerically solving nonlinear Volterra integral equations with fewer computations, SIAM J. Numer. Anal. 13 (1976), no. 5, 705–719. MR 433928, DOI 10.1137/0713058 J. BOWNDS & B. WOOD, "A faster numerical method for solving Volterra integral equations with convolution kernels." (Submitted.)
- J. M. Bownds and J. M. Cushing, A representation formula for linear Volterra integral equations, Bull. Amer. Math. Soc. 79 (1973), 532–536. MR 313731, DOI 10.1090/S0002-9904-1973-13189-1
- Michael A. Golberg, The conversion of Fredholm integral equations to equivalent Cauchy problems, Appl. Math. Comput. 2 (1976), no. 1, 1–18. MR 398134, DOI 10.1016/0096-3003(76)90016-3
- 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
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665 G. SZEGÖ, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939. MR 1, 14. T. SATŌ, "Sur l’équation intégrale nonlinéaire de Volterra," Compositio Math., v. 11, 1953, pp. 271-290. MR 15, 714.
- Frank de Hoog and Richard Weiss, High order methods for Volterra integral equations of the first kind, SIAM J. Numer. Anal. 10 (1973), 647–664. MR 373354, DOI 10.1137/0710057 M. ABRAMOWITZ & I. A. STEGUN, Editors, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, vol. 55, Supt. of Documents, U. S. Govt. Printing Office, Washington, D. C., 1964. MR 29 #4914.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp. 30 (1976), 747-757
- MSC: Primary 65R05; Secondary 45E10
- DOI: https://doi.org/10.1090/S0025-5718-1976-0438747-8
- MathSciNet review: 0438747