On solving weakly singular Volterra equations of the first kind with Galerkin approximations

Author:
John M. Bownds

Journal:
Math. Comp. **30** (1976), 747-757

MSC:
Primary 65R05; Secondary 45E10

DOI:
https://doi.org/10.1090/S0025-5718-1976-0438747-8

Corrigendum:
Math. Comp. **31** (1977), 808.

Corrigendum:
Math. Comp. **31** (1977), 808.

MathSciNet review:
0438747

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Richard Weiss,*Product integration for the generalized Abel equation*, Math. Comp.**26**(1972), 177–190. MR**299001**, https://doi.org/10.1090/S0025-5718-1972-0299001-7**[2]**G. KOWALEWSKI,*Integralgleichungen*, de Gruyter, Berlin, 1930.**[3]**Henry E. Fettis,*On the numerical solution of equations of the Abel type*, Math. Comp.**18**(1964), 491–496. MR**172477**, https://doi.org/10.1090/S0025-5718-1964-0172477-7**[4]**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**, https://doi.org/10.1007/BF01406681**[5]**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**, https://doi.org/10.2307/3212357**[6]**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**, https://doi.org/10.1137/0713058**[7]**J. BOWNDS & B. WOOD, "A faster numerical method for solving Volterra integral equations with convolution kernels." (Submitted.)**[8]**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**, https://doi.org/10.1090/S0002-9904-1973-13189-1**[9]**Michael A. Golberg,*The conversion of Fredholm integral equations to equivalent Cauchy problems*, Appl. Math. Comput.**2**(1976), no. 1, 1–18. MR**398134**, https://doi.org/10.1016/0096-3003(76)90016-3**[10]**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****[11]**F. G. Tricomi,*Integral equations*, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR**0094665****[12]**G. SZEGÖ,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939. MR**1**, 14.**[13]**T. SATŌ, "Sur l'équation intégrale nonlinéaire de Volterra,"*Compositio Math.*, v. 11, 1953, pp. 271-290. MR**15**, 714.**[14]**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**, https://doi.org/10.1137/0710057**[15]**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.

Retrieve articles in *Mathematics of Computation*
with MSC:
65R05,
45E10

Retrieve articles in all journals with MSC: 65R05, 45E10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0438747-8

Article copyright:
© Copyright 1976
American Mathematical Society