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

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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]**R. WEISS, "Product integration for the generalized Abel equation,"*Math. Comp.*, v. 26, 1972, pp. 177-190. MR**45**#8050. MR**0299001 (45:8050)****[2]**G. KOWALEWSKI,*Integralgleichungen*, de Gruyter, Berlin, 1930.**[3]**H. FETTIS, "On the numerical solution of equations of Abel type,"*Math. Comp.*, v. 18, 1964, pp. 491-496. MR**30**#2696. MR**0172477 (30:2696)****[4]**R. WEISS & R. S. ANDERSSEN, "A product integration method for a class of singular first kind Volterra equations,"*Numer. Math.*, v. 18, 1971/72, pp. 442-456. MR**47**#1314. MR**0312759 (47:1314)****[5]**R. S. ANDERSSEN, F. R. de HOOG & R. WEISS, "On the numerical solution of Brownian motion processes,"*J. Appl. Probability*, v. 10, 1973, pp. 409-418. MR**50**#3374. MR**0350882 (50:3374)****[6]**J. BOWNDS & B. WOOD, "On numerically solving nonlinear Volterra integral equations with fewer computations,"*SIAM J. Numer. Anal.*, v. 13, 1976. MR**0433928 (55:6898)****[7]**J. BOWNDS & B. WOOD, "A faster numerical method for solving Volterra integral equations with convolution kernels." (Submitted.)**[8]**J. M. BOWNDS & J. M. CUSHING, "A representation formula for linear Volterra integral equations,"*Bull. Amer. Math. Soc.*, v. 79, 1973, pp. 532-536. MR**47**#2285. MR**0313731 (47:2285)****[9]**M. GOLDBERG, "The conversion of Fredholm integral equations to equivalent Cauchy problems,"*Appl. Math. Comput.*(To appear.) MR**0398134 (53:1989)****[10]**L. V. KANTOROVIČ & V. I. KRYLOV,*Approximate Methods of Higher Analysis*, 3rd ed., GITTL, Moscow, 1950; English transl., C. Benster, Interscience, New York; Noordhoff, Groningen, 1958. MR**13**, 77;**21**#5268. MR**0106537 (21:5268)****[11]**F. G. TRICOMI,*Integral Equations*, Pure and Appl. Math., vol. 5, Interscience, New York and London, 1957. MR**20**#1177. MR**0094665 (20:1177)****[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]**F. R. de HOOG & R. WEISS, "High order methods for Volterra integral equations of the first kind,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 647-658. MR**0373354 (51:9554)****[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.

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

Article copyright:
© Copyright 1976
American Mathematical Society