On solving weakly singular Volterra equations of the first kind with Galerkin approximations
Author:
John M. Bownds
Journal:
Math. Comp. 30 (1976), 747757
MSC:
Primary 65R05; Secondary 45E10
Corrigendum:
Math. Comp. 31 (1977), 808.
Corrigendum:
Math. Comp. 31 (1977), 808.
MathSciNet review:
0438747
Fulltext PDF Free Access
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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 initialvalue problem. The method represents a significant decrease in the normal number of computations required to solve the integral equation.
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 R. WEISS, "Product integration for the generalized Abel equation," Math. Comp., v. 26, 1972, pp. 177190. MR 45 #8050. MR 0299001 (45:8050)
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 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. 491496. 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. 442456. MR 47 #1314. MR 0312759 (47:1314)
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 R. S. ANDERSSEN, F. R. de HOOG & R. WEISS, "On the numerical solution of Brownian motion processes," J. Appl. Probability, v. 10, 1973, pp. 409418. MR 50 #3374. MR 0350882 (50:3374)
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 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. 532536. MR 47 #2285. MR 0313731 (47:2285)
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 M. GOLDBERG, "The conversion of Fredholm integral equations to equivalent Cauchy problems," Appl. Math. Comput. (To appear.) MR 0398134 (53:1989)
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 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)
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 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. 271290. 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. 647658. 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197604387478
PII:
S 00255718(1976)04387478
Article copyright:
© Copyright 1976
American Mathematical Society
