Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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.


References [Enhancements On Off] (What's this?)

  • [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.

Similar Articles

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

American Mathematical Society