Abstract:Of concern here is the numerical solution of singular integral equations of Cauchy type; i.e., equations involving principal value integrals. The unknown function is expressed as the product of an appropriate weight function and a cubic spline. The problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots. An estimate is provided for the maximum error of the approximate solution. Numerical results from the spline method are compared with those obtained using other methods.
- J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The theory of splines and their applications, Academic Press, New York-London, 1967. MR 0239327
- F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quart. Appl. Math. 29 (1971/72), 525–534. MR 408277, DOI 10.1090/S0033-569X-1972-0408277-4 A. Gerasoulis, Product Integration Methods for the Solution of Singular Integral Equations of Cauchy type, Dept. of Comput. Sci. Report No. DCS-TR-86 Rutgers Univ., New Brunswick, 1979.
- A. Gerasoulis and R. P. Srivastav, A method for the numerical solution of singular integral equations with a principal value integral, Internat. J. Engrg. Sci. 19 (1981), no. 9, 1293–1298. MR 660561, DOI 10.1016/0020-7225(81)90148-8 I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
- Charles A. Hall and W. Weston Meyer, Optimal error bounds for cubic spline interpolation, J. Approximation Theory 16 (1976), no. 2, 105–122. MR 397247, DOI 10.1016/0021-9045(76)90040-x D. P. Rooke & I. N. Sneddon, "The crack energy and stress intensity factor for a cruciform crack deformed by internal pressure," Internat. J. Engrg. Sci., v. 7, 1969, pp. 1079-1089.
- P. S. Theocaris and N. I. Ioakimidis, Numerical integration methods for the solution of singular integral equations, Quart. Appl. Math. 35 (1977/78), no. 1, 173–187. MR 445873, DOI 10.1090/S0033-569X-1977-0445873-X
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 37 (1981), 417-423
- MSC: Primary 65R20; Secondary 41A15, 45E05
- DOI: https://doi.org/10.1090/S0025-5718-1981-0628705-4
- MathSciNet review: 628705