Cubic splines and approximate solution of singular integral equations

Authors:
Erica Jen and R. P. Srivastav

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**J. H. Ahlberg, E. H. Nilson and J. L. Walsh,*The Theory of Splines and Their Applications*, Academic Press, New York, 1967. MR**0239327 (39:684)****[2]**F. Erdogan & G. D. Gupta, "On the numerical solution of singular integral equation,"*Quart. Appl. Math.*, v. 30, 1972, pp. 525-534. MR**0408277 (53:12042)****[3]**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.**[4]**A. Gerasoulis & R. P. Srivastav, "A method for the numerical solution of singular integral equations with a principal value integral,"*Internat. J. Engrg. Sci.*(To appear.) MR**660561 (83e:65047)****[5]**I. S. Gradshteyn & I. M. Ryzhik,*Table of Integrals, Series, and Products*, Academic Press, New York, 1965.**[6]**C. A. Hall & W. W. Meyer "Optimal error bounds for cubic spline interpolation,"*J. Approx. Theory*, v. 16, 1976, pp. 105-122. MR**0397247 (53:1106)****[7]**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.**[8]**P. S. Theocaris & N. I. Ioakimidis, "Numerical integration methods for the solution of singular integral equations,"*Quart Appl. Math.*, v. 35, 1977, pp. 173-183. MR**0445873 (56:4206)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
41A15,
45E05

Retrieve articles in all journals with MSC: 65R20, 41A15, 45E05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628705-4

Keywords:
Spline approximation,
numerical solution of singular integral equations

Article copyright:
© Copyright 1981
American Mathematical Society