Cubic splines and approximate solution of singular integral equations
Authors:
Erica Jen and R. P. Srivastav
Journal:
Math. Comp. 37 (1981), 417423
MSC:
Primary 65R20; Secondary 41A15, 45E05
MathSciNet review:
628705
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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.
N. Nilson, and J.
L. Walsh, The theory of splines and their applications,
Academic Press, New York, 1967. MR 0239327
(39 #684)
 [2]
F.
Erdogan and G.
D. Gupta, On the numerical solution of singular integral
equations, Quart. Appl. Math. 29 (1971/72),
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. DCSTR86 Rutgers Univ., New Brunswick, 1979.
 [4]
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
(83e:65047), http://dx.doi.org/10.1016/00207225(81)901488
 [5]
I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
 [6]
Charles
A. Hall and W.
Weston Meyer, Optimal error bounds for cubic spline
interpolation, J. Approximation Theory 16 (1976),
no. 2, 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. 10791089.
 [8]
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 0445873
(56 #4206)
 [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. 525534. 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. DCSTR86 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. 105122. 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. 10791089.
 [8]
 P. S. Theocaris & N. I. Ioakimidis, "Numerical integration methods for the solution of singular integral equations," Quart Appl. Math., v. 35, 1977, pp. 173183. MR 0445873 (56:4206)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106287054
PII:
S 00255718(1981)06287054
Keywords:
Spline approximation,
numerical solution of singular integral equations
Article copyright:
© Copyright 1981 American Mathematical Society
