Some considerations on numerical methods for Cauchy singular integral equations on the real line
Abstract
Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation.
Keyword : Hilbert transform, singular integral equation, Hermite weight
How to Cite
Capobianco, M. R., & Criscuolo, G. (2024). Some considerations on numerical methods for Cauchy singular integral equations on the real line. Mathematical Modelling and Analysis, 29(2), 268–276. https://doi.org/10.3846/mma.2024.18688
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.R. Capobianco, G. Criscuolo and R. Giova. Approximation of the weighted Hilbert transform on the real line by an interpolatory process. BIT, Numerical Mathematics, 41(4):666–682, 2001. https://doi.org/10.1023/A:1021939801117
S.B. Damelin and K. Diethelm. Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line. J. Numerical Functional Analysis and Optimization, 22(1-2):13–54, 2001. https://doi.org/10.1081/NFA-100103786
M.C. De Bonis, B. Della Vecchia and G. Mastroianni. Approximation of the Hilbert transform on the real line using Hermite zeros. Mathematics of Computation,71(239):1169–1188,2002. https://doi.org/10.1090/S0025-5718-01-01338-2
Z. Ditzian and D.S. Lubinky. Jacson and smoothness theorems for Freud weights in Lp (0 < p ≤ ∞). Constr. Approx., 13(1):99–152, 1997. https://doi.org/10.1007/BF02678431
Z. Ditzian and V. Totik. Moduli of Smoothness, volume 9. SSCM, SpringerVerlag, 1987. https://doi.org/10.1007/978-1-4612-4778-4
I. Gohberg and N. Krupnik. One-dimensional linear singular integral equations, volume 54. Birkha¨user Verlag Basel, 1992. https://doi.org/10.1007/978-3-0348-8602-4
B.N. Mandal and A. Chakrabarti. Applied Singular Integral Equations. CRC Press, 2012.
S.G. Mikhlin and S. Prössdorf. Singular integral operators. Springer Berlin, Heidelberg, 1986.
N.I. Muskhelishvili. Singular integral equations. Springer Dordrecht, 1958.
S. Pössdorf and B. Silbermann. Numerical analysis for integral and related operator equations. Birkhauser Basel, 1991.
J. Szabados. Weighted Lagrange and Hermite-Fejer interpolation on the real line. J. of Inequal. and Appl., 1(2):99–123, 1997.
F.G. Tricomi. Integral Equations. Interscience Publishers, INC., New York, 1967.