Abstract
We discuss solvability properties of a nonlinear hypersingular integral equation of Prandtl’s type using monotonicity arguments together with different collocation iteration schemes for the numerical solution of such equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Berthold, W. Hoppe, B. Silbermann, A fast algorithm for solving the generalized airfoil equation, J. Comp. Appl. Math., 43 (1992), 185-219.
M.R. Capobianco, G. Criscuolo, P. Junghanns, A fast algorithm for Prandtl’s integrodi-differential equation, J. Comp. Appl. Math., 77 (1997), 103–128.
M.R. Capobianco, G. Criscuolo, P. Junghanns, U. Luther, Uniform convergence of the collocation method for Prandtl’s integro-differential equation, ANZIAM J., 42 (2000), 151–168.
M.R. Capobianco, G. Mastroianni, Uniform boundedness of Lagrange operator in some weighted Sobolev-type space, Math. Nachr., 187 (1997), 1–17.
I. Gohberg, N. Krupnik, One-Dimensional Linear Singular Integral Equations, Volume I, Birkhauser Verlag, 1992.
N.I. Ioakimidis, Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mech., 45 (1982), 31–47.
A.C. Kaya, F. Erdogan, On the solution of integral equations with strong singularities, Quart. Appl. Math., 45 (1987), 105–122.
G. Mastroianni, M.G. Russo, Lagrange interpolation in weighted Besov spaces, Constr. Approx., 152 (1999), 257–289.
G. Mastroianni, M.G. Russo, Weighted Marcinkiewicz inequalities and boundedness of the Lagrange operator, Math. Anal. Appl., (2000), 149–182.
S. Nemat-Nasser, M. Hori, Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites, Mech. Mat., 6 (1987), 245–269.
S. Nemat-Nasser, M. Hori, Asymptotic solution of a class of strongly singular integral equations, SIAM J. Appl. Math., 503 (1990), 716–725.
P. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc. 213, Providence, RI, 1979.
D. Oestreich, Approximated solution of a nonlinear singular equation of Prandtl’s type, Math. Nachr., 161 (1993), 95–105.
M.A. Sheshko, G.A. Rasol’ko, V.S. Mastyanitsa, On approximate solution of Prandtl’s integro-differential equation, Differential Equations, 29 (1993), 1345–1354.
G. Steidl, Fast radix-p discrete cosine transform, AAECC, 3 (1992), 39–46.
M. Tasche, Fast algorithms for discrete Chebyshev-Vandermonde transforms and applications, Numer. Algor., 5 (1993), 453–464.
L. von Wolfersdorf, Monotonicity methods for nonlinear singular integral and integro-differential equations, ZAMM, 63 (1983), 249–259.
E. Zeidler, Nonlinear Functional Analysis and its Applications, Part II, Springer Verlag, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Professor Israel Gohberg on the Occasion of his 75th Birthday
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Capobianco, M., Criscuolo, G., Junghanns, P. (2005). On the Numerical Solution of a Nonlinear Integral Equation of Prandtl’s Type. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_4
Download citation
DOI: https://doi.org/10.1007/3-7643-7398-9_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7290-3
Online ISBN: 978-3-7643-7398-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)