Summary
This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dang, D., Norrie, D.: A finite element method for the solution of singular integral equations. Comput. Math. Appl.4, 219–224 (1978)
Elschner, J.: Galerkin methods with splines for singular integral equations over (0,1). Numer. Math.43, 265–281 (1984)
Gerasoulis, A.: The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type. Comput Math. Appl.8, 15–22 (1982)
Gerasoulis, A., Srivastav, R.P.: A method for the numerical solution of singular integral equations with a principal value integral. Int. J. Eng. Sci.19, 1293–1298 (1981)
Gohberg, I., Krupnik, N.: Einführung in die Theorie der eindimensionalen singulären Integraloperatoren. Basel: Birkhäuser 1979
Jen, E., Srivastav, R.: Cubic splines and approximate solution of singular integral equations. Math. Comput.37, 417–423 (1981)
Junghanns, P., Silbermann, B.: Zur Theorie der Näherungsverfahren für singuläre Integralgleichungen auf Intervallen. Math. Nachr.103, 199–244 (1981)
Junghanns, P., Silbermann, B.: Local theory of the collocation method for the approximate solution of singular integral equations. I. Integral Equations. Oper. Theory7, 791–807 (1984)
Mus'chelischwili, N.I.: Singuläre Integralgleichungen. Berlin: Akademie-Verlag 1965
Prößdorf, S.: Ein Lokalisierungsprinzip in der Theorie der Splineapproximationen und einige Anwendungen. Math. Nachr.119, 239–255 (1984)
Prößdorf, S., Rathsfeld, A.: A spline colocation method for singular integral equations with piecewise continuous coefficients. Integral Equ. a. Operator Th.7, 536–560 (1984)
Prößdorf, S., Schmidt, G: A finite element collocation method for singular integral equations. Math. Nachr.100, 33–60 (1981)
Schmidt, G.: On spline collocation methods for boundary integral equations in the plane. Math. Methods in Appl. Sci.7, 74–89 (1985)
Schmidt, G.: On ε-collocation for pseudodifferential equations on a closed curve. Math. Nachr., to appear
Washizu, K., Ikegawa, M.: Finite element technique in lifting surface problems. In: International symposium on finite element methods in flow problems, Univ. of Wales, Swansea 1974
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schmidt, G. Spline collocation for singular integro-differential equations over (0.1). Numer. Math. 50, 337–352 (1986). https://doi.org/10.1007/BF01390710
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01390710