Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Approximation methods for the Muskhelishvili equation on smooth curves


Authors: V. Didenko and E. Venturino
Journal: Math. Comp. 76 (2007), 1317-1339
MSC (2000): Primary 65R20
DOI: https://doi.org/10.1090/S0025-5718-07-01971-0
Published electronically: February 23, 2007
MathSciNet review: 2299776
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the possibility of applying approximation methods to the famous Muskhelishvili equation on a simple closed smooth curve $ \Gamma $. Since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way. It is shown that the spline Galerkin, spline collocation and spline qualocation methods for the corrected equation are stable, and the corresponding approximate solutions converge to an exact solution of the Muskhelishvili equation in appropriate norms. Numerical experiments confirm the effectiveness of the proposed methods.


References [Enhancements On Off] (What's this?)

  • 1. Raymond H. Chan, Thomas K. Delillo, and Mark A. Horn, The numerical solution of the biharmonic equation by conformal mapping, SIAM J. Sci. Comput. 18 (1997), no. 6, 1571–1582. MR 1480625, https://doi.org/10.1137/S1064827595292710
  • 2. Raymond H. Chan, Thomas K. DeLillo, and Mark A. Horn, Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation, SIAM J. Sci. Comput. 19 (1998), no. 1, 139–147. Special issue on iterative methods (Copper Mountain, CO, 1996). MR 1616882, https://doi.org/10.1137/S1064827596303570
  • 3. J. M. Chuang and S. Z. Hu, Numerical computation of Muskhelishvili’s integral equation in plane elasticity, Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), 1996, pp. 123–138. MR 1393724, https://doi.org/10.1016/0377-0427(95)00190-5
  • 4. M. Costabel and J. Saranen, Boundary element analysis of a direct method for the biharmonic Dirichlet problem, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988) Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 77–95. MR 1038333
  • 5. M. Costabel, I. Lusikka, and J. Saranen, Comparison of three boundary element approaches for the solution of the clamped plate problem, Boundary elements IX, Vol. 2 (Stuttgart, 1987) Comput. Mech., Southampton, 1987, pp. 19–34. MR 965334
  • 6. Martin Costabel and Monique Dauge, Invertibility of the biharmonic single layer potential operator, Integral Equations Operator Theory 24 (1996), no. 1, 46–67. MR 1366540, https://doi.org/10.1007/BF01195484
  • 7. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
  • 8. V. D. Didenko and B. Silbermann, On stability of approximation methods for the Muskhelishvili equation, J. Comput. Appl. Math. 146 (2002), no. 2, 419–441. MR 1925971, https://doi.org/10.1016/S0377-0427(02)00393-X
  • 9. Victor D. Didenko and Bernd Silbermann, Spline approximation methods for the biharmonic Dirichlet problem on non-smooth domains, Toeplitz matrices and singular integral equations (Pobershau, 2001) Oper. Theory Adv. Appl., vol. 135, Birkhäuser, Basel, 2002, pp. 145–160. MR 1935762
  • 10. V. D. Didenko and G. L. Pel′ts, On the stability of the spline-qualocation method for singular integral equations with conjugation, Differentsial′nye Uravneniya 29 (1993), no. 9, 1593–1601, 1654 (Russian, with Russian summary); English transl., Differential Equations 29 (1993), no. 9, 1383–1391 (1994). MR 1278829
  • 11. R. Duduchava, On general singular integral operators of the plane theory of elasticity, Rend. Sem. Mat. Univ. Politec. Torino 42 (1984), no. 3, 15–41. MR 834780
  • 12. R. V. Duduchava, General singular integral equations and fundamental problems of the plane theory of elasticity, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 82 (1986), 45–89 (Russian). MR 884698
  • 13. I. Gohberg, N. Feldman, Convolution Equations and Projection Methods for their Solutions, Akademie Verlag, Berlin, 1974.
  • 14. I. Gohberg, N. Krupnik, Introduction to the theory of one-dimensional singular integral operators, Birkhäuser, Basel, Boston, Berlin, 1995.
  • 15. Rainer Kress, Linear integral equations, 2nd ed., Applied Mathematical Sciences, vol. 82, Springer-Verlag, New York, 1999. MR 1723850
  • 16. Jian-ke Lu, Complex variable methods in plane elasticity, Series in Pure Mathematics, vol. 22, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370444
  • 17. N.I. Muskhelishvili, Fundamental Problems in Theory of Elasticity, Nauka, Moscow, 1966 (in Russian).
  • 18. N. I. Muskhelishvili, \cyr Singulyarnye integral′nye uravneniya, Third, corrected and augmented edition, Izdat. “Nauka”, Moscow, 1968 (Russian). \cyr Granichnye zadachi teorii funktsiĭ i nekotorye ikh prilozheniya k matematicheskoĭ fizike. [Boundary value problems in the theory of function and some applications of them to mathematical physics]; With an appendix by B. Bojarski. MR 0355495
  • 19. V. Z. Parton and P. I. Perlin, \cyr Integral′nye uravneniya teorii uprugosti, “Nauka”, Moscow, 1977 (Russian). With an introduction by D. I. Šerman and appendices by S. F. Stupak and N. F. Andrianov. MR 509209
  • 20. P. I. Perlin and Ju. N. Šaljuhin, On the numerical solution of the integral equations of plane elasticity theory, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. 1 (1976), 86–88, 94 (Russian). MR 0455783
  • 21. P.I. Perlin and Yu.N. Shalyukhin, On the numerical solution of some plane problems in elasticity theory, Prikl. Mech. 15 (1977) p. 83-86 (in Russian).
  • 22. Siegfried Prössdorf and Bernd Silbermann, Numerical analysis for integral and related operator equations, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 84, Akademie-Verlag, Berlin, 1991 (English, with English and German summaries). MR 1206476
    Siegfried Prössdorf and Bernd Silbermann, Numerical analysis for integral and related operator equations, Operator Theory: Advances and Applications, vol. 52, Birkhäuser Verlag, Basel, 1991. MR 1193030
  • 23. Larry L. Schumaker, Spline functions: basic theory, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1993. Correlated reprint of the 1981 original. MR 1226234
  • 24. Ian H. Sloan, A quadrature-based approach to improving the collocation method, Numer. Math. 54 (1988), no. 1, 41–56. MR 960849, https://doi.org/10.1007/BF01403890

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65R20

Retrieve articles in all journals with MSC (2000): 65R20


Additional Information

V. Didenko
Affiliation: Mathematics Department, University of Brunei Darussalam, Tungku BE 1410, Brunei

E. Venturino
Affiliation: Dipartimento di Matematica, Universitá di Torino, via Carlo Alberto 10, 10123 Torino, Italy

DOI: https://doi.org/10.1090/S0025-5718-07-01971-0
Received by editor(s): January 26, 2006
Received by editor(s) in revised form: June 20, 2006
Published electronically: February 23, 2007
Additional Notes: The first author thanks INDAM for the support provided to him during his June 2002 visit to the University of Torino, where most of this research was carried out. He was also partially supported by UBD via Grant UBD/PNC2/2/RG/1(49)
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.