Numerical Solution of a Class of Integral Equations Arising in Two-Dimensional Aerodynamics

  • J. A. Fromme
  • M. A. Golberg
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 18)


We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This enables us to study the problem in terms of an equivalent integral equation of the second kind. Using Galerkin’s method, we are able to derive a convergent numerical algorithm for its solution. It is shown that this algorithm is numerically equivalent to Bland’s collocation method, which is then used as our method of computation. Extensive numerical calculations are presented establishing the validity of the theory.


Integral Equation Wind Tunnel Collocation Method Singular Integral Equation Lift Coefficient 
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  1. 1.
    Bland, S. R., The Two-Dimensional Oscillating Airfoil in a Wind Tunnel in Subsonic Compressible Flow, University of North Carolina, PhD Thesis, 1968.Google Scholar
  2. 2.
    Bland, S. R., The Two-Dimensional Oscillating Airfoil in a Wind Tunnel in Subsonic Flow, SIAM Journal on Applied Mathematics, Vol. 18, pp. 830–848, 1970.CrossRefGoogle Scholar
  3. 3.
    Fromme, J., and Golberg, M., Unsteady Two-Dimensional Airloads Acting on Thin Oscillating Airfoils in Subsonic Ventilated Wind Tunnels, NASA Contractor Draft Report, NSG-2140, 1977.Google Scholar
  4. 4.
    Golberg, M. A., Chapter 1, this volume; ELLIOT, D., Chapter 3, this volume.Google Scholar
  5. 5.
    Atkinson, K., A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1976.Google Scholar
  6. 6.
    Nashed, M. Z., Aspects of Generalized Inverses in Analysis and Regularization, Generalized Inverses and Applications, Edited by M. Z. Nashed, Academic Press, New York, New York, 1976.Google Scholar
  7. 7.
    Söhngen, H., Die Lösungen der Integralgleichung g(x)= (1 /2 rr) S% (f(6)/x -)(k und Deren Anflwendung in der Tra ügeltheorie, Matematische Zeitschrift, Vol. 45, pp. 245–264, 1939.CrossRefGoogle Scholar
  8. 8.
    Söhngen, H., Zur Theorie der Endlichen Hilbert-Transformation, MalÉematische Zeitschrift, Vol. 60, pp. 31–51, 1954.CrossRefGoogle Scholar
  9. 9.
    Mushkelishvili, N. I., Singular Integral Equations, Wolters-Noordhoff Publishing Company, Groningen, Holland, 1953.Google Scholar
  10. 10.
    Tricomi, F. G., On the Finite Hilbert Transformation, Quarterly of Applied Mathematics, Vol. 2, pp. 199–211, 1951.Google Scholar
  11. 11.
    Tricomi, F. G., Integral Equations, John Wiley and Sons, (Interscience Publishers), New York, New York, 1957.Google Scholar
  12. 12.
    Milne, R. D., Application of Integral Equations to Fluid Flows in Unbounded Regions, Finite Elements in Fluids, Vol. 2, Edited by R. H. Gallagher, J. T. Oden, C. Taylor, and O. C. Zienkiewicz, John Wiley and Sons, New York, New York, 1975.Google Scholar
  13. 13.
    Dunford, N., and Schwarz, J., Linear Operators-Part I, John Wiley and Sons (Interscience Publishers), New York, New York, 1967.Google Scholar
  14. 14.
    Agmon, S., Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, New York, New York, 1965.Google Scholar
  15. 15.
    Noble, B., Some Applications of the Numerical Solution of Integral Equations to Boundary Value Problems, Proceedings of the Conference on the Applications of Numerical Analysis, Edited by J. Morris, Springer-Verlag, New York, New York, 1971.Google Scholar
  16. 16.
    Hsu, P. T., Some Recent Developments in the Flutter Analysis of Low Aspect Ratio Wings, Proceedings of the National Specialists Meetings on Dynamics and Aeroelasticity, Fort Worth, Texas, 1958.Google Scholar
  17. 17.
    Küssner, H. G., Das Zweidimensionale Problem der Beliebig Bewegten Tragfläche unter Bereucksichtigung von Partialbewegugung“’der Flussigkeit, Luftfahrtforschung, Vol. 17, pp. 355–361, 1940.Google Scholar
  18. 18.
    Schwarz, L., Berußhung der Druckverteilung einer Harmonisch sich Verformerden Tragfläche in Ebener Strömung, Luftfahrtforschung, Vol. 17, pp. 379–386, 1940.Google Scholar
  19. 19.
    Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions, United States Government Printing Office, Washington, DC, 1964.Google Scholar
  20. 20.
    Sloan, I., Error Analysis for a Class of Degenerate Kernel Methods, Numerische Mathematik, Vol. 25, pp. 231–238, 1976.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • J. A. Fromme
    • 1
  • M. A. Golberg
    • 2
  1. 1.Department of MathematicsUniversity of Nevada at Las VegasLas VegasUSA
  2. 2.University of Nevada at Las VegasLas VegasUSA

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