Advertisement

A Survey of Numerical Methods for Integral Equations

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

Abstract

A brief survey of the existing literature on numerical methods for integral equations is given. Emphasis is placed on equations in one unknown, although it is noted that many methods can be carried over to multidimensional equations as well. Some discussion is presented on the relation of numerical analysis to applications, and areas are delineated for future research.

Keywords

Integral Equation Projection Method Singular Integral Equation Quadrature Method Fredholm Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anselone, P. M., Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.Google Scholar
  2. 2.
    Kagiwada, H., and Kalaba, R. E., Integral Equations Via Imbedding Methods, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.Google Scholar
  3. 3.
    Atkinson, K. E., 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
  4. 4.
    Baker, C. T. H., The Numerical Treatment of Integral Equations, Cambridge University Press, Oxford, England, 1977.Google Scholar
  5. 5.
    Ivanov, V. V., The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations, Translated by A. Ideh, edited by R. S. Anderssen and D. Elliott, Noordhoff International Publishing Company, Leyden, Holland, 1976 (Russian edition published in 1968 ).Google Scholar
  6. 6.
    Delves, L. M., and Walsh, J., Numerical Solution of Integral Equations, Clarendon Press, Oxford, England, 1974.Google Scholar
  7. 7.
    Lock, R. C., Methods for Elliptic Problems in External Aerodynamics, Computational Methods and Problems in Aeronautical Fluid Dynamics, Edited by B. L. Hewitt, C. R. Illingworth, R. C. Lock, K. W. Mangler, J. H. McDowell, C. Richards, and I. Walkden, Academic Press, New York, New York, 1976.Google Scholar
  8. 8.
    Rowe, W., Redman, M., Ehlers, F., and Sebastian, J., Prediction of Unsteady Loadings Caused by Leading and Trailing Edge Control Surface Motions in Subsonic Compressible Flow, National Aeronautics and Space Administration, Contractor Report No. 2543, 1975.Google Scholar
  9. 9.
    Noble, B., A Bibliography on Methods for Solving Integral Equations, Mathematics Research Center Technical Summary Report No. 1176, Madison, Wisconsin, 1971.Google Scholar
  10. 10.
    Defranco, R. J., Stability Results for Multiple Volterra Equations, University of Arizona, PhD Thesis, 1973.Google Scholar
  11. 11.
    Davies, B., Integral Transforms and Their Applications, Springer-Verlag, New York, New York, 1978.Google Scholar
  12. 12.
    Fromme, J., and Golberg, M., Unsteady Two Dimensional Airloads Acting on Oscillating Thin Airfoils in Subsonic Ventilated Wind Tunnels, National Aeronautics and Space Administration, Contractor Report No. 2967, 1978.Google Scholar
  13. 13.
    Fromme, J., and Golberg, M., Numerical Solution of a Class of Integral Equations Arising in Two Dimensional Aerodynamics, Journal of Optimization Theory and Applications, Vol. 24, No. 1, 1978.Google Scholar
  14. 14.
    Milne, R., Application of Integral Equations for Fluid Flows in Unbounded Domains, Finite Elements in Fluids, Vol. 2, Edited by R. H. Gallagher, J. T. Oden, and O. C. Zienkiewicz, John Wiley and Sons, New York, New York, 1975.Google Scholar
  15. 15.
    Nishiyama, T., Lifting Surface Theory of Fully Submerged Hydrofoils, Journal of Ship Research, Vol. 8, No. 4, 1965.Google Scholar
  16. 16.
    Dow, M. L., and Elliott, D., The Numerical Solution of Singular Integral Equations over [-1, 1] Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 16, No. 1, 1979.Google Scholar
  17. 17.
    Linz, P., An Analysis of a Method for Solving Singular Integral Equations, BIT, Vol. 17, pp. 329–337, 1977.CrossRefGoogle Scholar
  18. 18.
    Landahl, M. T., and Stark, V. J. F., Numerical Lifting Surface Theory—Problems and Progress, Journal of the American Institute of Aeronautics and Astronautics, Vol. 6, No. 11, 1968.Google Scholar
  19. 19.
    Allen, R. C., and Wing, G. M., A Method for Accelerating the Iterative Solution of a Class of Fredholm Integral Equations,Chapter 2, this volume.Google Scholar
  20. 20.
    Bellman, R., Kalaba, R., and Lockett, J. A., Numerical Inversion of the Laplace Transform, American Elsevier Publishing Company, New York, New York, 1966.Google Scholar
  21. 21.
    Stenger, F., Connection Between a Cauchy System Representation of Kalaba and Fourier Transforms, Applied Mathematics and Computation, Vol. 1, No. 1, 1975.Google Scholar
  22. 22.
    Golberg, M. A., The Conversion of Fredholm Integral Equations to Equivalent Cauchy Problems, Applied Mathematics and Computation, Vol. 2, pp. 1–18, 1976.CrossRefGoogle Scholar
  23. 23.
    Golberg, M. A., The Conversion of Fredholm Integral Equations to Equivalent Cauchy Problems—Computation of Resolvents, Applied Mathematics and Computation, Vol. 3, No. 1, 1977.Google Scholar
  24. 24.
    Aalto, S. K., Reduction of Fredholm Integral Equations with Green’s Function Kernels of Volterra Equations, Oregon State University, MS Thesis, 1966.Google Scholar
  25. 25.
    Sloan, I., Burn, B., and Datyner, N., A New Approach to the Numerical Solution of Integral Equations, Journal of Computational Physics, Vol. 18, No. 1, 1975.Google Scholar
  26. 26.
    Phillips, J., The Use of Collocation as a Projection Method for Solving Linear Operator Equations, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 9, pp. 14–27, 1972.CrossRefGoogle Scholar
  27. 27.
    Nyström, E. J., Ober die Praktische Auflösung von Integralgleichingen mit Anwerdungen auf Randwertanfgaben, Acta Mathematica, Vol. 54, pp. 185–204, 1930.CrossRefGoogle Scholar
  28. 28.
    Rall, L., Resolvent Kernels of Green’s Function Kernels and Other Finite Rank Modifications of Fredholm and Volterra Kernels, Journal of Optimization Theory and Applications, Vol. 24, No. 1, 1978.Google Scholar
  29. 29.
    Chandrasekhar, S., Radiative Transfer, Clarendon Press, Oxford, England, 1950.Google Scholar
  30. 30.
    Golberg, M. A., Initial Value Methods in the Theory of Fredholm Integral Equations, Journal of Optimization Theory and Applications, Vol. 9, pp. 112–119, 1972.CrossRefGoogle Scholar
  31. 31.
    Golberg, M. A., Convergence of an Initial Value Method for Solving Fredholm Integral Equations, Journal of Optimization Theory and Applications, Vol. 12, pp. 334–356, 1973.CrossRefGoogle Scholar
  32. 32.
    Bownds, J. M., and Wood, B., On Numerically Solving Non-linear Volterra Integral Equations with Fewer Computations, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 13, pp. 705–719, 1976.CrossRefGoogle Scholar
  33. 33.
    Bownds, J. M., On Solving Weakly Singular Volterra Equations of the First Kind with Galerkin Approximations, Mathematics of Computation, Vol. 30, pp. 747–757, 1976.CrossRefGoogle Scholar
  34. 34.
    Goursat, E., Determination de la Resolvante d’une Equation Volterra, Bulletin des Sciences et Mathematiques, Vol. 57, pp. 144–150, 1933.Google Scholar
  35. 35.
    Shumitzky, A., On the Equivalence Between Matrix Riccati Equations and Fredholm Resolvents, Journal of Computer and System Science, Vol. 3, pp. 76–87, 1968.CrossRefGoogle Scholar
  36. 36.
    Stenger, F., The Approximate Solution of Wiener—Hopf Integral Equations, Journal of Mathematical Analysis and Applications, Vol. 3, pp. 687–724, 1972.CrossRefGoogle Scholar
  37. 37.
    Kailath, T., Some New Algorithms for Recursive Estimation in Constant Linear Systems, Institute for Electrical and Electronic Engineering Transactions on Information Theory, Vol. 19, pp. 750–760, 2973.CrossRefGoogle Scholar
  38. 38.
    Vainikko, G. M., On the Stability and Convergence of the Collocation Method, Differential Equations, Vol. 1, pp. 186–195, 1965.Google Scholar
  39. 39.
    Walsh, J., Boundary-Value Problems in Ordinary Differential Equations, The State of the Art in Numerical Analysis, Edited by D. A. H. Jacobs, Academic Press, New York, New York, 1977.Google Scholar
  40. 40.
    Szégo, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Fourth Edition, Providence, Rhode Island, 1975.Google Scholar
  41. 41.
    Fromme, J. A., and Golberg, M. A., On the L2 Convergence of Collocation for the Generalized Airfoil Equation (to appear).Google Scholar
  42. 42.
    Fromme, J. A., and Golberg, M. A., Computation of Aerodynamic Interference Effects on Oscillating Airfoils with Flaps, National Aeronautics and Space Administration, Contractor Report No. 2967, May 1978.Google Scholar
  43. 43.
    Sloan, I. H., Improvement by Iteration for Compact Operator Equations, Mathematics of Computation, Vol. 30, pp. 758–764, 1976.CrossRefGoogle Scholar
  44. 44.
    Sloan, I. H., Iterated Galerkin Method for Eigenvalue Problems, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 13, pp. 753–760, 1976.CrossRefGoogle Scholar
  45. 45.
    Chatelin, F., Theorie de l’Approximation des Operateurs Lineaires—Application au Calcul del Valeurs Propres D’operateurs Differentiels et Integraux, Universite Scientifique et Medicale de Grenoble, Institute de Recherche en Mathematiques Avancees, Grenoble, France, 1977.Google Scholar
  46. 46.
    Brakhage, H., Ober die Numerische Behandlung von Integralgleichungen Nach der Quadrature Formelmethode, Numerische Mathematik, Vol. 2, pp. 183–196, 1960.CrossRefGoogle Scholar
  47. 47.
    Anselone, P. M., and Moore, R. H., Approximate Solutions of Integral and Operator Equations, Journal of Mathematical Analysis and Applications, Vol. 9, pp. 268–277, 1964.CrossRefGoogle Scholar
  48. 48.
    De Hoog, F., and Weiss, R., Asymptotic Expansions for Product Integration, Mathematics of Computation, Vol. 27, pp. 295–306, 1973.CrossRefGoogle Scholar
  49. 49.
    Atkinson, K., Iterative Variants of the Nyström Method for the Numerical Solution of Integral Equations. Numerische Mathematik, Vol. 22, pp. 17–31, 1973.CrossRefGoogle Scholar
  50. 50.
    Kantorovich, L. V., Functional Analysis and Applied Mathematics, Uspekhi Mathematika Nauk, Vol. 3, pp. 89–185, 1948.Google Scholar
  51. 51.
    Williams, M. H., The Resolvent of Singular Integral Equations, Quarterly of Applied Mathematics, Vol. 28, pp. 99–110, 1977.Google Scholar
  52. 52.
    Williams, M. H., Exact Solutions in Oscillating Airfoil Theory, Journal of the American Institute of Aeronautics and Astronautics, Vol. 15, pp. 875–877, 1977.CrossRefGoogle Scholar
  53. 53.
    Williams, M. H., The Solution of Singular Integral Equations by Jacobi Polynomials (to appear).Google Scholar
  54. 54.
    Anderssen, R. S., Application and Numerical Solution of Abel Type Integral Equations, Mathematics Research Center, University of Wisconsin, Madison, Technical Summary Report No. 1787, 1977.Google Scholar
  55. 55.
    Golberg, M. A., A Method of Adjoints for Solving Some Ill-posed Equations of the First Kind, Journal of Applied Math. and Computation, Vol. 5, No. 2, pp. 123–130, 1979.CrossRefGoogle Scholar
  56. 56.
    Golberg, M. A., Boundary and Initial-Value Methods for Solving Fredholm Equations with Semidegenerate Kernels, Journal of Optimization Theory and Applications, Vol. 24, No. 1, 1978.Google Scholar
  57. 57.
    Muskhelishvili, N. I., Singular Integral Equations, Noordhoff International Publishing Company, Amsterdam, Holland, 1953.Google Scholar
  58. 58.
    Noble, B., The Numerical Solution of Integral Equations, The State of the Art in Numerical Analysis, Edited by D. A. H. Jacobs, Academic Press, New York, New York, 1977.Google Scholar
  59. 59.
    Erdogan, F., Gupta, G. D., and Cook, T. S., Numerical Solution of Singular Integral Equations, Mechanics of Fracture, Vol. 1, pp. 368–425, 1973.Google Scholar
  60. 60.
    Söhngen, H., Zur Theorie der Endlichen Hilbert Transformation, Mathematische Zeitschift, Vol. 60, pp. 31–51, 1954.CrossRefGoogle Scholar
  61. 61.
    Tricomi, F., On the Finite Hilbert Transformation, Quarterly Journal of Mathematics, Vol. 2, pp. 199–211, 1951.CrossRefGoogle Scholar
  62. 62.
    Tricomi, F., Integral Equations, Interscience Publishers, New York, New York, 1957.Google Scholar
  63. 63.
    Ralston, A., and Rabinowitz, P., A First Course in Numerical Analysis, Second Edition, McGraw-Hill Book Company, New York, New York, 1978.Google Scholar
  64. 64.
    Goldman, A., and Visscher, W., Applications of Integral Equations in Particle Size Statistics,Chapter 6, this volume.Google Scholar
  65. 65.
    Lonseth, A. T., Sources and Applications of Integral Equations, Society for Industrial and Applied Mathematics Review, Vol. 19, No. 2, 1977.Google Scholar
  66. 66.
    Twomey, S., The Application of Numerical Filtering to the Solution of Integral Equations Encountered in Indirect Sensing Measurements, Journal of the Franklin Institute, Vol. 279, pp. 95–109, 1965.CrossRefGoogle Scholar
  67. 67.
    Tikhonov, A. N., On the Solution of Ill-posed Problems and the Method of Regularization, Soviet Mathematics, Vol. 4, pp. 1035–1038, 1963.Google Scholar
  68. 68.
    Wahba, G., Practical Approximate Solutions to Linear Operator Equations when the Data Are Noisy, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 14, No. 4, 1977.Google Scholar
  69. 69.
    Strand, O. N., Theory and Methods Related to the Singular Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 11, pp. 798–825, 1974.CrossRefGoogle Scholar
  70. 70.
    Rall, L. B., Computational Solution of Nonlinear Operator Equations, John Wiley and Sons, New York, New York, 1969.Google Scholar
  71. 71.
    Kraft, E. M., and Lo, C. F., Analytical Determination of the Blockage Effect in a Perforated Wall Wind Tunnel, Journal of the American Institute of Aeronautics and Astronautics, Vol. 15, No. 4, 1977.Google Scholar
  72. 72.
    Bellman, R., and Kalaba, R. E., Quasilinearization and Boundary Value Problems, American Elsevier Publishing Company, New York, New York, 1965.Google Scholar
  73. 73.
    Ortega, J. M., On Discretization and Differentiation of Operators with Applications to Newton’s Method, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 3, No. 1, 1966.Google Scholar
  74. 74.
    Vainikko, G. M., Galerkin’s Perturbation Method and the General Theory of Approximate Methods for Non-linear Equations, U.S.S.R. Computational Mathematics and Mathematical Physics, Vol. 7, pp. 1–41, 1967.Google Scholar
  75. 75.
    Garey, L., Solving Nonlinear Second Kind Volterra Equations by Modified Increment Methods, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 12, pp. 501–508, 1975.CrossRefGoogle Scholar
  76. 76.
    Pouzet, P., Methode l’ Integration Numerique des Equations Integrales et Integro-differentielles du type de Volterra de Second Espece-Formules de RungeKutta, Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-differential Equations, Birkhauser-Verlag, Basel, Switzerland, 1960.Google Scholar
  77. 77.
    Schmaedeke, W. W., Approximate Solutions of Volterra Integral Equations of the First Kind, Journal of Mathematical Analysis and Applications, Vol. 24, pp. 604–613, 1968.CrossRefGoogle Scholar
  78. 78.
    Miller, R. K., Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., Menlo Park, California, 1971.Google Scholar
  79. 79.
    Shampine, L. F., and Watts, H. A., Solving NonStiff Ordinary Differential Equations — The State of the Art, Society for Industrial and Applied Mathematics Review, Vol. 18, pp. 376–411, 1976.Google Scholar
  80. 80.
    Scott, M. R., and Watts, H. A., Computational Solution of Linear Two Point Boundary Value Problems via Orthonormalization, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 14, pp. 40–70, 1977.CrossRefGoogle Scholar
  81. 81.
    Hull, T. E., Enright, W. H., Fellen, B. M., and Sedgwick, A. F., Comparing Numerical Methods for Ordinary Differential Equations, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 9, pp. 603–637, 1972.CrossRefGoogle Scholar
  82. 82.
    Hull, T. E., and Enright, W. H., Test Results on Initial Value Methods for Non-stiff Ordinary Differential Equations, Society for Industrial and Applied Mathematics Journal on Numerical Analysis, Vol. 13, pp. 944–961, 1976.Google Scholar
  83. 83.
    Nixon, D., An Extended Integral Equation Method for the Unsteady Transonic Flow Past a Two-dimensional Airfoil, Computational Methods and Problems in Aeronautical Fluid Dynamics, Edited by B. L. Hewitt, C. R. Illingsworth, R. C. Lock, K. W. Mangler, J. H. McDonnel, C. Richards, and F. Walkden, Academic Press, New York, New York, 1976.Google Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • M. A. Golberg
    • 1
  1. 1.University of Nevada at Las VegasLas VegasUSA

Personalised recommendations