On a Method of Bownds for Solving Volterra Integral Equations

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

Abstract

An initial-value method of Bownds for solving Volterra integral equations is reexamined using a variable-step integrator to solve the differential equations. It is shown that such equations may be easily solved to an accuracy of O(10−8), the error depending essentially on that incurred in truncating expansions of the kernel to a degenerate one.

Keywords

Kernel Approximation Fredholm Integral Equation Volterra Integral Equation Volterra Equation Integration Error 
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References

  1. 1.
    Bownds, J. M., and Wood, B., On Numerically Solving Nonlinear Volterra Integral Equations with Fewer Computations, SIAM Journal of Numerical Analysis, Vol. 13, pp. 705–719, 1976.CrossRefGoogle Scholar
  2. 2.
    Bownds, J. M., and Wood, B., A Note on Solving Volterra Integral Equations with Convolution Kernels, Applied Mathematics and Computation, Vol. 3, pp. 307–315, 1977.CrossRefGoogle Scholar
  3. 3.
    Bownds, J. M., On Solving Weakly Singular Volterra Equations of the First Kind with Galerkin Approximations (to appear).Google Scholar
  4. 4.
    Noble, B., The Numerical Solution of Nonlinear Integral Equations and Related Topics, Nonlinear Integral Equations, Edited by P. M. Anselone, University of Wisconsin Press, Madison, Wisconsin, 1964.Google Scholar
  5. 5.
    Garey, L., Solving Nonlinear Second Kind Volterra Equations by Modified Increment Methods, SIAM Journal on Numerical Analysis, Vol. 12, No. 6, 1975.Google Scholar
  6. 6.
    Allen, R. C., and Shampine, L. F., Numerical Computing: An Introduction,W. B. Saunders Company, Philadelphia, Pennsylvania, 1973.Google Scholar
  7. 7.
    Golberg, M. A., The Conversion of Fredholm Integral Equations to Equivalent Cauchy Problems, Applied Mathematics and Computation, Vol. 2, No. 1, 1976.Google Scholar
  8. 8.
    Golberg, M. A., The Conversion of Fredholm Integral Equations to Equivalent Cauchy Problems, II: Computation of Resolvents, Applied Mathematics and Computation, Vol. 1, No. 1, 1977.Google Scholar
  9. 9.
    Kagiwada, H., and Kalaba, R. E., Integral Equations via Imbedding Methods, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.Google Scholar
  10. 10.
    Casti, J., and Kalaba, R. E., Imbedding Methods in Applied Mathematics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.Google Scholar
  11. 11.
    Shanipine, L. F., and Gordon, M. K., The Computer Solution of Ordinary Differential Equations: The Initial-Value Problem, W. H. Freeman, San Francisco, California, 1975.Google Scholar
  12. 12.
    Atkinson, K., A Survey of Numerical Methods for the Numerical Solution of Fredholm Integral Equations of the Second Kind, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1976.Google Scholar
  13. 13.
    Bownds, J. M., An Initial-Value Method for Quickly Solving Volterra Integral Equations, Journal of Optimization Theory and Applications, Vol. 24, No. 1, 1978.Google Scholar
  14. 14.
    Sloan, I. H., Error Analysis for Integral Equation Methods (to appear).Google Scholar
  15. 15.
    Golberg, M. A., Boundary and Initial-Value Methods for Solving Fredholm Integral Equations with Semidegenerate Kernels,Chapter 13, this volume.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

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