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An initial-value method for nonlinear integral equations

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Abstract

This paper concerns the nonlinear integral equation

$$u(t) = g(t) + \int_0^x {k(t,y,u(y))dy, 0 \leqslant t \leqslant x.} $$

By considering the dependence of the unknown functionu upon the interval lengthx, we derive a Cauchy system foru in whichx plays the role of the timelike variable. This is important computationally, for modern computers can solve such initial-value problems with speed and accuracy. Specialization to the linear Fredholm integral equation is considered. Remaining problems are sketched, and applications to optimal filtering, radiative transfer, and lateral inhibition are mentioned.

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References

  1. Kagiwada, H.,Kalaba, R., andVereeke, B.,The Invariant Imbedding Numerical Method for Fredholm Integral Equations with Degenerate Kernels, Journal of Approximation Theory, Vol. 1, No. 3, 1968.

  2. Kagiwada, H., andKalaba, R.,An Initial Value Theory for Fredholm Integral Equations with Semidegenerate Kernels, Journal of the Association for Computing Machinery, Vol. 17, No. 3, 1970.

  3. Kagiwada, H., andKalaba, R.,An Initial Value Method for Fredholm Integral Equations of Convolution Type, International Journal of Computer Mathematics, Vol. 2, No. 2, 1968.

  4. Kalaba, R., andZagustin, E.,An Initial Value Method for Dual Integral Equations, International Journal of Engineering Science, Vol. 10, No. 7, 1972.

  5. Van Trees, H.,Detection, Estimation and Modulation Theory, John Wiley and Sons, New York, New York, 1968.

    MATH  Google Scholar 

  6. Landahl, H.,Mathematical Methods in the Behavior of the Central Nervous System, Mathematical Problems in the Biological Sciences, Edited by R. Bellman, American Mathematical Society, Providence, Rhode Island, 1961.

    Google Scholar 

  7. Kagiwada, H., Kalaba, R., andYang, C. C.,Reduction of a Class of Nonlinear Integral Equations to a Cauchy System, Journal of Mathematical Physics, Vol. 13, No. 2, 1972.

  8. Buell, J., Fymat, A., andKalaba, R.,An Initial Value Method for the Ambarzumian Integral Equation, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 12, No. 5, 1972.

  9. Holbrow, W., andKalaba, R.,Numerical Solution for a Class of Nonlinear Integral Equations with Application to Neural Inhibition (to appear).

  10. Cochran, J.,Analysis of Linear Integral Equations, McGraw-Hill Book Company, New York, New York, 1972.

    MATH  Google Scholar 

  11. Kagiwada, H., andKalaba, R.,An Initial Value Method for Fredholm Resolvents (to appear).

  12. Mikhlin, S., andSmolitskiy, K.,Approximate Methods for Solution of Differential and Integral Equations, American Elsevier Publishing Company, New York, New York, 1967.

    MATH  Google Scholar 

  13. Buell, J.,et al., Computation of the Resolvent for the Auxiliary Equation of Radiative Transfer, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 8, No. 8, 1968.

  14. Casti, J., Kalaba, R., andUeno, S.,Reflection and Transmission Functions for Finite Isotropically Scattering Atmospheres with Specular Reflectors, Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 9, No. 5, 1969.

  15. Sobolev, V.,Rasseyanie Sveta v Atmosferakh Planet, Nauka, Moscow, USSR, 1972.

    Google Scholar 

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Authors and Affiliations

  1. Biomedical Engineering Program, University of Southern California, Los Angeles, California

    H. Kagiwada (Consulting Mathematician) & R. Kalaba (Visiting Professor)

Authors
  1. H. Kagiwada
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  2. R. Kalaba
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Kagiwada, H., Kalaba, R. An initial-value method for nonlinear integral equations. J Optim Theory Appl 12, 329–337 (1973). https://doi.org/10.1007/BF00940414

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  • DOI: https://doi.org/10.1007/BF00940414

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