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Imbedding Methods for Integral Equations with Applications

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Solution Methods for Integral Equations

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 18))

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Abstract

During the last decade or two, significant progress has been made in the development of imbedding methods for the analytical and computational treatment of integral equations. These methods are now well known in radiative transfer, neutron transport, optimal filtering, and other fields. In this review paper, we describe the current status of imbedding methods for integral equations. The paper emphasizes new analytical and computational developments in control and filtering, multiple scattering, inverse problems of wave propagation, and solid and fluid mechanics. Efficient computer programs for the determination of complex eigenvalues of integral operators, analytical investigations of stability for significant underlying Riccati integrodifferential equations, and comparisons against other methods are described.

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References

  1. Courant, R., and Hilbert, D., Methods of Mathematical Physics, Vol. 1, John Wiley and Sons (Interscience Publishers), New York, New York, 1953.

    Google Scholar 

  2. Noble, B., A Bibliography of Methods for Solving Integral Equations, University of Wisconsin, Mathematics Research Center, Report No. 73, 1971.

    Google Scholar 

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

    Google Scholar 

  4. Kagiwada, H., and Kalaba, R., Integral Equations Via Imbedding Methods, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.

    Google Scholar 

  5. Scott, M. R., A Bibliography on Invariant Imbedding and Related Topics, Sandia Laboratories, Report No. SC-71–0886, 1971.

    Google Scholar 

  6. Shampine, L., and Gordon, M. K., Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman Press, San Francisco, California, 1975.

    Google Scholar 

  7. Ficken, F. A., The Continuation Method for Functional Equations, Communications on Pure and Applied Mathematics, Vol. 4, pp. 435–455, 1951.

    Article  Google Scholar 

  8. Hille, E., and Phillips, R., Functional Analysis and Semi-Groups, American Mathematical Society, Providence, Rhode Island, 1957.

    Google Scholar 

  9. Sobolev, V. V., A Treatise on Radiative Transfer, D. Van Nostrand Company, Princeton, New Jersey, 1963.

    Google Scholar 

  10. Ambarzumian, V. A., Diffuse Reflection of Light by a Foggy Medium, Doklady Akademiia Nauk SSSR, Vol. 38, pp. 229, 1943.

    Google Scholar 

  11. Golberg, M., Initial-Value Methods in the Theory of Fredholm Integral Equations, Journal of Optimization Theory and Applications, Vol. 9, pp. 112–119, 1972.

    Article  Google Scholar 

  12. Nelson, W., Existence, Uniqueness, and Stability of Solutions to Chandrasekhar’s Integrodifferential Equation for X and Y Functions, Journal of Mathematical Analysis and Applications, Vol. 37, pp. 580–606, 1972.

    Article  Google Scholar 

  13. Cochran, J. A., The Existence of Eigenvalues for the Integral Equations of Laser Theory, Bell System Technical Journal, Vol. 44, pp. 77–88, 1965.

    Google Scholar 

  14. Fredholm, I., Oevres Completes de Ivar Fredholm, Malmo, Lund, Sweden, 1955.

    Google Scholar 

  15. Courant, R., and Hilbert, D., Methods of Mathematical Physics, Vol. 2, John Wiley and Sons (Interscience Publishers), New York, New York, 1965.

    Google Scholar 

  16. Kalaba, R., and Ruspini, E. H., Invariant Imbedding and Potential Theory, International Journal of Engineering Science, Vol. 7, pp. 1091–1101, 1969.

    Article  Google Scholar 

  17. Willers, I. M., A New Integration Algorithm for Ordinary Differential Equations Based on Continued Fraction Approximations, Communications of the ACM, Vol. 17, pp. 504–510, 1974.

    Article  Google Scholar 

  18. Cali, M., Casti, J., and Juncosa, M., Invariant Imbedding and the Solution of Fredholm Integral Equations with Displacement Kernels—Comparative Numerical Experiments, Applied Mathematics and Computation, Vol. 1, pp. 287–393, 1975.

    Article  Google Scholar 

  19. Kagiwada, H., Kalaba, R., and Ueno, S., Multiple Scattering Processes: Inverse and Direct, Addison-Wesley Publishing Company, Reading, Massachusetts, 1975.

    Google Scholar 

  20. Sobolev, V. V., Light Scattering in Planetary Atmospheres, Pergamon Press, Oxford, England, 1975.

    Google Scholar 

  21. Kagiwada, H., System Identification, Addison-Wesley Publishing Company, Reading, Massachusetts, 1974.

    Google Scholar 

  22. Kailath, T., Some New Algorithms for Recursive Estimation In Constant Linear Systems, IEEE Transactions on Information Theory, IT-19, pp. 750–760, 1973.

    Google Scholar 

  23. Sidhu, G., and Casti, J., A Rapprochement of the Theories of Radiative Transfer and Linear Stochastic Estimation, Applied Mathematics and Computation, Vol. 1, pp. 295–323, 1975.

    Article  Google Scholar 

  24. Conway, W., and Thomas, J., Free Streamline Problems and the Milne—Thompson Integral Equation, Journal of Mathematical and Physical Science, Vol. 8, pp. 67–92, 1975.

    Google Scholar 

  25. Koivo, H., On the Equivalence of Maximum Principle Open-Loop Controllers and the Caratheodory Feedback Controllers for Time-Delay Systems, Journal of Optimization Theory and Applications, Vol. 14, pp. 163–178, 1974.

    Article  Google Scholar 

  26. Fleming, H. E., and Smith, W. L., Inversion Techniques For Remote Sensing of Atmospheric Temperature Profiles, Paper Presented at the Fifth Symposium on Temperature, Washington, DC, 1971.

    Google Scholar 

  27. Duncan, L. D., An Improved Algorithm for the Iterated Minimal Information Solution for Remote Sounding of Temperature, United States Army Electronics Command, Report No. ECOM-5571, 1975.

    Google Scholar 

  28. Kagiwada, H., and Kalaba, R., Imbedding Methods for Temperature Retrieval, Nonlinear Analysis, Vol. 1, pp. 65–74, 1976.

    Article  Google Scholar 

Additional Bibliography

  1. Kagiwada, H., and Kalaba, R., Direct and Inverse Problems for Integral Equations via Initial Value Methods, SIAM—AMS Symposium on Transport Theory, American Mathematical Society, Providence, Rhode Island, 1967.

    Google Scholar 

  2. Kagiwada, H., and Kalaba, R., An Initial Value Method Suitable for the Computation of Certain Fredholm Resolvents, Journal of Mathematical and Physical Sciences, Vol. 1, pp. 109–122, 1967.

    Google Scholar 

  3. Kagiwada, H., and Kalaba, R., Initial Value Methods for the Basic Boundary Value Problem and Integral Equations of Radiative Transfer, Journal of Computational Physics, Vol. 1, pp. 322–329, 1967.

    Article  Google Scholar 

  4. Kagiwada, H., and Kalaba, R., A New Initial Value Method for Internal Intensities in Radiative Transfer, Astrophysical Journal, Vol. 147, pp. 301–319, 1967.

    Article  Google Scholar 

  5. Kagiwada, H., and Kalaba, R., An Initial Value Method for Fredholm Integral Equations of Convolution Type, International Journal of Computer Mathematics, Vol. 2, pp. 143–155, 1968.

    Article  Google Scholar 

  6. Kagiwada, H., and Kalaba, R., An Initial Value Method for Nonlinear Integral Equations, Journal of Optimization Theory and Applications, Vol. 12, pp. 329–337, 1973.

    Article  Google Scholar 

  7. Kagiwada, H., Kalaba, R., and Shumitzky, A., A Representation for the Solution of Fredholm Integral Equations, Proceedings of the American Mathematical Society, Vol. 23, pp. 37–40, 1969.

    Article  Google Scholar 

  8. Golberg, M., An Initial Value Method for the Computation of the Characteristic Values and Functions of an Integral Operator-II: Convergence, Journal of Mathematical Analysis and Applications, Vol. 49, pp. 773–781, 1975.

    Article  Google Scholar 

  9. Golberg, M., Convergence of an Initial Value Method for Solving Fredholm Integral Equations, Journal of Optimization Theory and Applications, Vol. 12, pp. 344–356, 1973.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. HFS Associates, Los Angeles, California, USA

    H. Kagiwada (President)

  2. Departments of Economics and Biomedical Engineering, University of Southern California, Los Angeles, California, USA

    R. Kalaba (Professor)

Authors
  1. H. Kagiwada
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  2. R. Kalaba
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Nevada at Las Vegas, Las Vegas, Nevada, USA

    Michael A. Golberg

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© 1979 Springer Science+Business Media New York

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Kagiwada, H., Kalaba, R. (1979). Imbedding Methods for Integral Equations with Applications. In: Golberg, M.A. (eds) Solution Methods for Integral Equations. Mathematical Concepts and Methods in Science and Engineering, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1466-1_8

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  • DOI: https://doi.org/10.1007/978-1-4757-1466-1_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1468-5

  • Online ISBN: 978-1-4757-1466-1

  • eBook Packages: Springer Book Archive

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