# Nonlinear Integral Equations

• Robert Kalaba
• Nima Rasakhoo
• Karl Spingarn
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 31)

## Abstract

Integral equations appear in many engineering and physics problems. Numerical methods of solution for integral equations have been largely developed within the last 20 years (References 1–4). In this chapter a development involving an imbedding method for obtaining the numerical solution of nonlinear integral equations is described (References 5, 6). The numerical solution is obtained automatically from the initial value imbedding equations via the automatic derivative evaluation method described in the previous chapters (Reference 7). The derivatives required for the solution are computed automatically. The user need only enter the two known functions in equation (6.1) into the program. None of the derivatives associated with the imbedding method need be derived by hand. The subroutines are written in Basic here, rather than in Fortran, because the former has several advantages in the manipulation of the vectors and matrices which occur in using the method of lines. The vectors consisting of the variables or functions of the variables, and all their derivatives are defined as derivative vectors (instead of just vectors) in this chapter to distinguish them from the other vectors and matrices.

## Keywords

Integral Equation Quadrature Formula Fredholm Integral Equation Nonlinear Integral Equation Grid Interval
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.

## Chapter 6

1. 1.
golberg, M. A., Solution Methods for Integral Equations, Theory and Applications ,Plenum Press, New York, 1979.
2. 2.
3. 3.
4. 4.
Kalaba, R., and Spingarn, K., Control, Identification, and Input Optimization ,Plenum Press, New York, 1982.
5. 5.
Kagiwada, H. H., and Kalaba, R., An initial value method for the nonlinear integral equation F(u(t), t ,&#955; ) = &#955; &#8747;01 k(t, y, u(y)) dy, Applied Mathematics and Computation ,Vol. 13, Nos. 1 and 2, pp. 117&#x2013;124, August 1983.
6. 6.
Kagiwada, H. H., Kalaba, R., and Spingarn, K., Automatic solution of nonlinear integral equations, Computers and Mathematics with Applications (in press).Google Scholar
7. 7.
Kalaba, R., and Spingarn, K., Automatic solution of Mh-order optimal control problems, IEEE Transactions on Aerospace and Electronic Systems ,Vol. 21, pp. 345&#x2013;350, May 1985.
8. 8.
Mikhlin, S., and Smolitskiy, K., Approximate Methods for Solution of Differential and Integral Equations ,American Elsevier, New York, 1967.
9. 9.
Kalaba, R. E. ,Spingarn, K., and Zagustin, E. ,On the integral equation method for buckling loads, Applied Mathematics and Computation ,Vol. 1, No. 3, pp. 253&#x2013;261, 1975.
10. 10.
Sobolev, V. V., Scattering of Light in Planetary Atmospheres ,Nauka, Moscow, 1972.Google Scholar

© Plenum Press, New York 1986