# Integral Equation Method

## Abstract

Each field problem done by the integral equation method has the following steps.

- 1.
Using Green’s theorem or the Helmholtz theorem (see Section 2.5), develop a suitable expression for the field to be computed as an integral over certain sources. These sources can be real or equivalent currents or charges, or fields or scalar potentials.

- 2.
Express these sources, in turn, in terms of the field to be computed. This leads to an integral equation either in terms of the field or in terms of the sources.

- 3.
Develop a partition of the volume or surface over which the integral is taken. That is, divide this volume or surface suitably into a number of subdivisions.

- 4.
Using this partition and the integral equation, derive a system of equations to be solved for the sources or the field over the volume or surface of integration. If the medium is linear, this is a linear system of equations. If the medium is nonlinear, this is a nonlinear system of equations. In any event, compute the sources over the volume or surface of integration.

- 5.
Using these sources or fields, the partition, and the integral expression developed in step 1, compute the field at any desired point in space.

## Keywords

Integral Equation Collocation Method Integral Equation Method Field Point Field Problem## Preview

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## References

- 1.Lindholm, Dennis, “Notes on Boundary Integral Equations for Three-Dimensional Magneto-statics,”
*IEEE Transaction on Magnetics*,November 1980, Vol. MAG-16, No. 6, pp. 1409–1413.Google Scholar - 2.Chari, M. V. K., and Silvester, P. P., Editors.
*Finite Elements in Electrical and Magnetic Field Problems*. New York: John Wiley & Sons, 1980.Google Scholar - 3.Jeng, G., and Wexler, A., “Isoparametric, Finite Element Variational Solution of Integral Equations for Three-Dimensional Fields,”
*International Journal for Numerical Methods in Engineering*, Vol. II, 1977, pp. 1455–1471.CrossRefGoogle Scholar - 4.Lindholm, Dennis, “Effect of Track Width and Side Shields on the Long Wavelength Response of Rectangular Magnetic Heads,”
*IEEE Transactions on Magnetics, March*1980, Vol. MAG-16, No. 2, pp. 430–435.CrossRefGoogle Scholar - 5.Rao, S. M., Glisson, A. W., Wilton, D. R., and Vidula, B. S., “A Simple Numerical Solution Procedure for Statics Problems Involving Arbitrary- Shaped Surfaces,”
*IEEE Transactions on Antennas and Propagation, September*1979, Vol. AP-27, No. 5, pp. 604–608.CrossRefGoogle Scholar - 6.Todhunter, I., and Leatham, J. G.
*Spherical Trigonometry*, 3d ed. London: mcmillan, 1911.Google Scholar - 7.Mcdonald, B. H., Friedman, M., and Wexler, A., “Variational Solution of Integral Equations,”
*IEEE Transactions on Microwave Theory and Techniques, March*1974, Vol. MTT-22, No. 3.MathSciNetGoogle Scholar - 8.Vichnevetsky, R.
*Computer Methods for Partial Differential Equations*, Vol. 1. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1981.Google Scholar - 9.Zienkiewicz, O. C.
*The Finite Element Method in Engineering Science*, 2d ed. London: McGraw-Hill, 1971.Google Scholar