A discussion on the three-dimensional boundary value problem for electromagnetic fields
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A discussion on the three-dimensional boundary value problem for electromagnetic fields

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      Three -dimensional boundary value problems are difficult to solve. Indeed, while the separation of the scalar wave equation can be effected in 11 different coordinate systems, an analytic solution requires that the boundaries, both external and internal, possess the same symmetry as the coordinate system. Numerical methods are thus of great importance for the solution of such problems; however, despite the availability of high-speed, large memory digital computers , the solution to a significant three -dimensional problem is by no means trivial. It is unfortunate that the results reported here by Jones are invalid. Jones discusses a model in which a three-dimensional island lies off a linear coastline where all the interfaces lie in the coordinate planes in a Cartesian system. A downward plane em wave polarized with the electric field parallel to the linear coastline, and two of the islands coast, is incident downward. This, of course, requires that the incident electric field is perpendicular to the other faces of the idealized Cartesian island. The significant equation used by Jones for his finite difference calculations is [ ] where n2 = ?µ? . is the conductivity, µ=µo is the permeability of the appropriate medium, and w is the angular frequency of a Fourier component of the incident field. We see that the displacement current term is omitted, which is an excellent approximation for the parameters of interest in this model. In the similar equation for the y- component of the diffusion equation, the same considerations which we are about to discuss are equally valid. It can be seen from the equation that the field components are related and thus any errors introduced will be propagated into all three components. The difficulty arises when Ex or E is perpendicular Y to one or other of the surfaces of discontinuity in ?. 
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