Chapter 15
NUMERICAL METHODS
The recipe for ignorance is: be satisfied with your opinions and content with your
knowledge.
—ELBERT HUBBARD
15.1 INTRODUCTION
In the preceding chapters we considered various analytic techniques for solving EM prob
lems and obtaining solutions in closed form. A
closed form solution
is one in the form of
an
explicit, algebraic equation in which values of the problem parameters can be substituted.
Some of these analytic solutions were obtained assuming certain situations, thereby
making the solutions applicable to those idealized situations. For example, in deriving the
formula for calculating the capacitance of a parallelplate capacitor, we assumed that the
fringing effect was negligible and that the separation distance was very small compared
with the width and length of the plates. Also, our application of Laplace's equation in
Chapter 6 was restricted to problems with boundaries coinciding with coordinate surfaces.
Analytic solutions have an inherent advantage of being exact. They also make it easy to
observe the behavior of the solution for variation in the problem parameters. However, an
alytic solutions are available only for problems with simple configurations.
When the complexities of theoretical formulas make analytic solution intractable, we
resort to nonanalytic methods, which include (1) graphical methods, (2) experimental
methods, (3) analog methods, and (4) numerical methods. Graphical, experimental, and
analog methods are applicable to solving relatively few problems. Numerical methods
have come into prominence and become more attractive with the advent of fast digital
computers. The three most commonly used simple numerical techniques in EM are
(1) moment method, (2) finite difference method, and (3) finite element method. Most EM
problems involve either partial differential equations or integral equations. Partial differ
ential equations are usually solved using the finite difference method or the finite element
method; integral equations are solved conveniently using the moment method. Although
numerical methods give approximate solutions, the solutions are sufficiently accurate for
engineering purposes. We should not get the impression that analytic techniques are out
dated because of numerical methods; rather they are complementary. As will be observed
later, every numerical method involves an analytic simplification to the point where it is
easy to apply the method.
The Matlab codes developed for computer implementation of the concepts developed
in this chapter are simplified and selfexplanatory for instructional purposes. The notations
660
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15.2
FIELD PLOTTING
•
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used in the programs are as close as possible to those used in the main text; some are
defined wherever necessary. These programs are by no means unique; there are several
ways of writing a computer program. Therefore, users may decide to modify the programs
to suit their objectives.
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 Fall '10
 DinavahiandIyer
 Numerical Analysis, Finite Element Method, Partial differential equation, finite difference, Finite difference method, Eqs

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