12 2. NUMERICAL AND ANALYTICAL METHODS ON BOUNDARY VALUE PROBLEMS
2.3.1 IMAGES METHOD
In general, we use this method to solve a special classes of electrostatic (and magnetostatic)
problems are extended by the addition of its mirror image with respect to a symmetry plane.
e key idea of this method is to transform the complex relevant problem and then replace it
with a finite number of chosen discrete points. erefore, this facilitated the solution of the
original problem since the given boundary conditions are automatically satisfied by their mirror
image. erefore, this is based on the uniqueness theorem—as long as the solution satisfies
Poisson’s or Laplace’s equation and the given boundary condition, the simplest solution should
be taken [59].
is method is a powerful technique for solving the applications involving charges (elec-
trical or magnetic) and conductors with simple geometries such as spheres, cylinders, ellipsoids,
and planes. e potentials resulting from the image charges are the solutions of the Laplace
equation when the potentials from other charges are the solutions of the Poisson equation [60].
Another application is two regions of dielectrics, with different permittivity [11].
e most important characteristic of this method is that the problem should be always
presented with two distinct regions of space because when you need to find the potential in one
region, the image charges should be placed in other region, and not to be allowed in the same
region. (Examples of this are the left and right of the plane or inside and outside of the object.)
e most important advantage of this method is that can apply for the multiple regions by
adding mirror image charges to the system. In addition, this method is closely related to Green’s
function method and can be used to find Green’s functions for the same simple geometries.
2.3.2 GREEN’S FUNCTION METHOD
e potential due to a single point charge with the boundary conditions and its image for a given
surface shape can be formulated in terms of a scalar function, known as Green’s function. So,
the method of images is used to generate Green’s functions for a problem. A Green’s function
is related to the inverse of a differential operator and can be used to express the solution to a
boundary value problem [61, 62].
Consider the general problem with n th order, inhomogeneous, differential equation as
Ly.x/ D ˛.x/; l < x < u; (2.29)
where L is a differential operator given by
L D
n
X
iD0
c
i
.x/
d
i
dx
i
(2.30)
with the coefficients ˛.x/ and c
i
.x/ are known continuous functions of x. en, (2.29) becomes
c
0
.x/ y.x/ C c
1
.x/ y
0
.x/ C C c
n
.x/ y
.n/
.x/ D ˛.x/: (2.31)
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