A Method of Moment Approach in Solving Boundary Value Problems

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A Method of Moment Approach in Solving Boundary
Value Problems
H. M. El Misilmani, K. Y. Kabalan , M. Abou Shahine
Electrical and Computer Eng. Dept.
American University of Beirut
Beirut, Lebanon
hilal.elmisilmani@ieee.org, kabalan@aub.edu.lb,
mya26@aub.edu.lb
M. Al-Husseini
Beirut Research and Innovation Center
Lebanese Center for Studies and Research
Beirut, Lebanon
husseini@ieee.org
Abstract— Several available methods, known in literatures, are
available for solving nth order differential equations and their
complexity differ based on the accuracy of the solution. A
successful method, known to researcher in the area of
computational electromagnetic and called the Method of Moment
(MoM) is found to have its way in this domain and can be used in
solving boundary value problems where differential equations
are resulting. A simplified version of this method is adopted in
this paper to address this problem, and two differential equations
examples are considered to clarify the approach and present the
simplicity of the method. As illustrated in this paper, this
approach can be introduced along with other methods, and can
be considered as an attractive way to solve differential equations
and other boundary value problems.
Boundary value problem, differential equations, method of
moment, Galerkin method, weight coefficient
I. INTRODUCTION
The design and analysis of electromagnetic devices and
structures before the computer invention were largely
depending on experimental procedures. With the development
of computers and programming languages, researches began
using them to solve the challenging electromagnetic problems
that could not be solved analytically. This led to a burst of
development in a new field called computational
electromagnetics (CEM), for which powerful numerical
analysis techniques, including the Method of Moment (MoM),
have been developed in this area in the last 50 years [1].
Harrington in [2] describes the MoM as a simple numerical
technique used to convert integro-differential equations into a
linear system that can be solved numerically using a computer.
When the order of the equation is small, MoM can analytically
solve the problem in a general and very clear manner.
Although MoM has been studied in a large number of
publications, it has been mainly a part of graduate courses on
computational electromagnetics, or aimed to help professionals
apply MoM in their field problems. In fact, it is infrequent to
see a work on MoM addressed to undergraduate students to aid
their understanding of this method, and help them apply it to
solve problems they face in their undergraduate courses.
A large number of publications address the use of MoM in
solving various problems. Djordjevic and Sarkar in [3] showed
that the inner product involved in MoM is usually an integral,
which is evaluated numerically by summing the integrand at
certain discrete points. Newman in [4] has presented three
simple examples on the use of MoM in electromagnetics.
These examples deal with the input impedance of a short
dipole, a plane wave scattering from a short dipole, and two
coupled short dipoles. The use of MoM in electromagnetic
field has been introduced by Serteller et al. in [5]. This is done
by presenting examples and software program, and also by
giving the curriculum needed to quickly learn the basic
concepts of numerical solutions.
As mentioned earlier, the idea in this paper is to introduce
the approach in a simple and straightforward manner to make it
understandable to students taking basic course in differential
equation. First, the method is defined and illustrated, a
conversion into respective integral equation is determined, and
then the unknown function is expanded into a sum of weighted
basis functions, where the weight coefficients are to be found.
The Galerkin method, which selects testing functions equal to
the basis functions, is adopted. The problem then becomes a
system of linear equations, which is solved analytically or
numerically to find the needed weight coefficients.
II. FAMILIARILZING STUDENTS WITH THE METHOD OF
MOMENTS
As the Method of Moments is based on expanding the
unknown solution of the differential equation into known
expansion and testing functions that satisfy the boundary value
constraints, it is advisable to first introduce this approach in the
solution. Accordingly, the sough of solution is expanded into a
sum of known function, each satisfying the boundary
conditions of the problem, with unknown coefficients to be
determined by the solution. This determination is done with
the help of testing function chosen similarly as the expansion
function for the simplicity of the solution. The resulting
equation is then converted into a linear system of equations by
enforcing the boundary conditions at a number of points. This
resulting linear system is then solved analytically for the
unknown coefficients. This approach is very simple and quite
interesting when applied to differential equation of order less
than 3, but it will get more complicated for equations of higher
order.

Accordingly, it is advisable to start with some basic
mathematical techniques for reducing functional equations to
matrix equations. A deterministic problem is considered,
which will be solved by reducing it to a suitable matrix
equation, and hence the solution could be found by matrix
inversion.
Simple examples using linear spaces and operators are
used. At first, it is recommended to introduce MoM and define
some terms related to first order non-homogeneous differential
equation. The choice of this equation is important only for
better understanding of the solution.
A general nth order linear differential equation, defined over a
domain D, has the form


 ... 
  (1)
In (1), the coefficients a,a,...,a,a and gx are known
quantities, and fx is the function whose solution is to be
determined. Equation (1) can be written in the form of an
operator equation
Lfxgx, (2)
where  is the operator equation, operating on , and given
by
La...
a...
... a...
a. (3)
The solution of (1) is based on defining the inner product
f,g, a scalar quantity valid over the domain of definition
of L, which is given by
〈,〉
 (4)
Similarly, we define
〈Lfx,gx〉L
fxgxdx (5)
The first step in calculating the integral, using Method of
Moments, is to expand f into a sum of weighted basis functions
f,f,f;…, in the domain of L, as
fx∑α f (6)
Testing functions denoted w,w,w;….are defined in the
range of L. These testing functions are used for all values of n.
Using the inner product defined in (5), we obtain
∑〈w,Lf〉
〈w,gx〉 form1,2,3,... (7)
Expanding (7) over the values of m and n=1,2,3,…, the
following matrix equation is then obtained
〈,〉 〈,〉 .... 〈,〉
〈,〉 〈,〉 .... 〈,〉
.... .... .... ....
〈,〉 〈,〉 .... 〈,〉

...

〈,〉
〈,〉
...
〈,〉 (8)
In a simpler form,
LαG, (9)
and the solution for the unknown coefficients is then
αLG (10)
In our calculations, the test function wm is chosen to be equal to
the basis functionf, which is known as Garlekin method. The
determination of matrix L is straightforward, and its
inverse is easy to obtain either analytically or numerically.
Once this is done, the α coefficients are obtained, and the
solution for f is found.
It is good to note here that choosing the appropriate basis/test
function is necessary to get fast to the accurate solution.
A. Example 1
Considering the following second order differential
equation defined by

x (11)
defined over the domain D0,1 with the following
boundary conditions f0f10. Starting by choosing
the basis function, let us choose
fxx (12)
It is clear from (12) that the chosen basis function meets the
boundary conditions and can be considered as a solution to the
problem. Substituting (12) into (6), the left-hand side elements
of (7), which are the elements of the matrixL, are found to
be
Lmn 〈w,Lf〉xxm+n+
dxdx


xxm+nn+1xdx

Lmn 
n+m+
(13)
In the same manner, we compute the elements of the matrix
G, defined in (7), which are found to be
G〈w,g〉xxxdx 
 (14)
Then, we start by choosing N = 1, for which nm1 in
L and , hence L1/3;G1/20,α3/20, and
f(x) is given by

fx
xx (15)
It is clear from (15), that the function fx does not meet the
original differential equation defined in (11). Accordingly, we
need to increase the value of N.
Let N = 3, and calculating the values of L,, and α:
L L L
L L L
L L L

g
g
g
=
1/3 1/2 3/5
1/2 4/5 1
4/5 1 9/7

1/20
1/12
3/28 (16)
Hence, α could be calculated as


 0
0
1/12 (17)
Finally, calculating fx
fx∑α fαfαfαf
 (18)
The function fx given in (18) meets the boundary conditions
defined in (11), and accordingly it is the correct solution of the
problem.
B. Example 2
Considering the following second order differential
equation defined by

8x2 (19)
defined over the domain D0,1 with the following
boundary conditions f0f10. Starting by choosing
the basis function, it was noted that the basis function used in
example 1 could also be used here
fxx (20)
The chosen basis functions meet the boundary conditions and
can be considered as a solution to the problem. L is
calculated as in example 1, since the same basis function is
used, and it is given by
Lmn 
n+m+ (21)
In the same manner, we compute the elements of the matrix
G, defined in (7), which are found to be
G〈w,g〉xx8x2dx 

(22)
Then, we start by choosing N = 1 for which nm1.
Accordingly, L1/3;G17/30,α21/30, and f(x)
is given by
fx
xx (23)
It is clear from (23), that the function fx does not meet the
original differential equation defined in (19). Accordingly, we
need to increase the value of N.
Let N = 3, and calculating the values of L,, and α:
L L L
L L L
L L L

g
g
g
=
1/3 1/2 3/5
1/2 4/5 1
4/519/7


17/30
14/15
25/21 (24)
Hence, α could be calculated as


0
1/3
2/3 (25)
Finally, calculating fx
fxα
fαfαfαf
1
32
3
2
3
3
(26)
The function fx given in (26) meets the boundary conditions
defined in (19), and accordingly it is the correct solution of the
problem.
III. CONCLUSION
As demonstrated, MoM approach could be easily used to
solve mathematical problems and equations. It can be easily
employed by undergraduate students. According to the type of
the equation, the solution of the Moment Method will vary to
accommodate for the change in the given problem.
REFERENCES
[1] W. C. Gibson, The Method of Moments in Electromagnetics, Chapman
& Hall/ CRC, Taylor & Francis group, 2008.
[2] R. F. Harrington, Field Computation by Moment Methods, Krieger
Publishing Co., Inc., 1968.
[3] A. R. Djordjevic and T. K. Sarkar, “A Theorem on the Moment
Methods," IEEE Transactions on Antennas and Propagation, vol. 35,
no. 3, pp. 353-355, March 1987.
[4] E. H. Newman, “Simple Examples of the Method of Moments in
Electromagnetics," IEEE Transactions on Education, vol. 31, no. 3, pp.
193-200, August 1998.
[5] N.F.O. Serteller, A.G. Ak, G. Kocyigit, and T.C. Akinci, “Experimental
Study of Moment Method for Undergraduates in Electromagnetic”,
Journal of Electronics and Electrical Engineering, vol. 3, no. 3, pp. 115-
118, 2011.