ArticlePDF Available

Abstract

Several available methods, known in literatures, are available for solving nth order differential equations and their complexities 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.
Journal of Electromagnetic Analysis and Applications, 2015, 7, 61-65
Published Online March 2015 in SciRes. http://www.scirp.org/journal/jemaa
http://dx.doi.org/10.4236/jemaa.2015.73007
How to cite this paper: El Misilmani, H.M., Kabalan, K.Y., Abou-Shahine, M.Y. and Al-Husseini, M. (2015) A Method of Mo-
ment Approach in Solving Boundary Value Problems. Journal of Electromagnetic Analysis and Applications, 7, 61-65.
http://dx.doi.org/10.4236/jemaa.2015.73007
A Method of Moment Approach in Solving
Boundary Value Problems
Hilal M. El Misilmani1, Karim Y. Kabalan1, Mohamad Y. Abou-Shahine1,
Mohammed Al-Husseini2
1Electrical and Computer Engineering Department, American University of Beirut, Beirut, Lebanon
2Beirut Research and Innovation Center, Lebanese Center for Studies and Research, Beirut, Lebanon
Email: hilal.elmisilmani@ieee.org, kabalan@aub.edu.lb, mya26@aub.edu.lb, husseini@ieee.org
Received 9 February 2015; accepted 2 March 2015; published 5 March 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
Several available methods, known in literatures, are available for solving nth order differential
equations and their complexities 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 Mo-
ment (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 consi-
dered 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.
Keywords
Boundary Value Problem, Differential Equations, Method of Moment, Galerkin Method, Weight
Coefficient
1. 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, re-
searches began using them to solve the challenging electromagnetic problems that could not be solved analyti-
cally. 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 con-
vert integro-differential equations into a linear system that can be solved numerically using a computer. When
H. M. El Misilmani et al.
62
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 giv-
ing 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 man-
ner 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.
2. Familiarizing 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 sa-
tisfying 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 equa-
tions to matrix equations. A deterministic problem is considered, which will be solved by reducing it to a suita-
ble 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
1 10
1
dd d
d
dd
nn
nn
nn
fx fx fx
a a a af x gx
x
xx
+ ++ + =
(1)
In Equation (1), the coefficients
1 10
,, ,,
nn
aa aa
and
( )
g x
are known quantities, and
( )
f x
is the func-
tion whose solution is to be determined. Equation (1) can be written in the form of an operator equation:
(2)
where
L
is the operator equation, operating on
( )
f x
, and given by:
1
1 10
1
dd d
d
dd
nn
nn
nn
La a a a
x
xx
= + ++ +

(3)
The solution of Equation (1) is based on defining the inner product
,fg
, a scalar quantity valid over the
domain of definition of
L
, which is given by:
( ) ( ) ( ) ( )
,d
D
fx gx f xgx x=
(4)
Similarly, we define:
H. M. El Misilmani et al.
63
( )
( )
( ) ( )
( )
( )
,d
D
Lfx gx Lfx gx x=
(5)
The first step in calculating the integral, using Method of Moments, is to expand
f
into a sum of weighted
basis functions
123
,,,fff
in the domain of
L
, as:
( )
nn
n
fx f
α
=
(6)
Testing functions denoted
123
,,,www
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:
( )
( )
, , for 1,2,3,
mn m
n
w Lf w gx m= =
(7)
Expanding Equation (7) over the values of m and
1, 2, 3,n=
, the following matrix equation is then ob-
tained:
( ) ()
( )
( ) ()
( )
( ) ( )
()
11 12 1 11
21 22 2 2 2
12
,, , ,
,, ,
,
,, ,
,
n
n
nm
m m mn
wLf wLf wLf wg
wLf wLf wLf wg
wg
w Lf w Lf w Lf
α
α
α









=











 
(8)
In a simpler form:
[ ][ ] [ ]
mn n m
LG
α
=
(9)
and the solution for the unknown coefficients is then:
[ ] [ ] [ ]
1
n mn m
LG
α
=
(10)
In our calculations, the test function
m
w
is chosen to be equal to the basis function fn, which is known as
Garlekin method. The determination of matrix
[ ]
mn
L
is straightforward, and its inverse is easy to obtain either
analytically or numerically. Once this is done, the
n
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.
2.1. Example 1
Considering the following second order differential equation defined by:
( )
22
2
d
d
fx x
x=
(11)
defined over the domain
[ ]
0,1D=
with the following boundary conditions
( ) ( )
0 10ff= =
. Starting by
choosing the basis function, let us choose:
1n
n
f xx
+
= −
(12)
It is clear from Equation (12) that the chosen basis function meets the boundary conditions and can be consi-
dered as a solution to the problem. Substituting Equation (12) into Equation (6), the left-hand side elements of
Equation (7), which are the elements of the matrix
[ ]
mn
L
, are found to be:
( )
( ) ( )
( )
( )
( )
21
112
0
111
0
d
,d
d
1 d,
.
1
n
m
mn m n
mn
mn
xx
L w Lf x x x
x
x x nn x x
mn
Lnm
+
+
+−

−−

= = − 

=−+
=++
(13)
H. M. El Misilmani et al.
64
In the same manner, we compute the elements of the matrix
[]
m
G
, defined in Equation (7), which are found
to be:
( )
( )
12
,d
44
m
mm
m
G wg xx xx m
+
==−=
+
(14)
Then, we start by choosing
1N=
, for which
1nm= =
in
mn
L
and
m
G
, hence
11
13L=
,
11 20G=
,
13 20
α
=
, and
( )
fx
is given by:
( )
( )
2
3
20
fx xx= −
(15)
It is clear from Equation (15), that the function
()
fx
does not meet the original differential equation de-
fined in (11). Accordingly, we need to increase the value of N.
Let
3N=
, and calculating the values of
mn
L
,
m
G
and
n
α
:
11 12 13 1 1 1
21 22 23 2 2 2
31 32 33 3 3 3
13 12 35 120
; 1 2 4 5 1 1 12
4 5 1 9 7 3 28
LLL g
LLL g
LLL g
αα
αα
αα
  
 
  
 
= =
  
 
  
 
 
  
(16)
Hence,
n
α
could be calculated as:
1
2
3
0
0
1 12
α
α
α


=




(17)
Finally, calculating
( )
fx
:
( )
( )
4
11 2 2 33
1
12
nn
n
fx f f f f xx
α αα α
= =++= −
(18)
The function
( )
fx
given in Equation (18) meets the boundary conditions defined in Equation (11), and ac-
cordingly it is the correct solution of the problem.
2.2. Example 2
Considering the following second order differential equation defined by:
( )
22
2
d82
d
fx xx
x= +
(19)
defined over the domain
[ ]
0,1D=
with the following boundary conditions
( ) ( )
0 10ff= =
. Starting by
choosing the basis function, it was noted that the basis function used in example 1 could also be used here:
1n
n
f xx
+
= −
(20)
The chosen basis functions meet the boundary conditions and can be considered as a solution to the problem.
[ ]
mn
L
is calculated as in Example 1, since the same basis function is used, and it is given by:
1
mn
mn
Lnm
=++
(21)
In the same manner, we compute the elements of the matrix
[ ]
m
G
, defined in Equation (7), which are found
to be:
( )( )
( )
( )( )
12
8 26
, 8 2d 33 4
m
mm
mm
G wg xx x xx mm
+
+
= = +=
++
(22)
Then, we start by choosing
1N=
for which
1nm= =
. Accordingly,
11
13L= −
,
1
17 30G=
,
1
21 30
α
=
,
and
( )
fx
is given by:
H. M. El Misilmani et al.
65
( )
( )
2
21
30
fx xx= −
(23)
It is clear from Equation (23), that the function
()
fx
does not meet the original differential equation de-
fined in Equation (19). Accordingly, we need to increase the value of N.
Let
3N=
, and calculating the values of
,
mn m
LG
, and
n
α
:
11 12 13 1 1 1
21 22 23 2 2 2
31 32 33 3 3 3
1 3 1 2 3 5 17 30
; 1 2 4 5 1 14 15
4 5 1 9 7 25 21
LLL g
LLL g
LLL g
αα
αα
αα
  
 
  
 
= =
  
 
  
 
 
  
(24)
Hence,
n
α
could be calculated as:
1
2
3
0
13
23
α
α
α


=




(25)
Finally, calculating
( )
fx
:
( )
( ) ( )
3
3 44
11 2 2 3 3
12 2
3 3 33
nn
n
x
fx f f f f xx xx x x
α αα α
= =++=+=+
(26)
The function
( )
fx
given in Equation (26) meets the boundary conditions defined in Equation (19), and ac-
cordingly it is the correct solution of the problem.
3. 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] Gibson, W.C. (2008) The Method of Moments in Electromagnetics. Chapman & Hall/CRC, Taylor & Francis Group,
UK.
[2] Harrington, R.F. (1968) Field Computation by Moment Methods. Krieger Publishing Co., Inc., Huntington.
[3] Djordjevic, A.R. and Sarkar, T.K. (1987) A Theorem on the Moment Methods. IEEE Transactions on Antennas and
Propagation, 35, 353-355. http://dx.doi.org/10.1109/TAP.1987.1144097
[4] Newman, E.H. (1998) Simple Examples of the Method of Moments in Electromagnetics. IEEE Transactions on Edu-
cation, 31, 193-200. http://dx.doi.org/10.1109/13.2311
[5] Serteller, N.F.O., Ak, A.G., Kocyigit, G. and Akinci, T.C. (2011) Experimental Study of Moment Method for Under-
graduates in Electromagnetic. Journal of Electronics and Electrical Engineering, 3, 115-118.
Article
Full-text available
A novel, low complexity approach for the analysis of nonuniform lossy substrate‐integrated waveguide transmission lines based on the method of moments is proposed. The approach uses frequency‐dependent basis functions derived from the structure's propagation characteristics. Two tapered structures are analyzed, fabricated, and measured to validate the proposed approach. The analytical results of the proposed approach for both structures are compared to those obtained by measurement and by three‐dimensional field simulation. Excellent agreement is observed between the three sets of results with simulation time savings on more than 98% and memory requirement reduction of more than 97%.
Chapter
In this chapter, the mathematics needed to understand the basic properties of nanophotonic systems is reviewed. These are basic techniques which have been developed for general applications in condensed matter physics and in studies of electrical engineering problems. They are important in nanophotonics as many of the systems of nanophotonics are similar to systems studied in the general physics of material science and in engineering applications. The primary difference being the length scales relevant to the definition of the problems being posed.
Article
Full-text available
The experimental study and Moment Method with is programmed in Mathematica are presented here to facilitate introductory instruction on numerical methods for undergraduate electrical education students. The Method of Moments (MOM) is introduced at the beginning level to prepare students for subsequent advanced topics in complex matrix method and linear vector space theory. The study is carried out with parallel plate capacitor. The different capacitor (dielectric) materials are tried and coincidental program of MOM are studied.
Book
The method of moments is a generic name given to projective methods in which a functional equation in an infinite dimensional function space is approximated by a matrix equation in a finite dimensional subspace. Any projective method can be put into the language and notation of the method of moments, hence the concept is very general. Any linear field problem can be formulated either by differential equations (Maxwell's equations plus boundary conditions) or by integral equations (Green's functions plus superposition). Furthermore, neither the differential formulation nor the integral formulation for any particular problem is unique. The method is applied to electromagnetic scattering from conducting bodies. Computational examples are given for a sphere to illustrate a numerical implementation of the method.
Book
From the Publisher: "An IEEE reprinting of this classic 1968 edition, FIELD COMPUTATION BY MOMENT METHODS is the first book to explore the computation of electromagnetic fields by the most popular method for the numerical solution to electromagnetic field problems. It presents a unified approach to moment methods by employing the concepts of linear spaces and functional analysis. Written especially for those who have a minimal amount of experience in electromagnetic theory, this book illustrates theoretical and mathematical concepts to prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems.Written especially for those who have a minimal amount of experience in electromagnetic theory, theoretical and mathematical concepts are illustrated by examples that prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems."
Article
Three simple examples of the use of the method of moments in electromagnetics, i.e. analysis of the input impedance of a short dipole, and plane-wave scattering from both a short dipole and two coupled short dipoles, are presented. The relative simplicity of the examples is a direct result of obtaining simple expressions for the elements in the method-of-moments impedance matrix
Article
The inner product involved in the moment methods is usually an integral, which is evaluated numerically by summing the integrand at certain discrete points. In connection with this inner product, a theorem is proved, which states that the overall number of points involved in the integration must not be smaller than the number of unknowns involved in the moment method. If these two numbers are equal, a point-matching solution is obtained, irrespective of whether one has started with Galerkin's method or the least squares method. If the number of points involved in the integration is larger than the number of the unknowns, a weighted point-matching solution is obtained.