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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 gx are known

quantities, and fx is the function whose solution is to be

determined. Equation (1) can be written in the form of an

operator equation

Lfxgx, (2)

where is the operator equation, operating on , and given

by

La...

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

〈Lfx,gx〉L

fxgxdx (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

fx∑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,Lf〉

〈w,gx〉 form1,2,3,... (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,

LαG, (9)

and the solution for the unknown coefficients is then

αLG (10)

In our calculations, the test function wm is chosen to be equal to

the basis functionf, 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 D0,1 with the following

boundary conditions f0f10. Starting by choosing

the basis function, let us choose

fxx (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 matrixL, are found to

be

Lmn 〈w,Lf〉xxm+n+

dxdx

xxm+nn+1xdx

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〉xxxdx

(14)

Then, we start by choosing N = 1, for which nm1 in

L and , hence L1/3;G1/20,α3/20, and

f(x) is given by

fx

xx (15)

It is clear from (15), that the function fx 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 fx

fx∑α fαfαfαf

(18)

The function fx 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

8x2 (19)

defined over the domain D0,1 with the following

boundary conditions f0f10. Starting by choosing

the basis function, it was noted that the basis function used in

example 1 could also be used here

fxx (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〉xx8x2dx

(22)

Then, we start by choosing N = 1 for which nm1.

Accordingly, L1/3;G17/30,α21/30, and f(x)

is given by

fx

xx (23)

It is clear from (23), that the function fx 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 fx

fxα

fαfαfαf

1

32

3

2

3

3

(26)

The function fx 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.