Progress In Electromagnetics Research B, Vol. 8, 59–76, 2008
NEW DIRECT METHOD TO SOLVE NONLINEAR
VOLTERRA-FREDHOLM INTEGRAL AND
INTEGRO-DIFFERENTIAL EQUATIONS USING
OPERATIONAL MATRIX WITH BLOCK-PULSE
FUNCTIONS
E. Babolian
Department of Mathematics
Tarbiat Moallem University
599 Taleghani Avenue, Tehran 15618, Iran
Z. Masouri
Department of Mathematics
Islamic Azad University
Science and Research Branch, Tehran, Iran
S. Hatamzadeh-Varmazyar
Department of Electrical Engineering
Islamic Azad University
Science and Research Branch, Tehran, Iran
Abstract—A new and effective direct method to determine the
numerical solution of specific nonlinear Volterra-Fredholm integral and
integro-differential equations is proposed. The method is based on
vector forms of block-pulse functions (BPFs). By using BPFs and
its operational matrix of integration, an integral or integro-differential
equation can be transformed to a nonlinear system of algebraic
equations. Some numerical examples are provided to illustrate
accuracy and computational efficiency of the method. Finally, the error
evaluation of this method is presented. The benefits of this method are
low cost of setting up the equations without applying any projection
method such as Galerkin, collocation, ... . Also, the nonlinear system
of algebraic equations is sparse.
60 Babolian, Masouri, and Hatamzadeh-Varmazyar
1. INTRODUCTION
Over several decades, numerical methods in Electromagnetics have
been the subject of extensive researches [1–11]. On the other hand,
many problems in Electromagnetics can be modeled by integral and
integro-differential equations (see [12–21]); for example, electric field
integral equation (EFIE) and magnetic field integral equation (MFIE).
In recent years, several numerical methods for solving linear and
nonlinear integro-differential equations have been presented. Some
authors use decomposition method [22, 23]. In most methods, a set of
basis functions and an appropriate projection method such as Galerkin,
collocation, ... or a direct method have been applied [24–29]. These
methods often transform an integral or integro-differential equation to
a linear or nonlinear system of algebraic equations which can be solved
by direct or iterative methods. In general, generating this system needs
calculation of a large number of integrations.
This paper considers specific cases of Volterra-Fredholm integral
and integro-differential equations of the forms
x(s)+λ
1
s
0
k
1
(s, t)F (x(t)) dt+λ
2
1
0
k
2
(s, t)G(x(t)) dt = y(s),
0 s<1,
(1)
and
x
(s)+q(s)x(s)+λ
1
s
0
k
1
(s, t)F (x(t)) dt
+ λ
2
1
0
k
2
(s, t) G(x(t)) dt = y(s),
x(0) = x
0
, 0 s<1,
(2)
where the functions F (x(t)) and G(x(t)) are polynomials of x(t) with
constant coefficients. For convenience, we put F (x(t))=[x(t)]
n
1
and
G(x(t))=[x(t)]
n
2
where, n
1
,n
2
are positive integers. Note that the
method presented in this article can be easily extended and applied to
any nonlinear integral and integro-differential equations of the forms
Eqs. (1) and (2). It is clear that for n
1
,n
2
= 1, Eqs. (1) and (2)
are linear integral and integro-differential equations respectively. Also,
without loss of generality, it is supposed that the interval of integration
is [0, 1), since any finite interval [a, b) can be transformed to interval
[0, 1) by linear maps [26].
For solving these equations, this paper uses the orthogonal block-
pulse functions (BPFs). By using vector forms of BPFs and its
Progress In Electromagnetics Research B, Vol. 8, 2008 61
operational matrix of integration, Eqs. (1) and (2) can be easily
reduced to a nonlinear system of algebraic equations.
Finally, we apply the proposed method on some examples to show
its accuracy and efficiency. Also, the error evaluation of this method
is presented.
2. REVIEW OF BLOCK-PULSE FUNCTIONS
Block-pulse functions are studied by many authors and applied for
solving different problems; for example, see [30, 31].
2.1. Definition
An m-set of block-pulse functions (BPFs) is defined over the interval
[0,T)as
φ
i
(t)=
1,
iT
m
t<
(i +1)T
m
,
0, otherwise,
(3)
where, i =0, 1,... ,m− 1, with a positive integer value for m. Also,
consider h = T/m, and φ
i
is the ith block-pulse function.
In this paper, it is assumed that T = 1, so BPFs are defined over
[0, 1), and h =1/m.
There are some properties for BPFs, the most important
properties are disjointness, orthogonality, and completeness.
The disjointness property can be clearly obtained from the
definition of BPFs:
φ
i
(t)φ
j
(t)=
φ
i
(t),i= j,
0,i= j,
(4)
where i, j =0, 1,... ,m− 1.
The other property is orthogonality. It is clear that
1
0
φ
i
(t)φ
j
(t)dt = hδ
ij
, (5)
where, δ
ij
is the Kronecker delta.
The third property is completeness. For every f ∈L
2
([0, 1)), when
m approaches to the infinity, Parseval’s identity holds:
1
0
f
2
(t)dt =
∞
i=0
f
2
i
φ
i
(t)
2
, (6)
62 Babolian, Masouri, and Hatamzadeh-Varmazyar
where,
f
i
=
1
h
1
0
f(t) φ
i
(t)dt. (7)
2.2. Vector Forms
Consider the m terms of BPFs and write them concisely as m-vector:
Φ(t)=[φ
0
(t),φ
1
(t),... ,φ
m−1
(t)]
T
,t∈ [0, 1). (8)
Above representation and disjointness property, follows:
Φ(t)Φ
T
(t)=
φ
0
(t)0... 0
0 φ
1
(t) ... 0
.
.
.
.
.
.
.
.
.
.
.
.
00... φ
m−1
(t)
, (9)
1
0
Φ(t)Φ
T
(t)dt =
h 0 ... 0
0 h ... 0
.
.
.
.
.
.
.
.
.
.
.
.
00... h
= hI
m×m
= D, (10)
Φ
T
(t)Φ(t)=1, (11)
Φ(t)Φ
T
(t)V =
˜
V Φ(t), (12)
where, V is an m-vector and
˜
V = diag(V ). Moreover, it can be clearly
concluded that for every m × m matrix B:
Φ
T
(t)BΦ(t)=
ˆ
B
T
Φ(t), (13)
where,
ˆ
B is an m-vector with elements equal to the diagonal entries of
matrix B.
2.3. BPFs Expansion
The expansion of a function f(t) over [0, 1), with respect to φ
i
(t),
i =0, 1,... ,m− 1 may be compactly written as
f(t)
m−1
i=0
f
i
φ
i
(t)=F
T
Φ(t)=Φ
T
(t)F, (14)
Progress In Electromagnetics Research B, Vol. 8, 2008 63
where, F =[f
0
,f
1
,...,f
m−1
]
T
and f
i
is defined by (7).
Now, assume k(t, s) is a function of two variables in L
2
([0, 1) ×
[0, 1)). It can be similarly expanded with respect to BPFs as
k(t, s) Φ
T
(t)KΨ(s), (15)
where, Φ(t) and Ψ(s) are m
1
and m
2
dimensional BPF vectors
respectively, and K is the m
1
× m
2
block-pulse coefficient matrix with
k
ij
, i =0, 1,...,m
1
− 1, j =0, 1,...,m
2
− 1 as follows:
k
ij
= m
1
m
2
1
0
1
0
k(t, s) φ
i
(t) ψ
j
(s)dtds. (16)
For convenience, we put m
1
= m
2
.
2.4. Operational Matrix
Computing
t
0
φ
i
(τ)dτ follows:
t
0
φ
i
(τ)dτ =
0, t < ih,
t − ih, ih t<(i +1)h,
h, (i +1)h t<1.
(17)
Note that t − ih, equals to h/2, at mid-point of [ih, (i +1)h]. So, we
can approximate t − ih, for ih t<(i +1)h,byh/2.
Now, expressing
t
0
φ
i
(τ)dτ, in terms of the BPFs follows:
t
0
φ
i
(τ)dτ
0,...,0,
h
2
,h,...,h
Φ(t), (18)
in which h/2, is ith component. Therefore,
t
0
Φ(τ)dτ P Φ(t), (19)
where, P
m×m
is called operational matrix of integration and can be
represented as
P =
h
2
122... 2
012... 2
001... 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
000... 1
. (20)
So, the integral of every function f can be approximated as follows:
t
0
f(τ)dτ
t
0
F
T
Φ(τ)dτ F
T
P Φ(t). (21)
