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Open Journal of Statistics, 2012, 2, 352-355
http://dx.doi.org/10.4236/ojs.2012.23043 Published Online July 2012 (http://www.SciRP.org/journal/ojs)
Numerical Solution of Integro-Differential Equations
with Local Polynomial Regression
Liyun Su1*, Tianshun Yan1, Yanyong Zhao1, Fenglan Li2, Ruihua Liu1
1School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, China
2Library, Chongqing University of Technology, Chongqing, China
Email: *[email protected]
Received April 18, 2012; revised May 20, 2012; accepted June 3, 2012
ABSTRACT
In this paper, we try to find numerical solution of
 
b
d, .
a
yxpxyxgxKxtyttyaax ba tb


 
d,. ,
a
yxpxyxgxKxtyttyaax ba tb


 
 

dxtytt
ya


a
or
 
x
by using Local polynomial regression (LPR) method. The numerical solution shows that this method is powerful in
solving integro-differential equations. The method will be tested on three model problems in order to demonstrate its
usefulness and accuracy.
Keywords: Integro-Differential Equations; Local Polynomial Regression; Kernel Functions
1. Introduction
In recent years, there has been a growing interest in the
Integro-Differential Equations (IDEs) which are a com-
bination of differential and Fredholm-Volterra integral
equations. IDEs play an important role in many branches
of linear and nonlinear functional analysis and their ap-
plications in the theory of engineering, mechanics, phys-
ics, chemistry, astronomy, biology, economics, potential
theory and electrostatics. The mentioned integro-differ-
ential equations are usually difficult to solve analytically,
so a numerical method is required. Many different meth-
ods are used to obtain the solution of the linear and non-
linear IDEs such as the successive approximations, A
domain decomposition, Homotopy perturbation method,
Chebyshev and Taylor collocation, Haar Wavelet, Tau
and Walsh series methods [1-8]. Recently, the authors [9],
have used local polynomial regression (LPR) method for
the numerical solution of linear and non-linear Fred-
holm and Volterra integral equations.
In this paper, we consider the linear IDEs,
 
x
a
yxpxyx gxK

(1)
where the upper limit of the integral is constant or vari-
able,
 
are constants,
g
xpx and the kernel

K
xt
y
are given functions, whereas x
0
needs to
be determined. The subject of this paper is to try to find
numerical solutions of integro-differential equations by
means of local polynomial regression method which is
presented firstly by Hikmat Caglar [9]. Finally, we show
the method to achieve the desired accuracy. Details of
the structure of the present method are explained in sec-
tions. We apply LPR method for IDEs. In Section 3, it’s
proved the efficiency of numerical method. Finally, Sec-
tion 4 contains some conclusions and directions for fu-
ture expectations and researches.
2. Numerical Method
In this section, we describe local polynomial regression
method to find the approximating solution of Equation
(1). The following is the mathematical formulation of the
local polynomial regression.
2.1. Local Polynomial Regression
First, we introduce the mathematical thoughts of local
polynomial regression. This idea was mentioned in [10-
14]. Since the form of regression function is not specified,
so the data points with long distance from
x
provide
*Corresponding author.
C
opyright © 2012 SciRes. OJS
L. Y. SU ET AL. 353
little information to 0

y
x. Therefore, we can only use
the local data points around 0
x
. We suppose that
y
x
has p + 1 derivative at 0
x
, for point
x
, located in the
neighborhood of this point 0
x
, we can use the p-order
multivariate polynomials to locally approximate
y
x,
and the surrounding local point of 0
x
, so we model

y
x
as:

.
0
0
p
j
j
j
yx
x
x
(2)
where parameter
j
depends on 0
x
so called local
pa-rameter. Obviously, the local parameter
j fits the local model with local data and
it can be minimized,


0
j
yx!j

2
0
j
ij
10
p
ij
ni
i
X
x
h
Y


K

Xx

 (3)
where controls the size of the bandwidth of local area.
Using matrix notation to represent the local polynomial
regression is more convenient. Below is the design ma-
trix corresponding to (3) with
h
X
and Y:


10
00
p
p
x
XY




,X
1
2
n
Y
Y
Y











WY
10
nn
Xx X
Xx Xx

1
1
. (4)


min
The weighted least squares problem (3) can be written
as
T
YX


0
,
n
x




1T
where,
10
K x
hh
KXWdiag X
so the solution vector can be written as
.
T
X
WY

0
ˆ
XW
X
(5)
Furthermore, we can get the estimation
y
x

1,
,

yx
01
E
ˆTT
X
WXX WY
where 1 is a column vector ( the same size of E
)
with the first element equal to 1, and the rest equal to
zero, that is,

111p

10 0E
. The selection of
does not influence the results much. We selected the
quadratic kernel as follows:
K


2
3if
4
0otherwise
u
1
1u
Ku




0
0
.
p
(6)
2.2. Illustration of Numerical Method
In this section, the LPR method for solving Equation (1)
is outlined. Let Equation (2) be an approximate solution
of IDEs (1):
j
j
j
yx
x
x
(7)
where, 12 n
X
aXX b

i
and it is required that
the approximate solution (7) satisfies the IDEs at the
point
x
X
. Putting (7) in (1), it follows that






1
00
10
0
0
d
pp
j
j
jj
jj
pxj
a
j
jpx
x
xxx
Kxtt gxtx




 









10
1
1
0
0
0
1,,0,,,
2, ,,1,,
,
,d
i
j
ji
j
ij i
j
ijii
Xj
ij i
a
ii
iaj pXx
yya
inaj jpXx
bpX
Xx
cKXtttx
ygX
 
 

 

10 111
2020 2021 21 21222
30 303031 3131333
000 111
p
ppp
ppp
nnnn n nnpnpnp
X
aa a
abc abcabc
abc abcabc
abcabc abc
  
(8)
This leads to the system
(9)
Consequently, the matrix form (4) can be written as
follows by using expression (9).
 
 
 


1
2
1n
n
y
y
Y
y
y
(10)
(11)
Putting expression (10) and expression (11) in Equa-
tion (5), then estimated set of coefficients i
are ob-
tained by solving matrix system solution. Therefore, ap-
proximate solution (7) can be obtained.
3. Simulation and Analysis
In this section, we consider some examples of IDEs. To
show the efficiency of the present method for our prob-
lem in comparison with the exact solution, we report
Copyright © 2012 SciRes. OJS
L. Y. SU ET AL.
354
absolute error which is defined by:
L
PR
Ey exact LPR
y y
,
where
L
PR is absolute error.
Ey
L
PR is LPR solution.
exact is exact solution. Calculations were all performed
by using MATLAB 7.0.
y
y
Example 1: First we consider the integro-differential
equation:



1
33
0
1
321
3
01,
x
yx eex
y
 

3dxtyt t

3
For which the exact solution is
x
yx e.
Some numerical results of these solutions are shown in
Table 1. We solve example 1 with n = 20, 30, 50 by
choosing p = 3 and various values of parameters h pre-
sented in Table 1.
L
PR gets up to value 0 which is
very accurate at point x = 0 given h = 0.04, n = 50.
Ey
L
PR also gets up to value 1.75 × 10−7 which is very
small at point x = 0.0 given h = 0.08, n = 20. Moreover,
it’s showed small absolute error at other point
Ey
x
given
different parameters and . More importantly, the
tabulated results indicate that the absolute errors present
decreases more rapidly when parameter n increases.
hn
Example 2: Consider the FIDE:
 

ln 1
d

1
20
11
21
1
21
ln
00,
y
xyx x
x
t
y

x
x
yt t


 
ln 1
For which the exact solution is
y
xx.
Some numerical results of these solutions are revealed
in Table 2. We solve example 2 with n = 20, 30, 50 by
choosing p = 3 and various values of parameters h pre-
sented in Table 2.
L
PR achieves value 0 which is so
accurate at point x = 0 given h = 0.08, n = 20.
Ey
L
PR
gets up to value which is very small at
point x = 0.0 given h = 0.04, n = 50. It’s also represented
small absolute errors at other point
Ey
12
287 10

x
given kinds of
parameters h and n. Further, the tabulated results indicate
that the absolute errors reduce rapidly when parameter n
increases approximately.
Example 3: At last, we consider the FIDE:
 
 

1
0
cos 2π2πsin 2π
1sin 4πsin 4π2
2
01,
yx yxx

πd
x
x
xtytt
 
ln 1
y


For which the exact solution is
y
Table 1. Absolute errors at point x with p = 3, different h,
Example 1.
xx. Some
numerical results of these solutions are also shown in
Table 3 which is similar to Tables 1 and 2.
x
008, 20
LPR
Ey
hn 0 055,30
LPR
Ey
hn 004, 50
LPR
Ey
hn 

0.0 7
175 10
 10
501 10
 0
0.2 5
391 10
 6
52810
 10
511 10

4
812 10
0.4
 6
118 10
 10
887 10

5
211 10
0.6
 6
622 10
 11
521 10

5
732 10
0.8
 5
35810
 9
364 10

4
659 10
1.0
 5
883 10
 9
639 10

Table 2. Absolute errors at point x with p = 3, different h,
Example 2.
x
008, 20
LPR
Ey
hn
 0 055,30
LPR
Ey
hn 004, 50
LPR
Ey
hn


0.00 9
349 10
 12
287 10

0.2 4
512 10
 5
487 10
 9
934 10

5
832 10
0.4
 5
172 10
 8
611 10

5
674 10
0.6
 6
596 10
 10
827 10

5
512 10
0.8
 6
412 10
 10
951 10

3
35510
1.0
 5
88510
 9
306 10

Table 3. Absolute errors at point x with p = 3, different h,
Example 3.
x
008, 20
LPR
Ey
hn
 0 055,30
LPR
Ey
hn 004, 50
LPR
Ey
hn


0.00 0 13
175 10

0.2 5
52310
 6
719 10
 10
287 10

5
187 10
0.4
 5
871 10
 10
876 10

4
562 10
0.6
 6
732 10
 11
93310

5
911 10
0.8
 6
158 10
 9
31810

5
496 10
1.0
 5
219 10
 10
682 10

In Table 3, some numerical results of these solutions
are also opened up. We solve example 2 with n = 20, 30,
50 by choosing p = 3 and different values of parameters h
presented in Table 3.
L
PR achieves value 0 at point
x = 0 given h = 0.08, n = 20. The equivalent result applys
to h = 0.0055, n = 30.
Ey
L
PR gets up to value Ey
13
175 10
 which is very small at point x = 0.0 given h
= 0.04, n = 50. It’s also showed that small absolute errors
at other point
x
given different parameters h and n.
From Table 3, we can conclude that the absolute errors
reduce approximately when parameter n increases.
Copyright © 2012 SciRes. OJS
L. Y. SU ET AL.
Copyright © 2012 SciRes. OJS
355
4. Conclusion
In this paper, we make use of LPR method to solve the
linear integro-differential equations. It’s showed that this
method is very convergent for solving linear integro-dif-
ferential equations. Moreover, the numerical results ap-
proximate the exact solution very well. The Method can
be extended to different parameters p, h and kinds of
kernel functions. LPR method can also solve nonlinear or
integro-differential equations which can be researched
and resolved.
5. Acknowledgements
This work was in part supported by Chongqing CSTC
foundations of China (Grant No. CSTC, 2010BB2310,
Grant No. CSTC, 2011jjA40033), Chongqing CMEC
foundations of China (Grant No. KJ080614, Grant No.
KJ100810, Grant No. KJ100818, KJ120829).
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