 Open Access
 Total Downloads : 370
 Authors : Taiwo O. A, Raji M. T
 Paper ID : IJERTV1IS9208
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Collocation Approximation Methods For The Numerical Solutions of General N^{th} Order Nonlinear IntegroDifferential Equations By Canonical Polynomial
1Taiwo O. A And 2Raji M. T
1Department of Mathematics, University of Ilorin
2Department of Mathematics and Statistics, The Poly., Ibadan
Abstract
In this Paper, a method based on the Tau method by canonical polynomials as the basis function is developed to find the numerical solutions of general nth order nonlinear integrodifferential equations. The differential parts appearing in the equation are used to construct the canonical polynomials and the nonlinear cases are linearized by the Newtons linearization scheme of order n and hence resulted to the use of iteration. Numerical examples are given to illustrate the effectiveness, convergence and the computational cost of the methods.
Introduction
Nonlinear differential equations are used in modelling many real life problems in science and engineering. Nonlinear ordinary differential equations mostly defy closed form solutions because the actual elegant theory valid for their linear counterparts often fails for them. Newtons linearization procedures leading to the use of iteration are commonly employed to facilitate provision of analytic solution.
This paper concerns the development of the Tau numerical method by canonical polynomials as the basis function (see Taiwo [8]) for the solution of nth order integrodifferential equation. The Tau numerical method by Chebyshev polynomials has found extensive application in recent year (see Taiwo and Evans [10], Taiwo [9], Taiwo and Ishola [11]) to mention a few for the case of numerical solution of ordinary differential equations. Applications of the Chebyshev Polynomials as basis function and their merits in solving ordinary differential equations numerically have been discussed by many authors (see Taiwo [7], Asady and Kajani [1], Danfu and Xafeng [3], RahimiArdabili and Shahmorad [4], Tavassoli et al. [12] and Behiry and Mohamed [2]). Many different approaches have also been proposed in the literature to handle integrodifferential equations numerically (Wang and He [6], Zhao and Corless [13], Shahmorad et al. [5]).
This paper is aimed, therefore, to work in this direction of extending the Tau numerical method by canonical polynomial as the basis function for the solution of general nth order integrodifferential equations. Finally, some results are presented to demostrate the efficiency of the new method compared with those results available in Behiry and Mohamed [2].
For the purpose of our discussion, we consider the nonlinear general nthorder ordinary integro differential equation of the form:
Gyn
y n (x) f (x, y(x), y(x)) y(x)
a k(x,t) y(t)dt g(x);
a x b.
(1)
b
together with the linear boundary conditions
A y(a) A
y(a) A y (a) A y(a) A
y n2
1 2
B y(b) B
3 4
y(b) B y (b) B y(b) B
n1
y n2
1 2
C1 y(c) C2
3
y(c) C3
4
y (c) C4
y(c) C
n1
n1
y n2
(2)
D1 y(d ) D2
y(d ) D3
y (d ) D4
y(d ) D
n1
y n2
Here,
A, B,C, D,, , and are constants and
y(x) are unknown functions,
g(x)
and
k(x,t)
are any given smooth function and in this case it can be linear or nonlinear and f is generally nonlinear.
Many numerical techniques have been used successively for equations (1) & (2) and in this section, we discussed in details a straight forward yet generally applicable techniques, the Tau numerical collocation method by canonical polynomial as the basis function. The Newtons scheme from the
Taylors series expansion is represented around xn ,tn , yn in the following form
G y G y G y G y n1 G y n G
y y
y
y n1
y n
n

b kx a
,tn
, yn
x xn
G x
x n
,tn
, yn
t tn
G x
t n
,tn
, yn

y yn
G x
y n
, tn
, yn
y(t)dt g(x) ;
a x b.
(3)
The integral parts of equation (3), where t is an independent variable, y is the dependent variable, are integrated with respect to t to obtain
G y G y G y G y n1 G y n G
n1
n
t kx
n
t
,tn
, yn
yt dt
y y
y
y y n
n
x x
G x
x n
,tn
, yn
t tn
G x
t n
,tn
, yn
y yn
G x
y n
, tn
, yn
nt

t t t
n
dt gx
(4)
Hence, from equation (1), we obtain the following
G
y
f y y,
k
k 1
k
and y j x y j x y j x,
G f
y y
G 1,
yn
j 1,2,3,
y f ,
.
(5)
Thus, substituting equation (5) into equation (4), after simplification, we obtain
y
n n1
(x) y
n1
(x) f

f yn
yy
(x) f yn
yy
n1

fn yn
yn (x)
n1
n

kx , t , y

x x
G x ,t , y b y
t dt
n n n
n y
n
n n n
a n1

kx ,t , y
G x ,t , y
b {t t
) ( y n

y n }y
t dt
n n n
n n n
yn
a n
n1
n n1
gx. (6)
Thus, equation (6) is the linearized form of equation (1).
In order to solve equation (6), we assumed an approximate solution of the form
N
yN ,n1 x aN ,n N ,n x;
n0
a x b,
(7)
where N ,n x,
n 0,1,2,…,tare the canonical basis functions generated below,
aN ,n are the
unknown constants to be determined, N is the degree of the approximant used and n is the number of iteration to be carriedout.
Construction of Canonical Polynomials for nthorder IDEs.
Consider the nthorder integrodifferential equation
P (x) yx P (x) yx P (x) y x P (x) y(x) P (x) y n (x)
b kx,tytdt gx(8)
0 1 2 3
n a
subject to the conditions
ya ya yn1 (a) A yb yb yn1 (b) B
(9)
(10)
Where P1(x); I = 0, 1, 2, . . . n can be variable or constants coefficients.
We define the following operator
d n
d n1 d
D Pn dxn Pn1 dxn1 P1 dx P0 .
(11)
A set of polynomials n x,
n 0,1,2,3…is defined by
n
D x xn .
(12)
Which is uniquely associated with the operated D and, which is obtained recursively as,
n x
1 {xn
P0
P1nn1 (x) P2 nn 1n2 x P3 n(n 1)(n 2)n3 (x)};
n 0.
(13)
Construction of Canonical polynomial for case n = 2
In order to generae the canonical polynomial, we consider the differential part of equation (6) i. e.
d 2 d
L P2 dx2 P1 dx P0
i
L x xi
Lxi P ii 1xi2 Pixi1 P xi
2 1 0
LLi x P2ii 1Li2 x P1iLi1 (x) P0 Li x
xi P ii 1 x Pi (x) P x
2 i2 1 i1 o i
x 1 xi P i (x) P ii 1
i
x,i 0; P 0
(14)
P
1 i1 2
0
i2 0
Thus, equation (14) is the recurrence relation
1
For i = 0:
0 x
P0
For i =1:
1 x
x P10
1
P0
(x)
x P1
P
P
2
0 0
For i = 2:
2 x
1 x2
P0
2P11
(x)
2P2 0
x2
x
P0
2x P1
P
2
0
P2
P
3
2 1
0
P2
P
2
2
0
3 x
1 x3 3P
1
2
P0
(x) 6P2 1
x
For i = 3:
x3 3x 2 P 6xP 2 6P3 6xP
3
(x)
1 1 1 2
P
P
P
P
P
2 3 4 2
0 0 0 0 0
For i = 4:
x 1 x 4 4P (x) 12P
x
4
P
1 3 2 2
0
x 4 4×3 P 12x 2 P 2 24xP 3 24P 4 48xP P 12x 2 P 24P 2 P 24P 2
2
3
4 5
3
1 1 1 1 1 2 2 1 2 2
P
P
P
P
P
P
P
P
P0 0 0 0 0 0
2 4 3
0 0 0
For the case n = 3, we define our operator as:
d 3 d 2 d
L P3 dx3 P2 dx2 P1 dx P0
i
L x xi
Lxi P i(i 1)(i 2)xi3 P ii 1xi2 Pixi1 P xi
3 2 1 0
LLi x P3i(i 1)(i 2)Li3 (x) P2ii 1Li2 x P1iLi1 (x) P0 Li x
xi P i(i 1)(i 2) (x) P ii 1
x Pi (x) P x
3 i3 2
i2
1 i1 o i
x 1 xi P i (x) P ii 1
i
x P i(i 1)(i 2)
(x),i 0; P 0
(15)
P
1 i1 2
0
i2 3
i3 0
Thus, equation (15) is the recurrence relation
1
For i = 0:
0 x
P0
For i =1:
1 x
x P10
1
P0
(x)
x P1
P
P
2
0 0
For i = 2:
2 x
1 x2
P0
2P11
(x)
2P2 0
x2
x
P0
2x P1
P
2
0
P2
P
3
2 1
0
P2
P
2
2
0
3
x 1 x3 3P (x) 6P x 6P
(x)
P
1 2 2 1 3 0
0
For i = 3:
x3 3x 2 P 6xP 2 6P3 6xP 6P
3
(x)
1 1 1 2 3
P
P
P
P
P
P
2 3
0 0 0
4 2 2
0 0 0
For i = 4:
4
x 1 x 4 4P (x) 12P x 24P
(x)
P
1 3 2 2 3 1
0
x 4 4×3 P 12x 2 P 2 24xP 3 24P 4 72P 2 P 48P P 12x 2 P 24xP 24P 2
2
3
1 1 1 1 1 2 1 3 2 3 2
P
P
P
P
P
P
P
P
P
P0 0 0
4 5 4
0 0 0
3 2 2 3
0 0 0 0
For the case n = 4, we define our operator as:
d 4 d 3
d 2 d
L P4 dx4 P3 dx3 P2 dx2 P1 dx P0
i
L x xi
Lxi P i(i 1)(i 2)(i 3)xi4 P i(i 1)(i 2)xi3 P ii 1xi2 Pixi1 P xi
4 3 2 1 0
LLi x P4i(i 1)(i 2)(i 3)Li4 (x) P3i(i 1)(i 2)Li3 (x) P2ii 1Li2 x
P1iLi1 (x) P0 Li x
xi P i(i 1)(i 2)(i 3) (x) P i(i 1)(i 2)
(x) P ii 1
x Pi (x) P x
4 i4 3
i3 2
i2
1 i1 o i
x
i
1 xi P i
(x) P ii 1
x P i(i 1)(i 2)
(x) P i(i 1)(i 2)(i 3)
(x),
P
1 i1 2
0
i2 3
i3
4
i 0; P0 0
(16)
i4
Thus, equation (16) is the recurrence relation
For i = 0:
0 x
1
P0
For i =1:
1 x
x P10
1
P0
(x)
x P1
P
P
2
0 0
For i = 2:
2 x
1 x2
P0
2P11
(x)
2P2 0
x2
x
P0
2x P1
P
2
0
P2
P
3
2 1
0
P2
P
2
2
0
3
x 1 x3 3P (x) 6P x 6P
(x)
P
1 2 2 1 3 0
0
For i = 3:
x3 3x 2 P 6xP 2 6P3 6xP 6P P 6P
3
(x)
1 1 1 2 1 2 3
P
P
P
P
P
P
P
2 3 4 2 3 2
0 0 0 0 0 0 0
For i = 4:
4 x
1 x 4 4P
1
3
P0
(x) 12P2 2
x 24P31
(x)
x 4 4×3 P 12x 2 P 2 24xP 3 24P3 48P P 24xP 24P 12x 2 P 24xP 24P 2 P
[ 1 1 1 1 1 2 2 3 2 1 1 2
P
P
P
P
P
P
P
P
P
P
P0
24P 2
2
0
24xP
3
0
24P P
4 5 4
0 0 0
24P
3 3 2 3 4
0 0 0 0 0
2 3 1 3 4 ]
P
P
P
P
3 2 3 2
0 0 0 0
DESCRIPTION OF METHODS PERTURBED COLLOCATION METHOD
In this section, we discuss the collocation Tau numerical solution for the solutions of the linearized equation (6).
In this method, after the evaluation of the integrals in equation (6), equation (7) is substituted into a slightly perturbed equation (6) to give
y
n
N ,n1
(x) yN n1
(x) f


f yn
y(x)y
(x) f
yN ,n
yn y
N n1
(x) fn
yN n
yN n
(x)
N n1
n
G x , t , y
x x
G x ,t , y b y
t dt
n
y
n n N n
n y
n
n n N n
a
N n1
G x
x n
,tn
, yN n
G x
y n
, tn
, yN n
b t t
n
a
) ( y
N n1

yN n
yN n1
t dt
n n
gx H N x. (17)
together with the boundary conditions
y
n1
N ,n1
y
n1
N ,n1

y

y
N n1 N n1
a
b
(18)
y
n1
N ,n1
( f ) y
N n1
f
Where
N
H N x iTN i1 xand N xis the Chebyshev polynomials of degree N valid in [a, b]
i0
and is defined by,
T x
1 2x b a ,
a x b.
(19)
N CosNCos
b a
The recurrence relation of equation (19) is given as:
2x b a
Tn1 x 2
b a
Tn x Tn1 x.
(20)
The Chebyshev polynomials oscillate with equal amplitude in the range under consideration and this makes the Chebyshev polynomials suitable in function approximation problem.
Thus, equation (17) is collocated at point x xk , hence, we get
y
n
N ,n1
(xk
) yN n1
xk
f



f yn
yxk
yN n1
xk
f
yN ,n
xk
yN
,n1
xk
n

f y , x
y , x G x ,t , y
, x
x
x G x ,t , y , x
n
b y ,
t dt
n N n k
N n1 k
y
n
n n N n k
k n y n n N n
k a N n1
G x
x n
,tn
, yN ,n
xk
G x
y n
, tn
, yN ,n
xk
b t t
n
a


yN ,
n1
yN ,n
yN ,
n1
t dt gxk
H N
xk
n n
(21)
Where for some obvious practical reasons, we have chosen the collocation points to be
x a b ak , k 1,2,3,…, N 1.
k N 2
Thus, we have N 1collocation equations in N 3unknowns a0 , a1 ,…, aN , 1 and
2 constants to be determined.
Other extra equations are obtained from equation (18).
Altogether, we have a total of N 3 algebraic linear system of equations in N 3 unknown constants. The N 3linear algebraic systems of equations are then solved by Gaussian elimination
method to obtain the unknown constants approximate solution given in equation (7).
ai (i 0)
which are then substituted back into the
STANDARD COLLOCATION METHOD
In this section, we discuss the collocation Tau numerical solution for the solutions of the linearized equation (6).
2
N n1
1
n
yn
N n1
yN ,n
n
N n1
n
N n
N n
In this method, after the evaluation of the integrals in equation (6), equation (7) is substituted into the linearized equation (6), to obtain
n
N ,n1
P yn
(x) P y
(x) P f f
y(x)y
(x) f
y (x) y
(x) f y
(x) y
(x)
G x , t , y
x x
G x ,t , y b y
t dt
n
y
n n N n
n y
n
n n N n
a
N n1
G x
x n
,tn
, yN n
G x
y n
, tn
, yN n
b t t
n
a
) ( y
N n1

yN n
yN n1
t dt
n n
gx. (22)
together with the boundary conditions
y
n1
N ,n1
y
n1
N ,n1

y

y
N n1 N n1
a
b
(23)
y
n1
N ,n1
( f ) y
N n1
f
Chebyshev polynomials of degree N valid in [a, b] and recurrence relation generated in equation (19) and (20) as above.
yn
Thus, equation (22) is collocated at point x xk , hence, we get
n
N ,n1
P yn
(x ) P y
x P f f
yx
y
x f
x y ,
x
k
2
N n1
k

f y , x
y , x G x ,t , y
1
n
, x
x
x G x ,t , y , x
n
k
N n1
k
yN ,n
k
N
n1
k
b y ,
t dt
n N n k
N n1 k
y
n
n n N n k
k n y n n N n
k a N n1
G x
x n
,tn
, yN ,n
xk
G x
y n
,tn
, yN ,n
xk
b t t
n
a


yN ,
n1
yN ,n
yN ,
n1
t dt gxk
n n
(24)
Where for some obvious practical reasons, we have chosen the collocation points to be
x a b ak , k 1,2,3,…, N 1.
k N
Thus, we have N collocation equations in N 1unknowns a0 , a1 ,…, aN ) constants to be determined.
Other extra equations are obtained from equation (23).
Altogether, we have a total of N 1
algebraic linear system of equations in N 1
unknown
constants. The N 1linear algebraic systems of equations are then solved by Gaussian elimination
method to obtain the unknown constants approximate solution given in equation (7).
ai (i 0)
which are then substituted back into the
Remark: All the above procedures have been automated by the use of symbolic algebraic program MATLAB 7.9 and no manual computation is required at any stage.
Error Estimation
In this section, we perform the estimating error for the IntegroDifferential Equations. Let us call
en (s) y(s) yN (s) the error function of the Tau approximation yN (s) to y(s).
where y(s) is the exact solution of
b
Dy(s) k(s, t)y(t)dt
a
f (s)
s [a,b],
(25)
together with the condition
d
n
[c1y (k i) (a) c2 y(k i) (b)] d ,
j 1,, n .
k 1
jk n
jk n j d
(26)
Therefore,
yN (s) satisfies the problem
b
Dyn (s) k (s, t) yn (t)dt
a
f (s) H n (s),
s [a, b],
(27)
together with the condition
d
n
[c1y (k i) (a) c2 y(k i) (b)] d ,
j 1,, n .
(28)
k 1
jk n
jk n j d
Hn(s) is a perturbation term associated with yN(s) and can be obtained by substituting yN(s) into the equation
b
H n (s) Dyn (s) k(s,t) yn (t)dt f (s).
a
We proceed to find an approximation en,N (s) to the eN (s) in the same way as we did before for the solution of problem (6). Subtracting equations (27) and (28) from (25) and (26), respectively, the error equation with the homogeneous condition is followed:
b
Den (s) k(s, t)en (t)dt H n (s),
a
s [a,b] , (29)
together with the condition
d
n
[c1y (k i) (a) c2 y(k i) (b)] d ,
j 1,, n .
(30)
k 1
jk n
jk n j d
and solving this problem in the same way, we get the approximation
en,N (s) . It should be noted
that in order to construct the Tau approximation
en,N (s)
to en, (s) , only the right hand side of
system (29) needs to be recomputed, the structure of the coefficient matrix Gn remains the same.
Numerical Examples
Numerical Experiments and Discussion
In this section, we present numerical results obtained with that obtained by Behiry and Mohammed [2], that considered these problems stated below as test problems and the problems are of orders 5, 6 and 8 nonlinear integrodifferential equations. We present tables of exact solutions, results of methods used and the results obtained by Behiry and Mohammed [2] for different values of the approximants.
Example 1: Consider the nonlinear integrodifferential equation.
3 (5)
4 3 x
(e2 1) x 1
2
x y (x) 2 y (x) xy(x) x
5x 4e
4 x 2 ty(t)dt xy
(t)dt, 0 x 1
0 0
together with the following initial conditions.
y0 0,
y0 1, y (0) 2, y(0) 3 and
y 4 (0) 4 .
The exact solution is given as
yx xe x . For favorable comparison, we have chosen our initial
guess
yN , k
xe x .Here k is the number of iterations in the new method and N is the degree of
approximant used.
X 
Exact value 
Standard Collocation method 
Perturbed Colocation Method 
Result Obtained by Behiry and Mohamed [2] 
0.0 
0.00000000 
0.00000000 
0.00000000 
0.00000000 
0.1 
0.1105170918 
0.1105171045 
0.1105170820 
0.1105170918 
0.2 
0.2442805516 
0.2442806261 
0.2442805427 
0.2442805516 
0.3 
0.4049576423 
0.4049577205 
0.4049576451 
0.4049576423 
0.4 
0.5967298791 
0.5967299147 
0.5967298684 
0.5967298791 
0.5 
0.8243606354 
0.8243607764 
0.8243606246 
0.8243606354 
0.6 
1.0932712800 
1.0932738510 
1.093272540 
1.0932712800 
0.7 
1.409626895 
1.409627468 
1.409626860 
1.409626895 
0.8 
1.780432743 
1.780433065 
1.780432654 
1.780432743 
0.9 
2.213642800 
2.213644521 
2.213643411 
2.213642800 
1.0 
2.718281828 
2.718281831 
2.718281825 
2.718281828 
Table 1b: Table of error for example 1
X 
Standard Collocation method 
Perturbed Collocation method 
Result Obtained by Behiry and Mohamed [2] 
0 
0.0000000E+00 
0.0000000E+00 
0.0000000E+00 
0.1 
1.2700000E08 
9.8000000E09 
1.2700000E08 
0.2 
7.4500000E08 
8.9000000E09 
7.4500000E08 
0.3 
7.8200000E08 
2.8000000E09 
7.8200000E08 
0.4 
3.5600000E08 
1.0700000E08 
3.5600000E08 
0.5 
1.4100000E07 
1.0800000E08 
1.4100000E07 
0.6 
2.5710000E06 
1.2600000E06 
2.5710000E06 
0.7 
5.7300000E07 
3.5000000E08 
5.7300000E07 
0.8 
3.2200000E07 
8.9000000E08 
3.2200000E07 
0.9 
1.7210000E06 
6.1100000E07 
1.7210000E06 
1.0 
3.0000002E09 
2.9999998E09 
3.0000002E09 
Example 2: Consider the nonlinear integrodifferential equation
x4 y (6) (x) y (3) (x) y(x) x4 cos x 0.5sin 2x 3x 0.4
x 1
0.1e{[cos(1) sin(1)][cos 2 (1) 3e]} 2[1 y 2 (t)dt et y3 (t)dt
0 0
together with the following initial conditions.
y0 1,
y0 0, y (0) 1,
y(0) 0,
y(4) (0) 1 y(5) (0) 0 .
and
The exact solution is
y(x) Cosx. the results of applying above methods with initial guess
yN , k
1 x are given as follows ( k denotes the number of iterations in new method and N the
degree of approximant used).
X 
Exact value 
Standard Collocation method 
Perturbed Collocation Method 
Result Obtained by Behiry and Mohamed [2] 
0.0 
1.00000000 
1.00000000 
1.00000000 
1.00000000 
0.1 
0.9950041653 
0.9950042014 
0.9950041476 
0.9950041653 
0.2 
0.9800665778 
0.9800666500 
0.9800665694 
0.9800665778 
0.3 
0.9553364891 
0.9553365047 
0.9553364781 
0.9553364891 
0.4 
0.921060994 
0.921061086 
0.921060852 
0.921060994 
0.5 
0.8775825619 
0.8775826457 
0.8775825569 
0.8775825619 
0.6 
0.8253356149 
0.8253357162 
0.8253356046 
0.8253356149 
0.7 
0.7648421873 
0.7648422641 
0.7648421764 
0.7648421873 
0.8 
0.6967067093 
0.6967067270 
0.6967066982 
0.6967067093 
0.9 
0.6216099683 
0.6216010098 
0.6216099512 
0.6216099683 
1.0 
0.5403023059 
0.5403023562 
0.5403023056 
0.5403023059 
Table 2b: Table of error for example 2
X 
Standard Collocation method 
Perturbed Collocation method 
Result Obtained by Behiry and Mohamed [2] 
0 
0.0000000E+00 
0.0000000E+00 
0.0000000E+00 
0.1 
3.6100000E08 
1.7700000E08 
3.6100000E08 
0.2 
7.2200000E08 
8.3999999E09 
7.2200000E08 
0.3 
1.5600000E08 
1.1000000E08 
1.5600000E08 
0.4 
9.2000000E08 
1.4200000E07 
9.2000000E08 
0.5 
8.3800000E08 
5.0000000E09 
8.3800000E08 
0.6 
1.0130000E07 
1.0300000E08 
1.0130000E07 
0.7 
7.6800000E08 
1.0900000E08 
7.6800000E08 
0.8 
1.7700000E08 
1.1100000E08 
1.7700000E08 
0.9 
8.9585000E06 
1.7100000E08 
8.9585000E06 
1.0 
5.0300002E08 
2.9999991E10 
5.0300000E08 
Example 3. Consider the nonlinear VolterraFredholm integrodifferential equation
2
x
(8) 8 x 2
sin(2x) 1
0
y (x)
y(x) y
0
(t)dt
2
cos(

y(t) y(t)dt,
0 x 1
With the initial conditions
y(0) 0,
y(0) ,
y (0) 0,
y (3) (0) 3 ,
y (4) (0) 0,
y (5) (0) 5 ,
y (6) (0) 0
and
y (7) (0) 7 .
The exact solution is
y(x) sin(x).
the results of applying above methods with initial guess
yN , k
sin(2x) are given as follows ( k denotes the number of iterations in new method and N the
2
degree of approximant used).
Table 3a: Table of solution for example 3k 5, N 5
X
Exact value
Standard Collocation method
Perturbed Collocation Method
Result Obtained by Behiry and
Mohamed [2]
0.0
/td>
0.00000000
0.000000000
0.000000000
0.000000000
0.1
0.3090169944
0.3090169992
0.3090169849
0.3090169944
0.2
0.5877852523
0.5877852684
0.5877852498
0.5877852523
0.3
0.8090169944
0.8090169989
0.8090169870
0.8090169944
0.4
0.9510565163
0.9510565248
0.9510565041
0.9510565163
0.5
1.0000000000
1.0000000000
1.0000000000
1.0000000000
0.6
0.9510565163
0.9510565248
0.9510565041
0.9510565163
0.7
0.8090169944
0.8090169989
0.8090169870
0.8090169944
0.8
0.5877852523
0.5877852684
0.5877852498
0.5877852523
0.9
0.3090169944
0.3090169992
0.3090169849
0.3090169944
1.0
0.0000000000
0.0000000000
0.0000000000
0.0000000000
Table 3b: Table of error for example 3
X
Standard Collocation method
Perturbed Collocation
method
Result Obtained by Behiry
and Mohamed [2]
0
0.0000000E+00
0.0000000E+00
0.0000000E+00
0.1
4.8000000E09
9.5000000E09
4.8000000E09
0.2
1.6100000E08
2.5000000E09
1.6100000E08
0.3
4.5000000E09
7.3999999E09
4.5000000E09
0.4
8.5000000E09
1.2200000E08
8.5000000E09
0.5
0.0000000E+00
0.0000000E+00
0.0000000E+00
0.6
8.5000000E09
1.2200000E08
8.5000000E09
0.7
4.5000000E09
7.3999999E09
4.5000000E09
0.8
1.6100000E08
2.5000000E09
1.6100000E08
0.9
4.8000000E09
9.5000000E09
4.8000000E09
1.0
0.0000000E+00
0.0000000E+00
0.0000000E+00
Conclusion
Higherorder nonlinear integrodifferential equations are usually difficult to solve analytically. In many cases, it is required to obtain the approximate solutions. For this purpose, the presented methods can be proposed. A considerable advantage of the methods is achieved as different approximate solutions are obtained by different values of N. Furthermore, after calculation of the approximate solutions, the approximate solution yN(x) can be easily evaluated for arbitrary values of x at low computation effort.
To get the best approximating solution of the equation, N (the degree of the approximating polynomial) must be chosen large enough. From the tabular points shown in Table 1, it is observed that the solution found for N=10 shows close agreement for various values of x. In particular, the solution of example 3, for N=10 shows a very close approximation to the analytical solution at the points in interval 0 x 1. An interesting feature of the Standard and Perturbed collocation methods is that we get an analytical solution in many cases, as demonstrated in examples 1, 2 and 3.
Reference

Asady, B. and Kajani M. T. (2005); Direct method for solving integrodifferential equations using hybrid Fourier and blockpulse functions, Int. J. Comput. Math. 82, 889895

Behiry, S. H. and Mohamed S. I.(2012); Solving highorder nonlinear Volterra Fredholm integrodifferential equations by differential transform method, Natural Science Vol. 4,(8), 581587

Han Danfu and Shang Xufeng(2007); Numerical solution of integro differential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comput, 194(2), 460466.

RahimiArdabili, M. Y. and Shahmorad, S.(2007); Iterative numerical solution of nonlinear integrodifferential equations by Tau method, Appl. Math. Comput. 193(2), 514522.

Shahmorad, S. Ivaz, K. Mostahkam, B. S.(2009); NewtonTau numerical solution of one dimensional nonlinear integrodifferential equations, Southeast Asian Bull. Math. 33(4), 733740.

ShuQiang Wang and JiHuan He(2007); Variational iteration method for solving integro differential equations Phys. Lett. A 367(3), 188191.

Taiwo, O. A. (2004);The application of cubic spline collocation Tau methods for the solution of second order nonlinear boundary value problems, J. Nigerian Math. Soc. 23, 2740

Taiwo, O. A.(2005); Comparison of collocation methods for the solution of second order non linear boundary value problems, Int. J. Comput. Math. 82(11), 13891401.

Taiwo, O. A. (2007); Comparison of two collocation methods for the solution of integral eqautions by cubic spline, Journal of Mathematical Association of Nigeria, (ABACUS) 34(2B), 289294.

Taiwo, O. A. and Evans, D. J.(1997); Collocation approximation for fourthorder boundary value problems, Int. J. Comput. Math. 63(12), 5766.

Taiwo O. A. and Ishola C. Y.(2009); Collocation approximation methods for the numerical solution of integral equations, Int. J. Pure Appl. Sci. 2, 2937.

Tavassoli Kajani, M.,Ghasemi, M. and Babolian, E.(2006); Numerical solution of linear integrodifferential equation by using sinecosine wavelets, appl. Math. Comput. 180(2), 569574.

Zhao, J. and Corless R. M. (2006); Compact finite difference method for integrodifferential equations, Appl. Math. Comput. 177(1), 271288.
