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1. Research Inventy: International Journal Of Engineering And Science
Vol.4, Issue 1 (January 2014), PP 46-51
Issn(e): 2278-4721, Issn(p):2319-6483, www.researchinventy.com
Numerical Solution of Second Order Nonlinear FredholmVolterra Integro Differential Equations by Canonical Basis
Function
1,
Taiwo, O . A, Ganiyu , 2,K. A and  Okperhie, E. P
Department Of Mathematics, University Of Ilorin, Nigeria

Department Of Mathematics And Statistics, Federal Polytechnic, Bida, Nigeria
ABSTRACT : This paper deals with the construction of Canonical Polynomials basis function and used to find
approximation solutions of Second order nonlinear Fredholm – Volterra Integro Differential Equations. The
solutions obtained are compared favorably with the solutions obtained by Cerdik-Yaslan et. al [1]. One of the
advantages of the method discussed is that solution is expressed as a truncated Canonicals series, then both the
exact and the approximate solutions are easily evaluated for arbitrary values of x in the intervals of
consideration to obtain numerical values for both solutions. Finally, some examples of second order nonlinear
Fredholm – Volterra Integro Differential Equations are presented to illustrate the method.
KEY WORDS: Canonical Polynomials, Nonlinear Fredholm – Volterra integro Differential
Equations,
Approximate solution
I.
INTRODUCTION
In recent years, there has been a growing interest in the Integro Differential Equations (IDEs). IDEs
play an important role in many branches of linear and nonlinear functional analysis and their applications in the
theory of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and
electrostatics. Higher order integral differential equations arise in mathematical, applied and engineering
sciences, astrophysics, solid state physics, astronomy, fluid dynamics, beam theory etc. to mention a few are
usually very difficulty to solve analytically, so numerical method is required. Variational Iteration Method
(VIM) is a simple and yet powerful method for solving a wide class of nonlinear problems, first envisioned by
He [ 2,3,4,5]. VIM has successfully been applied to many situations. For example, He [4] solved the classical
Blasius equation using VIM. He [2] used VIM to give approximate solutions for some well known nonlinear
problems. He [3] used VIM to solve the well known Blasius equation. He [5] solved strongly nonlinear equation
using VIM. Taiwo [6] used Canonical Polynomials to solve Singularly Perturbed Boundary Value Problems.
The idea reported in Taiwo [6] motivated the work reported in this paper in which Canonical Polynomials
obtained for Singularly Perturbed Boundary Value Problems are reformulated and used to solve nonlinear
Fredholm – Volterra Integro Differential Equations. Some of the advantages of Canonical Polynomials
constructed are it is generated recursively, it could be converted to any interval of consideration and it is easy to
use and user’s friendly in term of computer implementation.
GENERAL NONLINEAR FREDHOLM – VOLTERRA INTEGRO DIFFERENTIAL
EQUATION CONSIDERED
II.
The general nonlinear Fredholm – Volterra Differential Equation considered in the paper is of the form:
m
P y
k 0
k
k
1
x
1
1
( x)  g ( x)  1  F ( x, t ) y (t )dt  2  K ( x, t ) y (t )dt
(1)
With the mixed conditions
m 1
[ a
j 0
ij
y ( j ) (1)  bij y ( j ) (1)  cij y ( j ) (c) ]   i i  0(1) n
y( x) and y(t ) are unknown functions, g ( x), K ( x, t ), F ( x, t ) are smooth functions,
Pk , aij , bij , cij , 1 , 2 ,  i are constants or transcendental or hyperbolic functions and P0  0
Where
46
(2)

2. Numerical Solution Of Second Order….
III
CONSTRUCTION OF CANONICAL POLYNOMIALS
The construction of Canonical Polynomials is carried out by expanding the left hand side of (1). Thus
from (1), we write the left hand side in an operator form as:
Ly  P0  P
1
d
d2
 P2 2 
dx
dx
...  Pm
dm
dx m
Let,
LQr ( x)  x k
(3)
Then,
Lx k  P0 x k  P kxk 1  P2 k (k  1) x k 2  P3k (k  1)( k  2) x k 3  ..........
1
(4)
Thus, making use of (3), we have
L[ LQk ( x)]  P0 LQk ( x)  P kLQk 1 ( x)  P2 k (k  1) LQk 2 ( x)
1
(5)
Hence, making use of (3) in (5), we have
Qk ( x) 
1 k
[ x  P kQk 1 ( x)  P2 k (k  1)Qk 2 ( x)  .......... ]
1
P0
(6)
Thus, (6) is the recurrence relation of the Canonical Polynomials.
IV.
DESCRIPTION OF STANDARD METHOD BY CANONICAL POLYNOMIALS
In this section, the Standard method by Canonical Polynomials for solving (1) and (2) is discussed. The
method assumed an approximate solution of the form:
y ( x )  yn  x  
n
 a Q ( x)
r 0
r
(7)
r
Where n is the degree of approximant, ar (r
Polynomials generated recursively by (6).
Substituting (7) into (1), we obtain
 0) are constants to be determined and Qr (x) are the Canonical
n
n
n
r 0
r 0
r 0
P0  a r Qr ( x)  P1  a r Q ' ( x)  P2  a r Qr ' ' ( x)  .....
n
n
(8)
 g ( x)  1  F ( x, t ) a r Qr (t )dt  2  K ( x, t ) a r Qr (t )dt
1
1
r 0
x
1
r 0
This is further simplified to give
PO aoQO x   a1Q1 x   a2Q2 x   a3Q3  ...

 P a Q  x   a Q  x   a Q  x   a Q

 P aoQ 'O  x   a1Q '1  x   a2Q ' 2  x   a3Q '3  ...
1
''
2
o
O
1
''
1
''
2
2
3
''
3

 ...



 Pm aoQ m O  x   a1Q m1  x   a2Q m 2  x   a3Q m 3  ...
 g x   1  F x, t aoQ0 t   a1Q1 t   a2Q2 t   ...  a N QN t dt
1
1
  2  k  x, t a o Q0 t   a1Q1 t   a 2 Q2 t   ...  a N Q N t dt
1
1
47
(9)

3. Numerical Solution Of Second Order….
p Q x  p Q x  ...  p Q x    F x, t Q t dt    k x, t Q t  a
  p Q  x   p Q  x   ...  p Q  x     F x, t Q t dt    k x, t Q t  dta
  p Q  x   p Q  x   ...  p Q  x     F  x, t Q t dt    k  x, t Q t  dt a
Collect like terms in (9), we obtain
o
o
1
1
2
'
o
1
'
1
1
2
1
1
1
2
o
1
1
1
1
0
x
2
1
1
1
m
2
m
x
o
1
m
1
m
'
2
2
1
m
o
m
1
x
2
1
2
2
1
2



 p Q x   p Q ' x   p Q x   ... p Q m x    1 F x, t Q t dt 
1 N
2 N
m N
1
N
 o N

1

aN  g ( x)
x
 2  k x, t QN (t )dt

1


x  xi , we obtain
Thus, (10) is collocated at point
p Q x   p Q x   ...  p Q x     F x , t Q t dt    k x , t Q t  a
  p Q  x   p Q  x   ...  p Q  x     F  x , t Q t dt    k x , t Q t  a
  p Q  x   p Q x   ...  p Q  x     F x , t Q t dt    k  x , t Q t  a
'
o
o
i
1
o
i
1
m
o
m
i
1
i
1
'
1
i
m
m
1
i
1
2
i
1
'
2
i
m
m
2
i
1
2
i
o
0
2
2
1
i
1
i
2
1
x
1
1
N m2
1
1
1
b  a i
i
1
x

xi  a 
2
i
1
'
m
 p o Q N  xi   p1Q N  xi   ...  p m Q N  xi   1 
Where,
o
1
1
o



x
i
1
o
(10)
1
2

F  xi , t Q N t dt  2  k  xi , t Q N t  a N (11)
x
1
, i  1, 2, 3, ...,N - m  1
(12)
Thus, substituting (12) into (11) gives rise to N-m+1 algebraic linear equation in (N+1) unknown constants
ar r  0. hence, extra m equations are obtained from (2). Altogether, we have (N+1) algebraic linear equation
in (N+1) unknown constants. These equations are then solved by Gaussian Elimination Method to obtain the
unknown constants which are then substituted into (7) to obtain the approximate solution for the value of N.
V.
Demonstration of Standard Method by Canonical Basis Function on Examples
We solved the Fredholm-Volterra nonlinear Integro Differential equation of the form:
Example 1:
1
y ' '  x   xy ' x   xy x   g  x   
1
xtyt dt  
x
1
Where
g x  
2 6 1 4
23
5
x  x  x3  2 x 2  x 
15
3
15
3
together with the conditions
48
x  2t  y 2 t dt
(13)

4. Numerical Solution Of Second Order….
y 0  1
and ,
y ' 0  1
For N = 2, we seek the approximate solution
as a truncated Canonical series.
The values of
Therefore,
Equation (13) now becomes
2a 2  x ( a1  2a 2 x)  x(a0  a1 x  a 2 ( x 2  2))  g ( x)
1
  xt (a0  a1 t  a 2 (t 2  2))dt 
1

x
1
( x  2t )(a0  a1t  a 2 (t 2  2)) 2 dt
(14)
Thus, (14) is then simplified further and we obtain
2 6 1 4
5
x  x  x3  2x 2 
15
2
2
2
a a x
4a a a a
10a 0 a 2 x
a2 x4
2
2
2
 a1 x  a 0 x  a 0  0 2  a 0 a1 x  0 1  0 2 x 4 
 3a 0 a 2  1
3
3
3
3
3
6
2
2
5
2
2 6
2 4
a x a
3a a x
2a a x
3a a x 28a1 a 2 2a 2 x
2a x
 1  2  1 2  1 2  1 2 

 2
3
2
10
3
2
15
15
3
2
2
43a 2 x 7 a 2


(15)
15
3
2a 2  x ( a1  2a 2 x)  x(a 0  a1 x  a 2 ( x 2  2)) 
Comparing the powers of x in (15), we obtain the following:
For the constant terms:
(16)
49

5. Numerical Solution Of Second Order….
For the power of x:
(17)
For the power of
:
(18)
By using the condition equations, we obtain.
For
(19)
For
:
Implies
These equations are then solved and the results obtained are substituted into the approximate solution and after
simplification gives the exact solution
y 2 ( x)  x
2
1
Example 2
We solved the Fredholm – Volterra Integro differential equation given as
y ' ( x)  xy ( x)  g ( x)   x  t y t dt   x  t y 2 t dt
1
0
Where g x  
x
0
1 6 1 4
5
4
x  x  x 3  2x 2  x 
30
3
3
3
(20)
and y0  2
Thus, the approximate solution given in (7) for case N=2 becomes
2
y 2  x    a r Qr  x 
(21)
r 0
Hence,
y 2 x   a0 Q0 x   a1Q1 x   a 2 Q2 x 
(22)
For this example,
Qi  x   x i  iQi 1 x ; i  0
(23)
We obtain the following
Q0  x   1
Q1 x   x  1
Q2  x   x 2  2 x  2
Q3  x   x 3  3 x 2  6 x  6
(24)
Substituting, (24) into (22), we obtained


y 2  x   a 0   x  1a1  x 2  2 x  2 a 2
(25)
Putting (25) into (20), after simplification we obtain




a1  2 x  2 a 2  xa 0  x 2  x a1  x 3  2 x 2  2 x a 2 
50

6. Numerical Solution Of Second Order….

1 6 1 4
5 4 1
x  x  x 3  2 x 2     x  t 
30
3
3 3 0
Then simplified further, we obtain




1
1
a1  2 x  2a2  xa0  x 2  x a1  x 3  2 x 2  2 x a2   30 x 6  x 4  x 3  2 x 2
3
2
2
4
x 2a 2 a 0  a1  2a 2   a1  2a 2 
5
4
2 a 0  a1  2a 2 
 x  xa 2 
x
3
3
12
2
2 2
5
a  2a2 a0  a1  2a2 
a x
x a 2 a1  2a 2 
 2 
 x3 1
30
10
3


We then collected like terms of x and the following sets of equations are obtained.
For constant term:
a1  2a 2 
a0 1
7
4
 a1  a 2 
2 6
12
3
For power of x:
3
4
5
2a 2  2a0  a1  2a 2  a 2  
2
3
3
Therefore, using the condition, we obtain
a 0  a1  2a 2  2
These equations are then solved and the results obtained are substituted into the approximation solution and
after simplification gives the exact solution
y 2 x   x 2  2
VI
CONCLUSION
The method used in this paper is highly accurate and efficient. At lower degree of N ( degree of
approximant used ), the method gives the exact solution for the two examples considered without extra
computational cost. Also, the Canonical Polynomials generated are easily programmed and user friendly.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Cerdik-Yaslan, H. and Akyus-Dascioglu, A. (2006). Chebyshev Polynomial solution of nonlinear Fredholm Integro Differential
Equation, Juornal of Arts and Sciences. Vol. 5, Pp. 89-101.
He J . H. (1999). Variational iteration method. A kind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34(4)
Pp. 699-708
He. J . H. (2000) Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115 -123
He. J. H. (2003. A simple perturbation approach to Blasius equations. Apll. Math. Comput., 140, 217 -222
He. J. H. (2-006). Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Physics. B20, 1141 1199
Taiwo, O .A. (1991). Collocation approximation for singurly perturbed boundary value problems. Ph.D thesis (unpublished) ,
University of Ilorin, Nigeria.
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