A novel numerical approach for odd higher order boundary value problems

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.5, 2014
1
A Novel Numerical Approach for Odd Higher Order Boundary
Value Problems
Md. Bellal Hossain1
and Md. Shafiqul Islam2
*
1
Department of Mathematics, Patuakhali Science and Technology University,
Dumki, Patuakhali-8602, Bangladesh. Email: bellal77pstu@yahoo.com
2
Department of Mathematics, University of Dhaka, Dhaka – 1000, Bangladesh.
*
Email of the corresponding author: mdshafiqul_mat@du.ac.bd
Abstract
In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth,
seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary
conditions. We exploit Galerkin weighted residual method with Legendre polynomials as basis functions.
Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the
essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several
numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the
reliability and efficiency of the proposed method. The present method is quite efficient and yields better results
when compared with the existing methods.
Keywords: Galerkin method, fifth, seventh and ninth order linear and nonlinear BVPs, Legendre Polynomials.
1. Introduction
From the literature on the numerical solutions of BVPs, it is observed that the higher order differential equations
arise in some branches of applied mathematics, engineering and many other fields of advanced physical sciences.
Particularly, the solutions of fifth order BVPs arise in the mathematical modeling of viscoelastic flows in (Davis,
Karageorghis & Philips, 1988), the seventh order BVPs arise in modeling induction motors with two rotor
circuits in (Richards & Sarma, 1994), and the ninth order boundary value problems are known to arise in
hydrodynamic, hydro magnetic stability and applied sciences. The performance of the induction motor behavior
is modeled by a fifth order differential equation (Siddiqi, Akram & Iftikhar, 2012). This model is constructed
with two stator state variables, two rotor state variables and one shaft speed. Generally, two more variables must
be added to account for the effects of a second rotor circuit representing deep bars, a starting cage or rotor
distributed parameters (Siddiqi & Iftikhar, 2013) . For neglecting the computational burden of additional state
variables when additional rotor circuits are needed, the model is often bounded to the fifth order. The rotor
impedance is done under the assumption that the frequency of rotor currents depends on rotor speed. This
process is appropriate for the steady state response with sinusoidal voltage, but it does not hold up during the
transient conditions, when the rotor frequency is not a single value (Siddiqi, Akram & Iftikhar, 2012). So this
behavior is modeled in the seventh order differential equation . Recently, the BVPs of ninth order have been
developed due to their mathematical importance and the potential for applications in hydrodynamic, hydro
magnetic stability. Agarwal (1986) discussed extensively the existence and uniqueness theorem of solutions of
such BVPs in his book without any numerical examples. Caglar & Twizell (1999) used sixth degree B-spline
functions for the numerical solution of fifth order BVPs where their approach is divergent and unexpected
situation is found near the boundaries of the interval. The spline methods have been discussed for the solution of
other higher order BVPs. Siddiqi & Twizell (1997, 1998, 1996) applied twelfth, tenth, eighth and sixth degree
splines for solving linear BVPs of orders 12, 10, 8 and 6 successively. Kasi Viswanadham, Murali Krishna &
Prabhakara (2010) presented the numerical solution of fifth order BVPs by collocation method with sixth order
B-splines. Lamnii, Mraoui, Sbibih & Tijini (2008) derived sextic spline solution of fifth order BVPs. The
numerical solution of fifth order BVPs by the decomposition method was developed in (Wazwaz, 2001).
Recently, Erturk (2007) has investigated differential transformation method for solving nonlinear fifth order
BVPs. Siddiqi, Akram & Iftikhar (2012) presented the solutions of seventh order BVPs by the differential
transformation method and variational iteration technique respectively. Very recently the solution of seventh
order BVPs was developed in (Siddiqi & Iftikhar, 2013) using variation of parameters method. The modified
variational iteration method has been applied for solving tenth and ninth order BVPs in (Mohyud-Din &

Vol.4, No.5, 2014
2
Yildirim, 2010). Nadjafi & Zahmatkesh (2010) also investigated the homotopy perturbation method for solving
higher order BVPs.
In the present paper, first we shall employ the Galerkin weighted residual method (Reddy, 1991) with Legendre
polynomials (Davis &Robinowitz, 2007) as basis functions for the numerical solution of fifth, seventh and ninth
order linear BVPs of the following form:
5 4 3 2
5 4 3 2 1 05 4 3 2
,
d u d u d u d u du
c c c c c c u r a x b
dxdx dx dx dx
        (1a)
21100 )(,)(,)(,)(,)( AauBbuAauBbuAau  (1b)
7 6 5 4 3 2
7 6 5 4 3 2 1 07 6 5 4 3 2
,
d u d u d u d u d u d u du
c c c c c c c c u r a x b
dxdx dx dx dx dx dx
          (2a)
3221100 )(,)(,)(,)(,)(,)(,)( AauBbuAauBbuAauBbuAau  (2b)
and
9 8 7 6 5 4 3 2
9 8 7 6 5 4 3 2 1 09 8 7 6 5 4 3 2
,
d u d u d u d u d u d u d u d u du
c c c c c c c c c c u r
dxdx dx dx dx dx dx dx dx
a x b
         
 
(3a)
0 0 1 1 2 2 3 3
( )
4
( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) ,
( )iv
u a A u b B u a A u b B u a A u b B u a A u b B
u a A
            

(3b)
where 4,3,2,1,0, iAi and 3,2,1,0, jBj are finite real constants and 9,1,0, ici and r are all continuous
functions defined on the interval [a, b].
However, in section 2 of this paper, we give a short description on Legendre polynomials. In section 3, the
formulations are presented for solving linear fifth, seventh and ninth order BVPs by the Galerkin weighted
residual method. Numerical examples and results for both linear and nonlinear BVPs are considered to verify the
proposed formulation and the solutions are compared with the existing methods in the literature in section 4.
2. Legendre Polynomials
The general form of the Legendre polynomials of degree n is defined by
1],)1[(
)!(2
)1(
)( 2


 nx
dx
d
n
xP n
n
n
n
n
n
Now we modify the above Legendre polynomials as
  1),1()1(
!
1
)( 2









 nxxx
dx
d
n
xL nn
n
n
n
We write first few modified Legendre polynomials over the interval [0, 1]:
)1(2)(1  xxxL , 2
2 )1(6)(  xxxL , )61510)(1(2)( 2
3  xxxxxL ,
5432
4 7021023011020)( xxxxxxL 
65432
5 252882119077024030)( xxxxxxxL  ,
765432
6 924369659224830210046242)( xxxxxxxxL 
8765432
7 343215444286442818215750495681256)( xxxxxxxxxL 
98765432
8 12870643501355641561561067224389010500133272)( xxxxxxxxxxL 

Vol.4, No.5, 2014
3
10
98765432
9
48620
26741063063083226067267234234210857020460207090)(
x
xxxxxxxxxxL


1110
98765432
10
1847561108536
2892890430287040154402438436966966244530372903080110)(
xx
xxxxxxxxxxL


Si
nce the modified Legendre polynomials have special properties at 0x and 1x : 0)0( nL and
1,0)1(  nLn respectively, so that they can be used as set of basis function to satisfy the corresponding
homogeneous form of the Dirichlet boundary conditions to derive the matrix formulation in the Galerkin method
to solve a BVP over the interval [0, 1].
3. Matrix Formulation
In this section, we first derive the matrix formulation rigorously for fifth-order linear BVP and then we extend
our idea for solving seventh and ninth order linear BVPs. To solve the BVPs (1) by the Galerkin weighted
residual method we approximate )(xu as
1,)()()(~
1
,0  
nxLaxxu
n
i
nii (4)
Here )(0 x is specified by the essential boundary conditions, )(, xL ni are the Legendre polynomials which must
satisfy the corresponding homogeneous boundary conditions such that 0)()( ,,  bLaL nini for each
.,,3,2,1 ni 
Using eqn. (4) into eqn. (1a), the Galerkin weighted residual equations are
0)(~
~~~~~
,012
2
23
3
34
4
45
5
5 







 dxxLruc
dx
ud
c
dx
ud
c
dx
ud
c
dx
ud
c
dx
ud
c nj
b
a
(5)
Integrating by parts the terms up to second derivative on the left hand side of (5), we get
  









b
a
b
a
nj
b
a
njnj dx
dx
ud
xLc
dx
d
dx
ud
xLcdxxL
dx
ud
c 4
4
,54
4
,5,5
5
5
~
)(
~
)()(
~
    dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
b
a
nj
b
a
nj 3
3
,52
2
3
3
,5
~
)(
~
)( 








 [Since 0)()( ,,  bLaL njnj ]
      dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
b
a
nj
b
a
nj
b
a
nj 2
2
,53
3
2
2
,52
2
3
3
,5
~
)(
~
)(
~
)( 


















3 2 2 3 4
5 , 5 , 5 , 5 ,3 2 2 3 4
( ) ( ) ( ) ( )
b b b
j n j n j n j n
aa a
d d u d d u d du d du
c L x c L x c L x c L x dx
dx dx dxdx dx dx dx dx
     
                       
     
 (6)
In the same way of equation (6), we have
      dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dxxL
dx
ud
c
b
a
nj
b
a
nj
b
a
njnj
b
a
~
)(
~
)(
~
)()(
~
,43
3
,42
2
2
2
,4,4
4
4  

















 (7)
    






b
a
b
a
nj
b
a
njnj dx
dx
ud
xLc
dx
d
dx
ud
xLc
dx
d
dxxL
dx
ud
c
~
)(
~
)()(
~
,32
2
,3,3
3
3 (8)
  
b
a
b
a
njnj dx
dx
ud
xLc
dx
d
dxxL
dx
ud
c
~
)()(
~
,2,2
2
2 (9)
Substituting eqns. (6) – (9) into eqn. (5) and using approximation for )(~ xu given in eqn. (4) and after imposing
the boundary conditions given in eqn. (1b) and rearranging the terms for the resulting equations we get a system
of equations in matrix form as
njFaD ji
n
i
ji ,,2,1,
1
, 
(10a)
where

Vol.4, No.5, 2014
4
4 3 2
, 5 , 4 , 3 , 2 , 1 , ,4 3 2
( ) ( ) ( ) ( ) ( ) [ ( )]
b
i j j n j n j n j n j n i n
a
d d d d d
D c L x c L x c L x c L x c L x L x
dx dxdx dx dx

                    



3 3
0 , , 5 , , 5 , ,3 3
( ) ( ) ( ) [ ( )] ( ) [ ( )]i n j n j n i n j n i n
x b x a
d d d d
c L x L x dx c L x L x c L x L x
dx dxdx dx 
   
           
   
2 2 2
5 , , 4 , ,2 2 2
( ) [ ( )] ( ) [ ( )]j n i n j n i n
x b x b
d d d d
c L x L x c L x L x
dxdx dx dx 
   
          
   
(10b)
       )()()()()( ,2,32
2
,43
3
,54
4
, xLc
dx
d
xLc
dx
d
xLc
dx
d
xLc
dx
d
xrLF njnjnj
b
a
njnjj 









 
    
ax
nj
bx
njnjnj
dx
d
xLc
dx
d
dx
d
xLc
dx
d
dxxLc
dx
d
xLc
 




















 3
0
3
,53
0
3
,5,00
0
,1 )()()()(



      2,52
2
2
0
2
,42
0
2
,52
2
)()()( AxLc
dx
d
dx
d
xLc
dx
d
dx
d
xLc
dx
d
ax
nj
bx
nj
bx
nj 





























      2,41,53
3
1,53
3
)()()( AxLc
dx
d
AxLc
dx
d
BxLc
dx
d
ax
nj
ax
nj
bx
nj 

























      1,31,42
2
1,42
2
)()()( BxLc
dx
d
AxLc
dx
d
BxLc
dx
d
bx
nj
ax
nj
bx
nj 

























  1,3 )( AxLc
dx
d
ax
nj 







(10c)
Solving the system (10a), we obtain the values of the parameters ia and then substituting these parameters into
eqn. (4), we get the approximate solution of the desired BVP (1).
Similarly, for seventh order linear BVP given in eqn. (2), and for ninth order linear BVP given in eqn. (3), we
can obtain equivalent system of equations in matrix form. For nonlinear BVP, we first compute the initial values
on neglecting the nonlinear terms and using the system (10). Then using the Newton’s iterative method we find
the numerical approximations for desired nonlinear BVP. This formulation is described through the numerical
examples in the next section.
4. Numerical Examples and Results
To test the applicability of the proposed method, we consider both linear and nonlinear problems which are
available in the literature. For all the examples, the solutions obtained by the proposed method are compared
with the exact solutions. All the calculations are performed by MATLAB 10. The convergence of linear BVP is
calculated by
  )(~)(~
1 xuxuE nn
where )(~ xnu denotes the approximate solution using n-th polynomials and  (depends on the problem) which
is less than 12
10 .
In addition, the convergence of nonlinear BVP is calculated by the absolute error of two
consecutive iterations such that
 N
n
N
n uu ~~ 1
, where  is less than 10
10
and N is the Newton’s iteration number.
Example 1: Consider the linear BVP of fifth order (Viswanadham, Krishna & Prabhakara, 2010):
11,sin2sin41cos2cos19 23
5
5
 xxxxxxxxxu
dx
ud
(11a)
1sin81cos3)1(,1sin1cos4)1()1(,1cos)1()1(  uuuuu (11b)
The analytic solution of the above system is .cos)12()( 2
xxxu 
In Table 1, we list the maximum absolute errors for this problem to compare with the existing methods.

Vol.4, No.5, 2014
5
Table 1: Maximum absolute errors for the example 1
x Exact Results 12 Legendre Polynomials
Approximate Abs. Error
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5403023059
0.1950778786
-0.2310939722
-0.6263214759
-0.9016612516
-1.0000000000
-0.9016612516
-0.6263214759
-0.2310939722
0.1950778786
0.5403023059
0.5403023059
0.1950778786
-0.2310939722
-0.6263214759
-0.9016612516
-1.0000000000
-0.9016612516
-0.6263214759
-0.2310939722
0.1950778786
0.5403023059
0.0000000E+000
4.7337134E-013
3.3925640E-013
6.9122486E-013
1.3414825E-012
1.5528689E-012
6.3637984E-013
6.4226402E-013
1.0645096E-012
6.7196249E-013
0.0000000E+000
On the contrary, the maximum absolute error has been obtained by Kasi Viswanadham, Murali Krishna &
Prabhakara ( 2010) is 4
105162.4 
 .
Now the exact and approximate solutions are depicted in Fig. 1 of example 1 for 12n .
Fig. 1: Graphical representation of exact and approximate solutions of example 1.
Example 2: Consider the linear BVP of seventh order [Siddiqi & Iftikhar, 2013]
10),35122( 2
7
7
 xxxeu
dx
ud x
(12a)
3)0(,4)1(,0)0(,)1(,1)0(,0)1()0(  ueuueuuuu . (12b)
The analytic solution of the above system is x
exxxu )1()(  .
The maximum absolute errors for this problem are shown in Table 2 to compare with the existing methods.
On the other hand, the maximum absolute error has been found by Siddiqi & Iftikhar (2013) is .10482.7 10
 In
Fig. 2, the exact and approximate solutions are given for example 2.

Vol.4, No.5, 2014
6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0000000000
0.0994653826
0.1954244413
0.2834703496
0.3580379274
0.4121803177
0.4373085121
0.4228880686
0.3560865486
0.2213642800
0.0000000000
0.0000000000
0.0994653826
0.1954244413
0.2834703496
0.3580379274
0.4121803177
0.4373085121
0.4228880686
0.3560865486
0.2213642800
0.0000000000
6.2597190E-026
5.8911209E-014
1.0269563E-014
1.0608181E-013
6.8500761E-014
9.6977981E-014
7.1442852E-014
6.1228800E-014
5.0015547E-014
5.9674488E-015
0.0000000E+000
Example 3: Consider the following ninth-order linear BVP (Nadjafi and Zahmatkesh (2010), Mohy-ud-Din &
Yildirim (2010), Wazwaz, 2000)
10,99
9
 xeu
dx
ud x
(13a)
3)0(,3)1(,2)0(,2)1(,1)0(,)1(,0)0(,0)1(,1)0( )(
 iv
ueuueuueuuuu (13b)
x Exact Results 12 Legendre Polunomials
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
0.9946538263
0.9771222065
0.9449011653
0.8950948186
0.8243606354
0.7288475202
0.6041258122
0.4451081857
0.2459603111
0.0000000000
1.0000000000
0.9946538263
0.9771222065
0.9449011653
0.8950948186
0.8243606354
0.7288475202
0.6041258122
0.4451081857
0.2459603111
0.0000000000
0.0000000E+000
1.4432899E-015
3.2196468E-015
4.5519144E-015
4.8849813E-015
3.1086245E-015
5.6621374E-015
4.4408921E-016
3.1641356E-015
4.4408921E-016
0.0000000E+000
The analytic solution of the above system is x
exxu )1()(  . The numerical results for this problem are
summarized in Table 3. On the other hand, the accuracy is found nearly the order 10
10
by Mohy-ud-Din and
Yildirim (2010) and by Wazwaz (2000) and nearly the order 9
10
by Nadjafi and Zahmatkesh. In Fig. 3, the
exact and approximate solutions are given of example 3 for 12n .

Vol.4, No.5, 2014
7
Example 4: Consider the nonlinear BVP of fifth order (Erturk, 2007 and Wazwaz, 2001)
10,2
5
5
 
xeu
dx
ud x
(14a)
1)0(,)1(,1)0(,)1(,1)0(  ueuueuu (14b)
The exact solution of this BVP is .)( x
exu 
Consider the approximate solution of )(xu as
1,)()()(~
1
,0  
nxLaxxu
n
i
nii (15)
Here )1(1)(0 exx  is specified by the essential boundary conditions in (14b). Also 0)1()0( ,,  nini LL
for each ni ,,2,1  . Using eqn. (15) into eqn. (14a), the Galerkin weighted residual equations are
 








 
1
0
,
2
5
5
,,2,1,0)(~
~
nkdxxLeu
dx
ud
nk
x
 (16)
Integrating first term of (16) by parts, we obtain
dx
dx
ud
dx
xLd
dx
ud
dx
xLd
dx
ud
dx
xLd
dx
ud
dx
xdL
dxxL
dx
ud nknknknk
nk
~)(~)(~)(~)(
)(
~ 1
0
4
,
41
0
1
0
3
,
3
1
0
2
2
2
,
21
0
3
3
,
,5
5
 


























 (17)
Putting eqn. (17) into eqn. (16) and using approximation for )(~ xu given in eqn. (15) and after applying the
boundary conditions given in eqn. (14b) and rearranging the terms for the resulting equations, we obtain
dxexLxLxLaxLxLe
dx
xdL
dx
xLdn
i
n
j
x
nknjnijnkni
xnink
   














1
1
0 1
,,,,,0
,
4
,
4
))()()(()()(2
)()(

i
x
nink
x
nink
x
nink
a
dx
xLd
dx
xLd
dx
xLd
dx
xdL
dx
xLd
dx
xdL































 1
2
,
2
2
,
2
0
3
,
3
,
1
3
,
3
, )()()()()()(

0
3
0
3
,
1
3
0
3
,
1
0
,
2
0
0
4
,
4
)()(
)(
)(
























  x
nk
x
nk
nk
xnk
dx
d
dx
xdL
dx
d
dx
xdL
dxxLe
dx
d
dx
xLd 


0
3
,
3
1
3
,
3
0
2
,
2
1
2
0
2
2
,
2
)()()()(





































x
nk
x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
dx
d
dx
xLd 
(18)
The above equation (18) is equivalent to matrix form as
GABD  )( (19a)
where the elements of A, B, D, G are kikii dba ,, ,, and kg respectively, given by

Vol.4, No.5, 2014
8

1
3
,
3
,
1
0
,,0
,
4
,
4
,
)()(
)()(2
)()(















 
x
nink
nkni
xnink
ki
dx
xLd
dx
xdL
dxxLxLe
dx
xdL
dx
xLd
d 
1
2
,
2
2
,
2
0
3
,
3
, )()()()(



















x
nink
x
nink
dx
xLd
dx
xLd
dx
xLd
dx
xdL
(19b)
 


n
j
x
nknjnijki dxexLxLxLab
1
1
0
,,,, ))()()(( (19c)

0
3
0
3
,
1
3
0
3
,
1
0
,
2
0
0
4
,
4
)()(
)(
)(
























  x
nk
x
nk
nk
xnk
k
dx
d
dx
xdL
dx
d
dx
xdL
dxxLe
dx
d
dx
xLd
g



0
3
,
3
1
3
,
3
0
2
,
2
1
2
0
2
2
,
2
)()()()(





































x
nk
x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
dx
d
dx
xLd 
(19d)
The initial values of these coefficients ia are obtained by applying Galerkin method to the BVP neglecting the
nonlinear term in (14a). That is, to find initial coefficients we solve the system
GDA  (20a)
whose matrices are constructed from
0
3
,
3
,
1
3
,
3
,
1
0
,
4
,
4
,
)()()()()()(


















 
x
nink
x
ninknink
ki
dx
xLd
dx
xdL
dx
xLd
dx
xdL
dx
dx
xdL
dx
xLd
d
1
2
,
2
2
,
2
)()(










x
nink
dx
xLd
dx
xLd
(20b)
1
2
0
2
2
,
2
0
3
0
3
,
1
3
0
3
,
1
0
0
4
,
4
)()()()(



























 
x
nk
x
nk
x
nknk
k
dx
d
dx
xLd
dx
d
dx
xdL
dx
d
dx
xdL
dx
dx
d
dx
xLd
g

0
3
,
3
1
3
,
3
0
2
,
2
)()()(




























x
nk
x
nk
x
nk
dx
xLd
e
dx
xLd
dx
xLd
(20c)
Once the initial values of ia are obtained from eqn. (20a), they are substituted into eqn. (19a) to obtain new
estimates for the values of ia . This iteration process continues until the converged values of the unknown
parameters are obtained. Substituting the final values of the parameters into eqn. (15), we obtain an approximate
solution of the BVP (14). The maximum absolute errors for this problem are shown in Table 4 with 6 iterations.
On the other hand, the maximum absolute errors have been obtained by Wazwaz (2001) and Erturk (2007) are
8
101.4 
 and ,1052.1 10
 respectively. We depict the exact and approximate solutions in Fig. 4 of example 4
for 12n .
Example 5: Consider the nonlinear BVP of seventh order (Siddiqi, Akram & Iftikhar, 2012)
10,2
7
7
 
xeu
dx
ud x
(21a)
1)0(,)1(,1)0(,)1(,1)0(,)1(,1)0(  ueuueuueuu . (21b)
The exact solution of this BVP is x
exu )( . Following the proposed method illustrated in section 3 as well as in
example 4; the maximum absolute errors for this problem are summarized in Table 5.

Vol.4, No.5, 2014
9
Table 4: Maximum absolute errors of example 4 using 6 iterations
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
0.0000000E+000
2.9918024E-012
3.4379395E-011
1.1878055E-012
2.3980112E-011
3.4213951E-012
3.6519182E-012
2.8570975E-012
1.4395748E-011
2.8618543E-011
1.2759373E-000
Table 5: Maximum absolute errors of example 5 using 6 iterations
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0000000000
1.1051709181
1.2214027582
1.3498588076
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409285
2.4596031112
2.7182818285
1.0000000000
1.1051709181
1.2214027580
1.3498588075
1.4918246976
1.6487212707
1.8221188004
2.0137527075
2.2255409282
2.4596031112
2.7182818285
0.0000000E+000
1.6084911E-011
1.3298251E-011
5.0746074E-011
7.1547213E-012
6.2061467E-011
2.9898306E-012
1.8429702E-013
6.5281114E-011
1.6604496E-011
0.0000000E+000
On the contrary, the maximum absolute error has been obtained by Siddiqi , Akram & Iftikhar (2012) is
10
10586.7 
 . We depict the exact and approximate solutions in Fig. 5 of example 5 for 12n .

Vol.4, No.5, 2014
10
Fig. 5: Graphical representation of exact and approximate solutions of example 5 for.
5. Conclusions
In this paper, Galerkin method has been used for finding the numerical solutions of fifth, seventh and ninth order
linear and nonlinear BVPs with Legendre polynomials as basis functions. The numerical examples available in
the literature have been considered to verify the proposed method. We see from the tables that the numerical
results obtained by our method are better than other existing methods. It may also notice that the numerical
solutions coincide with the exact solution even Legendre polynomials are used in the approximation which are
shown in Figs. [1-5]. The algorithm can be coded easily and may be used for solving any higher order BVP.
6. References
[1] Agarwal, R.P. (1986). Boundary Value Problems for Higher Order Differential Equations, World
Scientific, Singapore.
[2] Caglar H.N., Caglar S.H., & Twizell E.H. (1999). The numerical solution of fifth order boundary value
problems with sixth degree B-spline functions, Appl. Math. Lett., 12, 25-30.
[3] Davis P.J., & Robinowitz P. (2007). Methods of Numerical Integration, Dover Publications, Inc, 2nd
Edition.
[4] Davis A.R., Karageorghis A., & Phillips T.N. (1988). Spectral Galerkin methods for the primary two point
boundary value problem in modeling viscoelastic flows, International J. Numer. Methods Eng. 26, 647-662.
[5] Karageorghis A., Phillips T.N. & Davis A.R., (1988). Spectral Collocation methods for the primary two
point boundary value problem in modeling viscoelastic flows, International J. Numer. Methods Eng. 26,
805-813.
[6] Erturk V.S. (2007). Solving nonlinear fifth order boundary value problems by differential transformation
method, Selcuk J. Appl. Math., 8 (1), 45-49.
[7] Kasi Viswanadham K.N.S., Murali Krishna P., & Prabhakara Rao, C. (2010). Numerical solution of fifth
order boundary value problems by collocation method with sixth order B-splines, International Journal of
Applied Science and Engneering, 8 (2), 119-125.
[8] Lamnii A., Mraoui H., Sbibih D., & Tijini A. (2008). Sextic spline solution of fifth order boundary value
problems, Math. Computer Simul., 77, 237-246.
[9] Mohyud-Din S.T., & Yildirim A. (2010). Solutions of tenth and ninth order boundary value problems by
modified variational iteration method, Application Applied Math., 5 (1), 11-25.
[10] Nadjafi, J.S., & Zahmatkesh, S. (2010). Homotopy perturbation method (HPM) for solving higher order
boundary value problems, Applied Math. Comput. Sci., 1 (2), 199-224.

Vol.4, No.5, 2014
11
[11] Reddy, J.N. (1991). Applied Functional Analysis and Variational Methods in Engineering, Krieger
Publishing Co. Malabar, Florida.
[12] Richards, G. & Sarma, P.R.R. (1994). Reduced order models for induction motors with two rotor circuits,
IEEE Transactions Energy Conversion, 9 (4), 673-678.
[13] Siddiqi, S. S., & Twizell E.H. (1997). Spline solutions of linear twelfth order boundary value problems, J.
Comp. Appl. Maths. 78, 371-390.
[14] Siddiqi, S. S., Twizell, E.H. (1998). Spline solutions of linear tenth order boundary value problems, Intern.
J. Computer Math. 68 (3), 345-362.
[15] Siddiqi, S. S., & Twizell, E.H. (1996). Spline solutions of linear eighth order boundary value problems,
Comp. Meth. Appl. Mech. Eng. 131, 309-325.
[16] Siddiqi, S. S., & Twizell, E.H. (1996). Spline solutions of linear sixth order boundary value problems,
Intern. J. Computer Math. 60 (3), 295-304.
[17] Siddiqi, S. S., Ghazala Akram & Muzammal Iftikhar (2012). Solution of seventh order boundary value
problem by differential transformation method, World Applied Sciences Journal, 16 (11), 1521-1526.
[18] Siddiqi S. S., Ghazala Akram & Muzammal Iftikhar (2012). Solution of seventh order boundary value
problems by variational iteration technique, Applied Mathematical Sciences, 6, (94), 4663-4672.
[19] Siddiqi, S. S. & Muzammal Iftikhar (2013). Solution of seventh order boundary value problem by variation
of parameters method, Research Journal of Applied Sciences, Engineering and Technology, 5 (1), 176-179.
[20] Wazwaz, A.M. (2001). The numerical solution of fifth order boundary value problems by the decomposition
method, J. Comput. Appl. Math., 136, 259-270.
[21] Wazwaz A.M. (2000). Approximate solutions to boundary value problems of higher order by the modified
decomposition method, Comput. Math. Appl., 40, 679-691.

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