Applied Mathematics, 2011, 2, 959-964
doi:10.4236/am.2011.28132 Published Online August 2011 (http://www.SciRP.org/journal/am)
Numerical Treatment of Nonlinear Third Order
Boundary Value Problem
Pankaj Kumar Srivastava, Manoj Kumar
Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, India
Received January 25, 2011; revised June 20, 2011; accepted June 27, 2011
Abstract
In this paper, the boundary value problems for nonlinear third order differential equations are treated. A ge-
neric approach based on nonpolynomial quintic spline is developed to solve such boundary value problem.
We show that the approximate solutions of such problems obtained by the numerical algorithm developed
using nonpolynomial quintic spline functions are better than those produced by other numerical methods.
The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.
Keywords: Nonlinear Third Order Boundary Value Problem, Nonpolynomial Quintic Spline, Draining and
Coating Flows
1. Introduction
Engineering problems that are time-dependent are often
described in terms of differential equations with condi-
tions imposed at single point (initial/final value prob-
lems); while engineering problems that are position de-
pendent are often described in terms of differential equa-
tions with conditions imposed at more than one point
(boundary value problems). Boundary value problems
are encountered in many engineering fields including
optimal control, beam deflections, heat flow, draining
and coating flows, and various dynamic systems. In this
paper we are concerned with general third-order nonlin-
ear boundary value problems, such problems arise in the
study of draining and coating flows. Many authors have
studied and solved such type of third order boundary
value problems with various types of boundary condi-
tions. A. Khan and Tariq Aziz [1] solved a third-order
linear and non-linear boundary value problem of the type
 
,,
y
xfxyax
 b
3
(1)
Subject to
 
12
,,
y
akyak ybk


(2)
by deriving a fourth order method using polynomial
quintic splines. S. Valarmathi and N. Ramanujam [2], T.
Y. Na [3], N. S. Asaithambi [4], Xueqin Li and Minggen
Cui [5] and N. H. Shuaib et al. [6] used some other
computational methods for solving boundary value pro-
blems for third-order ordinary differential equations. A.
Cabada et al. [7] studied external solutions for third-
order nonlinear problems with upper and lower solutions
in reversed order.
In this paper Nonpolynomial quintic spline functions
are applied to obtain a numerical solution of the follow-
ing nonlinear third order two-point boundary value pro-
blems

 

2
,, ,0,1,yfxyyygxyx
 

3
A
(3)
subject to the boundary conditions:





11
12
1,0 ,1yAy Ay


3
(4)
where ,1,2,
i
Ai

are finite real constants.
The existence theorems for the solution of (3) sub-
jected to boundary condition (4) are derived by Xueqin
Li and Minggen Cui [5]. M. Pei et al. and F. M. Minhos
[8,9] also discussed the existence of nonlinear third order
boundary value problems. Xueqin Li and Minggen Cui
consider the problem in the reproducing kernel Hilbert
space


10,1W. In order to derive the existence theorems
for solution of (3) they make the following assumptions
:
H




3
(1) ,,,0,1
H
fxyzw XR is completely continu-
ous function;




(2),,, ,,,, ,,,,
xy

H
fxyzwfxyzw fxyzw
and


,,,
w
f
xyzw are bounded;


(3) ,,,0Hfxyzw on


3
0,1
X
R,

960 P. K. SRIVASTAVA ET AL.
where
 

1
,,, 0,1fxyzw W as
 

10,1 ,yyx W
 

10,1 ,zzx W
 

10,1 ,wwx W

01, ,,xyzw

Here the reproducing kernel Hilbert space


10,1W is
the inner product space


10,1W which is defined by,


  


 
1
2
0,1:
0,1,0,1
Wuxuxis absolutely continuous real
valuefunction inuxL


The inner product and norm in


10,1W are given,
respectively, by
 
1
1
0
,00
W
uxvxuvu xvxx


d
 
1
1/2
,
W
uuxux
M. Cui et al. and C. I. Li et al. [10,11] had proved that


10,1W is a complete reproducing kernel space. That is,
there exists a reproducing kernel function
 



10,1 ,0,1 x
Qy Wy
for each fixed


0,1x and any



10,1uy W, such
that The reproducing kernel
can be denoted by
 
,x uy Qx

y

1
W
y u
x
Q

1,
1,
x
yyx
Qy ,
.
x
yx




In the present paper, our main objective is to apply
non-polynomial quintic spline function [12-14] that has a
polynomial and trigonometric parts to develop a new
numerical method for obtaining smooth approximations
to the solution of nonlinear third-order differential equa-
tions of the system of form (3) subjected to (4). Here
algorithms are developed and the approximate solutions
obtained by these algorithms are compared with the solu-
tions obtained by iterative method [5]. The paper is or-
ganized as follows—In Section 2, we have given a brief
introduction of nonpolynomial quintic spline. In Section
3, we give a brief derivation of this non-polynomial
quintic spline. We present the spline relations to be used
for discretization of the given system (3). In Section 4,
we present our numerical method for a system of non-
linear third-order boundary-value problems and develop-
ment of boundary conditions, truncation error and class
of the method are discussed, in Section 5, numerical
evidence is included to compare and demonstrate the
efficiency of the methods, in which we have shown that
our algorithm performs better than an iterative method.
Finally, in Section 6 we have concluded the paper with
some remarks.
2. Nonpolynomial Quintic Spline
A quintic spline function , interpolating to a func-
tion

Sx



ux defined on [, is such that ]ab
1) In each subinterval 1
[,]
j
j
x
x
, is a poly-
nomial of degree at most five.

Sx

2) The first, second, third and fourth derivatives of


Sx
 are continuous on . [,]ab
To be able to deal effectively with such problems we
introduce “spline functions” containing a parameter

.
These are “non-polynomial splines” defined through the
solution of a differential equation in each subinterval.
The arbitrary constants are being chosen to satisfy cer-
tain smoothness conditions at the joints. These “splines”
belong to the class and reduce into polynomial
splines as parameter
2
C
0

. The exact form of the
spline depends upon the manner in which the parameter
is introduced. We have studied parametric spline func-
tions: spline under compression, spline under tension and
adaptive spline. A number of spline relations have been
obtained for subsequent use.
A function


,Sx

 of class which inter-
polate
4[,]Cab


ux at the mesh points {}
j
x
depends on a
parameter

, reduces to ordinary quintic spline


Sx

in as [,ab]0

 is termed as parametric quintic
spline function. The three parametric quintic splines de-
rived from quintic spline by introducing the parameter in
three different ways are termed as “parametric quintic
spline-I”, “parametric quintic spline-II” and “adaptive
quintic spline”.
The spline function we propose in this paper has the
following form




23
23
2345
span1, ,,,sin,cos,
orspan 1,,,,sinh,cosh,
orspan1,,,,,, where0
xx xxx
x
xxx x
xx xxx




The above fact is evident when correlation between
polynomial and non-polynomial splines basis is investi-
gated in the following manner:


 


23
5
2
23
4
3
5
span1,,,,sin(),cos() ,
24
span1, ,,,cos1,
2
120 sin 6
Txxxxx
x
xx xx
x
xx

















  






P. K. SRIVASTAVA ET AL.
961
From the above equation it follows that

2345
05
lim1, ,,,,Txxxxx
 
where

is the frequency of the trigonometric part of
the splines function which can be real or pure imaginary
and which will be used to raise the accuracy of the
method. This approach has the advantage over finite dif-
ference methods that it provides continuous approxima-
tion to not only for

y
x
, but also for ,yy

 and
higher derivatives at every point of the range of integra-
tion. Also, the differentiability of the trigonomet-
ric part of non-polynomial splines compensates for the
loss of smoothness inherited by polynomial splines in
this paper.
C
3. Development of the Method
Without loss of generality in order to develop the nu-
merical method for approximating solution of a differen-
tial Equation (3), we consider a uniform mesh

with
nodal points i
x
on


,ab such that
0123
:N axx xxxb 
,0,1,2,,
i
x
aih iN 
where, ba hN

.
Let us consider a non-polynomial function


Sx
 of
class


4,Cab which interpolates

y
x at the mesh
points i
x
, depends on a parameter 0,1, 2,,,iN

,
and reduces to ordinary quintic spline in
as

x[,S
]ab
0.


For each segment the
non-polynomial, , define by
1
[,],0,1, 2,,1,
ii
xx iN


x

S

 
 
23
sin cos
0,1, 2,,1
ii ii iii
iii i
Sx abxxcxxdxx
exxfxx
iN

 

(5)
where and
,,, ,
iii ii
abcdei
f
are constants and

is
arbitrary parameter.
Let i be an approximation to y

i
y
x, obtained by
the segment of the mixed splines function pass-
ing through the points

Sx


,
ii

x
y and 11i

,
i

x
y , to ob-
tain the necessary conditions for the coefficients intro-
duced in (5), we do not only require that


xS
 satis-
fies interpolatory conditions at i
x
and 1i
x
 , but also
the continuity of first, second and third derivatives at
the common nodes

,

ii
x
y are fulfilled.
To derive expression for the coefficients of (5) in
terms of ,,, ,
i
y1i y,1 ,
ii i
DD T
1i
Ti
F
and1i
F
we first
denote:


 
 
 
11
1
(3) (3)
11
(4) (4)
11
,
,
,
,
iii i
iiii
iii i
iiii
Sx ySxy
Sx DSxD
SxT SxT
Sx FSxF

 1



 

 





(6)
From algebraic manipulation we get the following ex-
pression:

4
1
3
11
2
1
1
4
4
,
cos ,
sin
2,
2
cossin ,
6(1cos )
cos ,
sin
.
i
ii
ii
ii
iii
i
ii i
i
ii
i
i
i
F
ay
FF
bD
yyy
ch
TT F
d
FF
e
F
f
























(7)
where h



and 0,1,2,,1iN

.
Using the continuity of the first and third derivatives at


,
ii
,
x
y that is


ii ii
SxSx
 



and




i
x
ii
Sx

 i
S


, we obtain the following relations:





21 1
3
11 1
2
11 11
33
1
2,
iiii
iiii i
iii ii
yyyy
h
hF T TTT
h
TTThFFi N



 
 
 
111





 
(8)
The operator Λ is defined by for any function

 
22 11iii ii
wpwwqw wsw
 
 i
for any function evaluated at the mesh points. Then
we have the following relations connecting and its
derivatives:
w
y
1) i T






22 11
3
124
ii ii
yy yy
h
 
 

  


(9)
2) 4
4
1
ii
F
y
h


where 1,
6
p

 

11
1
22
6
q   ,








962 P. K. SRIVASTAVA ET AL.




11
2
2
12
1
24 2
6
11,
11,
11 ,
3
,
ii
s
csc
cot
h
TSx
 

























 
,
and

(4)
ii
F
Sx

.
4. Description of the Method and
Development of Boundary Conditions
At the mesh point i
x
the proposed differential equation
 

2
,,, '',,yfxyyygxyxab
 
 (10)
subjected to boundary conditions (4), may be discretized
by
2
ii ii
Tfgy (11)
where and

i
TSx


i

ii
g
gx.
Using the spline relation (9) (i), in (11) we have







2
21
22
2221
3222 2
22 1111
2
22
24 24
,212
ii
iiii i
i iiiiiii
ii
yy hqf
ph fyhpfqfsf
hpgyqgysg yqgy
pg yiN
 


 
 

 
 
 

11ii
y

(12)
To obtain unique solution we need two more equations
to be associated with (12) so that we use the following
boundary conditions:
1) To obtain the second-order boundary formula we
define:
 

 
1234
3
2332
33
,1
yy yy
hy yhyyi



 
 
 



321
3
211 2
33
,
1
NNNN
NNN N
yyyy
hyyhyy
iN



 

 
 

(13)
for any choice ofand,12α



. Using Equation
(3) we have:





1234
33
2233
32 3
23
33
ii
yy yy
hfhfhfh f
hhgy hhgy




 

 

2





321
33
2233
32 3
22 1
33
NNNN
NNNN
NN NN
yyyy
hfh fhfhf
hhgyhhgy





2
1



 
 
 
 

2) To obtain the fourth-order boundary formula we
define:




 
 

1234
3
233 2
321
3
211 2
33
,1,
28
33
,1
28
NNNN
NNNN
yy yy
hh
yyyyi
yyyy
hh
yyy yiN





 

 
 

  

 

(14)
for any choice ofand,12α

  . Using Equation
(3) we have:




1234
33
2233
33
22
23
33
2828
28 28
ii
yy yy
hhhh
ffff
hh hh
g
yg





 



 


y




321
33
2211
33
22
22 1
33
2828
28 28
NNNN
NNNN
NN NN
yyyy
hhhh
ffff
hh hh
2
g
yg



 

y




 


 
 
 
 

By expanding (8) in Taylor series about xi, we obtain
the following local truncation error:







4
4
6
6
8
8
9
193 4
6
11
33 316
1806 12
11
1613 274
131040 360
ii
i
i
Tpqhy
pqhy
pqhy
Oh
















(15)
For any choice of α and β, provided that 12

  .
Remark 1): Second-order method
For 14, 14



,

P. K. SRIVASTAVA ET AL.
963
and
0.04063489941134321703,p
0.25412730690212937985,
0.41047570631347259688.
q
s


gives

4.
i
TOh
Remark 2): Fourth-order method
For 11 12 ,, ,
6312012
pq   
6
0
,
and 66
120
s, gives .

6
i
TOh
Clearly, the family of numerical methods is described
by the Equation (12), boundary equations and the solu-
tion vector , T denoting transpose, is

12
,,, T
N
Yyy y

obtained by solving a non-linear algebraic system of or-
der N.
To ensure cost effectiveness, better accuracy and sim-
ple applicability of the new method, the best way is to
find the unknown parameters α and β, which are the ex-
pressions containing the actual parameter

. The hall
mark of the new approach is that it gives family of
fourth- and second-order methods by running the code
once and also skips the multiplications involved in the
expressions α and β.
5. Numerical Example
We now consider a numerical example illustrating the
comparative performance of nonpolynomial quintic
spline algorithms over an iterative method [5].
Example: Consider the boundary value problem
 
()x
yxyxyxxyxy
 

2
x
0
(16)
under the boundary condition

101yy y

 (17)
The analytic solution of (16) is



1yx xx (18)
Nonpolynomial Quintic Spline Solution of Example
The maximum observed errors (in absolute value) by
our algorithm (of second order) and iterative method
(Xueqin Li et al. [5]) for the example considered are
presented in Table 1.
6. Discussion and Conclusions
In this paper we used a nonpolynomial Quintic spline
function to develop numerical algorithms of system of
nonlinear third order boundary value problems. Here the
result obtained by our algorithm is better than that ob-
tained by some other method as compared in Tables 1
Table 1. Comparison of our algorithm of second order with
iterative method.
(Maximum Absolute Error)
(Our Method)
Node
(Maximum Abso-
lute Error)
(By Xueqin Li et
al. [5])
N = 10 N = 10 N = 20
0.1 4.24272E-04 2.52718E-04 6.01631E-05
0.2 1.97294E-04 1.25174E-04 3.2713E-05
0.3 3.66041E-04 2.52833E-04 5.3529E-05
0.4 1.020626E-03 2.62819E-04 6.52289E-05
0.5 1.52747E-03 3.82917E-04 8.82427E-05
Table 2. Comparison of our algorithm of fourth order with
iterative method.
(Maximum Absolute Error)
(Our Method)
Node
(Maximum Abso-
lute Error)
(By Xueqin Li et
al. [5])
N = 10 N = 10 N = 20
0.1 4.24272E-04 6.28192E-05 3.58192E-06
0.2 1.97294E-04 4.62816E-05 2.23147E-06
0.3 3.66041E-04 3.52842E-05 2.18372E-06
0.4 1.020626E-03 8.16252E-05 4.98326E-06
0.5 1.52747E-03 1.93165E-05 1.28429E-06
and 2. The approximate solutions obtained by the present
algorithms are very encouraging and it is a powerful tool
for solution of nonlinear third order boundary value
problems.
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