Research Journal of Applied Sciences, Engineering and Technology 7(23): 4859-4863, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: October 03, 2013
Accepted: November 26, 2013
Published: June 20, 2014
Quartic Non-polynomial Spline Solution of a Third Order Singularly Perturbed
Boundary Value Problem
Ghazala Akram and Imran Talib
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
Abstract: In this study, the non-polynomial spline function is used to find the numerical solution of the third order
singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to
have second order convergence. The order of convergence is improved up to fourth order using the improved end
conditions. Numerical results are given to describe the efficiency of the method and compared with the method
developed by Akram (2012), which shows that the present method is better.
Keywords: Boundary layers, monotone matrices, quartic non-polynomial spline, singularly perturbed boundary
value problems, uniform convergence
INTRODUCTION
The purpose of the study is to develop a new spline
method for the solution of third order singularly
perturbed boundary value problem. The method
depends on a non-polynomial spline function which has
a trigonometric part and a polynomial part. The
following third order self adjoint singularly perturbed
boundary value problem is considered, as:
L ( y ( x ) ) = − ε y (3) ( x ) + p ( x) y ( =
x ) f ( x) ,
=
y (0) α=
α 1=
, y (0) α 2
0 , y (1)
(1)
p ( x) ≥ 0 


(1)
or,
L ( y ( x ) ) = − ε y (3) ( x ) + p ( x) y ( x=
) f ( x) , p ( x) ≥ 0 

(2)
=
y (0) α=
, y (0) α 3
 (2)
0 , y (1) α 1 =
where, α 0 , α 1 , α 2 and α 3 are constants and ε is a small
positive parameter (0<ε<<1), also f (x) and p (x) are
smooth functions and p (x) = p = constant. The spline
function has the form T n = {1, x, x2, cos (kx), (kx), sin
(kx)} It is to be noted that k can be real or imaginary.
The theory of singularly perturbed problems frequently
occur in many branches of engineering and applied
sciences, for instance in geophysics, fluid dynamics,
Newtonian fluid mechanics, quantum mechanics, gas
dynamics, chemical reactions, optimal control theory
etc. The numerical treatment of singularly perturbation
problems yields major computational difficulties and
the usual numerical methods fail to produce accurate
results for all independent values of x when ε is very
small, owing to the multi-scale character of the solution
of singularly perturbation problems. That is there are
thin transition layers, where the solution varies rapidly,
while away from the layers the solution behaves
frequently and varies slowly. Three principle
approaches are frequently used to solve such kind of
problems numerically, namely the finite difference
methods, the finite element methods and spline
approximation methods. In this study the third one,
namely, the spline approximation method is used.
Howers (1976), Kelevedjiev (2002) and Roos et al.
(1996) discussed the existence and uniqueness of
singularly perturbed Boundary Value Problems (BVPs).
Lie (2008) constructed a computational method for
singularly perturbed two point BVP in the form of
series in reproducing Kernel space. Rashidinia and
Mahmoodi (2007) developed the classes of methods for
the numerical solution of singularly perturbed two point
BVP using non polynomial cubic spline and the method
is second order as well as fourth order accurate. Khan
et al. (2006) used sextic spline to solve second order
singularly perturbed BVP and the method is fifth order
accurate. Yao and Cui (2007) developed a new
algorithm for a class of singularly BVPs in the
reproducing Kernel space. Akram (2012) presented a
quartic spline solution of a third order singularly
perturbed BVP and the method is second order
accurate. Akram and Mehak (2012) proposed a quintic
spline technique to solve fourth order singularly
perturbed BVP.
Consistency relations: To develop the consistency
relations the following fourth degree non polynomial
spline is considered:
Q=
ai cos k ( x − xi ) + bi sin k ( x − xi ) +
i ( x)
+ ci ( x − xi ) 2 + di ( x − xi ) + ei
Corresponding Author: Ghazala Akram, Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
4859
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Res. J. App. Sci. Eng. Technol., 7(23): 4859-4863, 2014
Defined on [a, b], where 𝑥𝑥ϵ[𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖+1 ] with equally
𝑏𝑏−𝑎𝑎
spaced knots, x i = a + ih, i = 0, 1, …, n and ℎ =
.
𝑛𝑛
Using the following notations:
Qi ( xi ) y=
Qi ( xi +1 ) yi +1 , 
=
i,

(3)
=
Qi ( xi ) T=
Qi (3) ( xi +1 ) Ti +1 , 
i,

Qi (1) ( xi ) = N i .

(4)
 1 − cos θ 1 − 2 cos θ 
β  3
=
+
.
2θ sin θ 
 θ sin θ
It is to be noted that, if α = 0 and 𝛽𝛽 =
truncation error of the above equations is −
Using Eq. (1) and (6) can be rewritten as:
2
1
240
then the
ℎ7 𝑦𝑦𝑖𝑖7 .
(-ε + α ph3 ) yi +1 + (3ε + β ph3 ) yi - (3ε β- ph3 ) yi −1 + (ε + α ph3 ) yi -2 

i
2,3,..., n -1.
= h3 (α fi +1 + β fi + β fi −1α fi -2 ) ,
=

The coefficients in (3) are determined as:
(7)
 T − T cos θ 
ai = h3  i +1 3 i
,
 θ sin θ 
− h3Ti
,
bi =
3
End conditions: The system Eq. (7) consists of (n-2)
equations with (n-1) unknowns, so one more equation is
required. Following Akram and Siddiqi (2006), the
required end condition can be written as:
θ
yi +1 − yi h(cos θ − 1)(Ti +1 + Ti ) N iθ 2 + h 2Ti
,
ci =
−
−
h2
hθ 2
θ 3 sin θ
N θ 2 + h 2Ti
,
di = i
2
T0 + a1 T1 + a2 T
=
2 + T3
θ
e=
yi −
i
1
h (Ti +1 − Ti cos θ )
.
θ 3 sin θ
3
Applying the first and second derivative continuity
at knots, that is, Q ( µ )i −1 ( xi ) = Q ( µ )i ( xi ), for μ = 1, 2, the
following relations are derived:
1  4

b y + c0 hy0(1)  ,
3 ∑ i i
h  i =0

(8)
where, a 1 , a 2 , c 0 and b i , i = 0, 1, 2, 3 are arbitrary
parameters which are to be calculated using method of
undetermined coefficients. The end condition of O (h5)
can be calculated, as:
1 2
7

θ 2 ( Ni + Ni −1 )
2θ 2 ( yi −1 − yi ) 2(Ti + Ti −1 )(cos θ − 1)
T0 + T1 + T2=
+ T3
y0 + 3 y1 − 6 y2 + y3 + 2hy (1) 0  (9)
3 
(
T
T
)
0,
+
+
+
+
=
h 3
3
i −1
i

2
3
h
h
θ sin θ
2
2
2h (Ti −1 − Ti ) h (−Ti −1 + 2Ti cos θ − Ti +1 )
2( N i −1 − N i ) +
+
Using Eq. (1), Eq. (9) can be rewritten as:
θ2
θ sin θ
2
2( yi +1 − 2 yi + yi −1 ) 2h (cos θ − 1)(Ti −1 − Ti +1 )
0,
+
+
=
3
7
h
θ 3 sin θ
(−3ε + ph3 ) y1 + (6ε ) y2 + (− ε + ph ) y3 − h3 ( f 0 + f1 + f3 )
3
2
(10)
which leads the following consistency relation in terms
+ (− ε − ph3 )α 0 − 2ε hy (1)
=0.
0
3
of T and y :
i
i
− yi − 2 + 3 yi −1 − 3 yi + yi +1
1  3  1 − cos θ 1 − 2 cos θ 
 cos θ − 1
= h3Ti -2  3
+
+
 + h Ti −1  3

2θ sin θ 
 θ sin θ 2θ sin θ 
 θ sin θ
1 
 1 − cos θ 1 − 2 cos θ  3  cos θ − 1
;
+ h3Ti  3
+
+
 + h Ti +1  3
sin
2
sin
sin
2
sin
θ
θ
θ
θ
θ
θ
θ
θ 



=
i 2,3,..., n − 1.









(5)
T0 + 2 T1 + T2 + T3 =
3 
 135

 141

 49
ε + 2 ph3  y1 +  ε + ph3  y2 +  − ε + ph  y3 − h3 ( f 0 + 2 f1 + f 2 + f3
−
 11

 11

 11

6 2 (2)
 43
3
0.
+  ε − ph  α 0 + ε h y0 =
11
 11

(6)
where,
and,
135
141
49
6 2 ( 2) 
 43
 − 11 y0 + 11 y1 − 11 y2 + 11 y3 − 11 h y 0  ,
Using Eq. (2), Eq. (11) can be rewritten as:
=
i 2,3,..., n − 1.
1
 cos θ − 1

α  3
=
+

θ
sin
θ
2
θ
sin
θ


1
h3
(11)
Equation (5) can also be written, as:
yi +1 − 3 yi + 3 yi −1 − y=
α h3Ti +1 + β h3Ti + β h3Ti −1 + α h3Ti -2
i −2
Similarly the end condition of O (h5) for the system
(2) can be calculated, as:

)




(12)
The order of truncation error of end conditions can
be improved. Following Akram and Siddiqi (2006), the
improved end conditions of O (h8) for the system (1)
can be determined, as:
4860
Res. J. App. Sci. Eng. Technol., 7(23): 4859-4863, 2014
T0 +
97
69
1  99
176
309
96
7

T1 + T=
y0 +
y1 −
y2 +
y3 + y4 + 15hy (1) 0  ,
2 + T3
7
7
h3  28
7
7
7
4

Also,
(13)
 99
3
(1)
3
c1 =
 ε − ph  α 0 + 15ε hy0 + h f 0 ,
 28

Using Eq. (1), Eq. (13) can be rewritten as:
c2 = ( −ε − α ph3 ) α 0 + h3α f 0 ,
=
ci 0,=
i 3, 4,..., n − 2,
3 
97
69 3 
 176

 309
 96
7  
ph  y2 +  − ε + ph  y3 -  ε  y4 
ε + ph3  y1 + 
ε +
−
7
7
 7

 7

 7

4  

97
69

  99


f 2 +f3  +  − ε - ph3  α 0 + 15ε hy (1)
− h3  f 0 + f1 +
=0.
0

7
7

  28

and
(ε − α ph3 )α1 + α h3 f n .
cn −1 =
(14)
Similarly end condition of O (h8) for the system (2)
can be calculated, as:
T0 +
If Y = [ y ( x1 ), y ( x2 ),..., y ( xN − 2 ), y ( xN −1 ) ]T denotes
the exact solution of BVP (1) and Y be the approximate
solution then, Eq. (17) can be written, as:
265
213
1  607
944
921
224
113
90

T1 +
T2 + T3 =3  −
y0 +
y1 −
y2 +
y3 +
y4 − h 2 y ( 2 ) 0  ,
31
31
h  62
31
31
31
62
31

AY − h3 DF =
T + C,
(18)
(15)
where T = (t 1, t 2,..., t n −1)T with:
Using Eq. (2), Eq. (15) can be rewritten as:
23
3 
=
t1
ε h 8 y (8) (ξ 1),
265 3 
213 3 

 944
 921
 224
ph  y1 + 
ph  y2 +  −
ε+
ε +
ε + ph  y3
840
−

31
31
31
31
31








and
265
213
 113 

  607


f1 +
f 2 +f 3  + 
0
− 
=
ε  y4 − h 3  f 0 +
ε + ph3  α 0 + 15ε h 2 y (2)
0

31
31
1
 62 

  62


7 (7)
(16)
CONVERGENCE OF THE METHOD
−
ti =
εh y
240
A (Y − Y ) = A E = T ,
α ph3 + ε
(21)
To determine the error bound the row sums S 1 , S 2 ,
S n-1 of matrix A are calculated, as:





β ph3 + 3ε α ph3 − ε
,




β ph3 − 3ε β ph3 + 3ε α ph3 − ε

α ph3 + ε
β ph3 − 3ε β ph3 + 3ε 
S=
1
∑a=
1, j
j
Si
=∑ a2, j




,
β β α

  

α β β α 

α β β 
69
1
7
β α
C = (c1 , c2 ,..., cN -2 , cN -1 )T ,
=
− ε + ( 2 β + α ) p h3 ,
j
j
and
S n −1 =
∑ an−1, j =ε + ( 2β + α ) p h3 .






3, 4,..., n − 2 
i=






(22)
Since the matrix A is observed to be irreducible and
monotone, A-1 exists and its elements are non negative,
therefore from Eq. (20), it can be written as:
E = ( A−1 ) T
(23)
Also, from the theory of matrices it can be written as:
Y = ( y1 , y2 ,..., y N -2 , y N -1 )T
N −1
=
k 1 , 2 ,..., N − 1,
∑ ak−,1i S i 1 =
and
F = ( f1 , f 2 ,..., f N -2 , f N -1 )T .
99 173 3
ε+
ph ,
7
7
S N −2 =
( 2β + 2α ) p h3
∑ an−2, j =
j
 97
7

β
D= α





(19)
(20)
E = Y − Y = (e1 , e2 ,..., eN −1 )T .
69 3 309
96
7
ph +
ε ph3 − ε − ε
7
7
7
4
β ph3 + 3ε
α ph3 − ε
β ph3 − 3ε

x i − 2 < (ξ i ) < x i +1,
(17)
where, A = (a ij ) is a tetradiagonal matrix of order n-1:
 97 3 176
 7 ph − 7 ε

3
 β ph − 3ε

3
A = α ph + ε





(ξ i ),





2,3,..., n − 1.
i=

From Eq. (17) and (18), it follows that:
The system of Eq. (7) and (14) provides the
required quartic non polynomial spline solution of the
BVP (1), which can be written in matrix form, as:
AY − h3 DF =
C,
x 0 < (ξ 1) < x 4
i =1
−1
is the (k, i)th element of the matrix A-1.
where 𝑎𝑎𝑘𝑘,𝑖𝑖
4861
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Res. J. App. Sci. Eng. Technol., 7(23): 4859-4863, 2014
Similarly, it can be proved that ‖𝐸𝐸‖ = 𝑂𝑂(ℎ2 ) for
the solution of BVP (2) given by Eq. (7) and (12).
These results are summarized in the following
theorems.
From Eq. (22), it follows that:
n −1
∑a
i =1
−1
k ,i
≤
1
1
=
min Si (h3 Bi0 )
(25)
Theorem 3: The method given by Eq. (7) and (10) for
solving the boundary value problem (1) for sufficiently
small h gives a second order convergent solution.
where,
=
Bi0
1
min Si > 0
h3
(26)
For some i 0 between 1 and n-1.
From Eq. (23) it can be written as:
=
ek
n −1
∑a
i =1
−1
k ,i
Ti ,=
k 1, 2,..., n − 1.
(27)
Theorem 4: The method given by Eq. (7) and (12) for
solving the boundary value problem (2) for sufficiently
small h gives a second order convergent solution.
To illustrate the implementation of the method and
error analysis of the BVP (1) and (2), two examples are
discussed in the following section.
NUMERICAL EXAMPLES
Using Eq. (19) in (27), the following result is
obtained:
ek ≤ lh 4 / Bi0 ,
k=
1, 2,..., n − 1,
Example 1: Consider the following boundary value
problem:
(28)
where, l is a constant independent of h. From Eq. (28),
it follows that:
Similarly, it can be proved that ‖𝐸𝐸‖ = 𝑂𝑂(ℎ4 ) for
the solution of BVP (2) given by Eq. (7) and (16).
These results are summarized in the following
theorems.
Theorem 1: The method given by Eq. (7) and (14) for
solving the boundary value problem (1) for sufficiently
small h gives a fourth order convergent solution.
Theorem 2: The method given by Eq. (7) and (16) for
solving the boundary value problem (2) for sufficiently
small h gives a fourth order convergent solution.
On the same fashion, truncation errors of the
Eq. (10) and (7) are calculated as:
x 0 < (ξ 1) < x 4
and
1
−
ti =
ε h 7 y (7) (ξ i ),
240
x i − 2 < (ξ i ) < x i +1,





2,3,..., n − 1.
i=

0
−ε y (3) ( x) +
=
y ( x) 81ε 2 cos ( 3 x ) + 3ε sin ( 3 x ) , 

(2)
=
y (0) 0, y=
(0) 9=
ε , y (1) 3ε sin ( 3) ,

(33)
k=
1, 2,..., n − 1,
The analytical solution of system (32) and (33) is:
y ( x ) = 3ε sin ( 3 x )
The observed maximum errors (in absolute values)
associated with y i , for the problem (32) and (33)
1 1 1
corresponding to different values of 𝜀𝜀 = , ,
are
16 32 64
summarized in Table 1 and 2, respectively. Similarly,
the observed maximum errors (in absolute values)
associated with y i , for the problem (32) and (33)
corresponding to improved end conditions are
summarized in Table 3 and 4, respectively.
(29)
Using Eq. (29) in (27), the following result is
obtained:
ek ≤ lh 2 / Bi ,
(32)
and,
E = O ( h 4 ).
37
−
t1 =
ε h 5 y (5) (ξ 1),
20
−ε y (3) ( x) +=
y ( x) 81ε 2 cos ( 3 x ) + 3ε sin ( 3 x ) , 

(1)
=
y (0) 0, y=
(0) 9=
ε , y (1) 3ε sin ( 3) ,

(30)
Remark: The comparison of Table 1 and 3 with 5
shows that the errors in absolute are better than that
developed by Akram (2012), corresponding to the
problem (32). Similarly, the comparison of Table 2 and
4 with 6 shows that the errors in absolute are better than
that developed by Akram (2012), corresponding to the
problem (33).
Example 2:
where, l is a constant independent of h From Eq. (30), it
follows that:
E = O (h 2 ).
(31)
(34)
where,
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Res. J. App. Sci. Eng. Technol., 7(23): 4859-4863, 2014
Table 1: Maximum absolute errors for problem (32) in y i
ε
N = 10
N = 20
1
7.39×10-4
5.09×10-5
16
1
32
1
64
10
3.16×10-4
2.16×10-5
1.39×10-6
1.30×10-4
8.77×10-6
5.60×10-7
Table 2: Maximum absolute errors for problem (33) in y i
ε
N = 10
N = 20
1
1.02×10-2
1.40×10-3
16
1
32
1
64
N = 40
1.73×10-4
3.80×10-3
4.84×10-4
6.15×10-5
1.40×10-3
1.00×10-4
2.00×10-5
2.49×10-7
2.52×10-8
1.75×10-9
1.00×10-7
9.90×10-9
6.81×10-10
1.28×10-6
1.27×10-8
2.41×10-9
4.25×10-7
5.33×10-9
8.04×10-10
Table 5: Maximum absolute errors for problem (32) in yi developed
by Akram (2012)
ε
N = 10
N = 20
N = 40
1
6.9×10-5
3.1×10-5
5.4×10-6
16
1
32
1
64
3.1×10-5
1.8×10-5
2.8×10-6
4.9×10-5
9.9×10-6
1.4×10-7
Table 6: Maximum absolute errors for problem (33) in yi developed
by Akram (2012)
ε
N = 10
N = 20
N = 40
1
2.5×10-3
1.9×10-4
1.4×10-5
16
1
32
1
64
6.8×10-4
5.7×10-5
5.0×10-6
1.2×10-4
1.3×10-5
1.6×10-6
Table 7: Maximum absolute errors for problem (34) in yi
corresponding to improved end condition
ε
N = 10
N = 20
N = 40
1
2.18×10-12
3.65×10-15
4.62×10-18
16
1
32
1
64
9
sin ( ε x ) .
The observed maximum errors (in absolute values)
associated with yi, for the problem (34) corresponding
1 1 1
to different values of 𝜀𝜀 = , ,
are summarized in
16 32 64
Table 7.
Quartic non polynomial spline function is used to
develop a numerical method for solving third order
singularly perturbed boundary value problem. The
method is second order convergent and fourth order
convergent as well using the improved end conditions.
The numerical illustration shows that the developed
method maintains a very remarkable high accuracy that
makes it very encouraging for dealing with the solution
of singularly perturbed boundary value problems.
REFERENCES
Table 4: Maximum absolute errors for problem (33) in yi
corresponding to improved end condition
ε
N = 10
N = 20
N = 40
1
3.54×10-6
2.94×10-8
6.57×10-9
16
1
32
1
64
( x)
CONCLUSION
Table 3: Maximum absolute errors for problem (32) in yi
corresponding to improved end condition
ε
N = 10
N = 20
N = 40
1
5.70×10-7
5.97×10-8
4.14×10-9
16
1
32
1
64
y ( x )= (1 − x )
N = 40
3.26×10-6
1.09×10-12
1.83×10-15
2.31×10-18
5.45×10-13
9.13×10-16
1.16×10-18
The analytical solution of the above problem is:
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