ANZIAM J. 53 (E) pp.E44–E58, 2012 E44
Quartic spline solution of a third order singularly perturbed boundary value problem
Ghazala Akram
1(Received 21 July 2011; revised 16 April 2012)
Abstract
Singularly perturbed boundary value problems are solved using various techniques. The spline of degree four is used for the approximate solution of a third order self adjoint singularly perturbed boundary value problem. Convergence analysis is given and the method is proved to be second order convergent. Two examples numerically illustrate the method.
Contents
1 Introduction E45
2 Consistency relations E47
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1 Introduction E45
3 Convergence of the method E49
4 Numerical results E53
5 Conclusion E55
References E56
1 Introduction
The solution of a singularly perturbed boundary value problem exhibits a multiscale character. There is a thin transition layer, where the solution varies swiftly, while away from the layer the solution behaves regularly and varies gradually, therefore many complications are faced in solving singularly perturbed boundary value problems using standard numerical methods. In recent years, a large number of special purpose methods have been established to provide accurate results.
Consider the self adjoint singularly perturbed boundary value problem of the form
Ly(x) = −εy
(3)(x) + p(x)y(x) = f(x), p(x) > 0 ,
y(0) = α
0, y(1) = α
1, y
(1)(0) = α
2, (1) or
Ly(x) = −εy
(3)(x) + p(x)y(x) = f(x), p(x) > 0 ,
y(0) = α
0, y(1) = α
1, y
(2)(0) = α
3, (2)
where α
0, α
1, α
2and α
3are constants and ε is a small positive parameter
(0 < ε 6 1), also f(x) and p(x) are smooth functions. In this problem, we
take p(x) = p = constant. Singularly perturbed problems usually crop up
in chemical reactions, quantum mechanics, fluid mechanics, optical control,
1 Introduction E46
etc. Three principal approaches are available to solve such problems, namely, finite difference methods, finite element methods and spline approximation methods.
The existence and uniqueness of singularly perturbed boundary value prob- lems (bvps) was discussed by Howers, Kelevedjiev, and Roos et al. [ 3, 5, 10].
A numerical method for a class of second order singularly perturbed two point
boundary value problems on a uniform mesh using a compressed spline was
proposed by Mohanty and Jha [8]. Quartic non-polynomial spline functions
were used to develop a class of numerical methods for solving self adjoint sec-
ond order singularly perturbed two point boundary value problems by Tirmizi
et al. [13]. Difference schemes were developed for the numerical solution of
second order two point singularly perturbed boundary value problems using a
tension spline by Kadalbajoo and Patidar [4]. A sextic spline has been used
to solve second order singularly perturbed boundary value problems by Khan
et al. [6] and the method proved to be fifth order accurate. The method has
been presented to solve a class of second order singularly perturbed two point
boundary value problems for certain ordinary differential equations having
singular coefficients [14]. A numerical technique has been derived for a class
of singularly perturbed two point boundary value problems on an uniform
mesh using polynomial cubic spline by Rashidinia [9] and was given to be
second order convergent. A fourth order finite difference scheme based on a
nonuniform mesh for a class of singular two point perturbed boundary value
problem was described by Kumar [7] and shown to be of order four. This
method has also solved a class of third order singularly perturbed boundary
value problems [1]. The singularly perturbed boundary value problem for
quasilinear third order ordinary differential equations involving two small pa-
rameters has been considered by Su-rang et al. [11] and asymptotic solutions
constructed by the method of two step expansions. The derivation of solutions
to a singular two point boundary value problem for third order nonlinear
differential systems by employing the method of descent has been discussed
by Du [2]. Numerical solutions of fifth order two point singularly perturbed
boundary value problems have been considered by Syam and Attili [12].
2 Consistency relations E47
This paper is organized in four sections. Section 2 determines the consistency relation and end condition required for the solution of bvp ( 1) and (2).
Section 3 discusses the convergence analysis of the quartic spline method.
Finally, Section 4 discusses the results of two numerical examples.
2 Consistency relations
To develop the consistency relations the following fourth degree spline is considered:
Q
i(x) = a
i(x − x
i)
4+ b
i(x − x
i)
3+ c
i(x − x
i)
2+ d
i(x − x
i) + e
i(3) defined on [a, b], where x ∈ [x
i, x
i+1] with equally spaced knots, x
i= a + ih , i = 0, 1, . . . , N , and h = (b − a)/N . Using the following notations
Q
i(x
i) = y
i, Q
i(x
i+1) = y
i+1, Q
(3)i(x
i) = m
i, Q
(3)i(x
i+1) = m
i+1, Q
(1)i(x
i) = n
i,
the coefficients in (3) are determined to be a
i= (m
i− m
i+1)/24h ,
b
i= m
i/6 ,
c
i= −(3h
3m
i+ h
3m
i+1+ 24hn
i+ 24y
i− 24y
i+1)/24h
2, d
i= n
i,
e
i= y
i.
2 Consistency relations E48
Applying the first and second derivative continuities at knots, that is, Q
(µ)i−1(x
i) = Q
(µ)i(x
i), for µ = 1 and 2, the following relations are derived:
1
12h h
3m
i−1+ h
3m
i− 12(hn
i−1+ hn
i+ 2y
i−1− 2y
i) = 0 , 1
12h
2h
3m
i−1+ 8h
3m
i+ h
3m
i+1− 24hn
i−1+ 24hn
i− 24y
i−1+ 48y
i− 24y
i+1] = 0 , which leads the following consistency relation in terms of m
iand y
im
i−2+ 11m
i−1+ 11m
i+ m
i+1= 24
h
3(−y
i−2+ 3y
i−1− 3y
i+ y
i+1) , (4) for i = 2, 3, . . . , N − 1 . Using Equation (1), Equation (4) is
(24ε + ph
3)y
i−2+ (−72ε + 11ph
3)y
i−1+ (72ε + 11ph
3)y
i+ (−24ε + ph
3)y
i+1= h
3(f
i−2+ 11f
i−1+ 11f
i+ f
i+1), (5) for i = 2, 3, . . . , N − 1 . Since the above system consists of (N − 2) equations with (N − 1) unknowns, one more equation is required. The following relation describing truncation error is used in this regard:
T
0= −h
3(a
0m
−1+ a
1m
0+ a
2m
1) +
a
3y
−1+ a
4y
0+ a
5y
1+ a
6y
2+ a
7hy
(1)0. (6)
Moreover, the values of y
−1and m
−1can be calculated and are y
−1= (5y
0− 10y
1+ 10y
2− 5y
3+ y
4),
m
−1= (5m
0− 10m
1+ 10m
2− 5m
3+ m
4).
Using Taylor series of the right-hand side of Equation (6) along with the coefficients of y
0, hy
(1)0, h
2y
(2)0, h
3y
(3)0, h
4y
(4)0, the value of a
is are calculated as
a
0= 1 , a
1= − 5
3 , a
2= 3
2 , a
3= − 1 2 , a
4= 3 , a
5= − 7
2 , a
6= 1 , a
7= 1 .
3 Convergence of the method E49
Using the values of a
is, y
−1and m
−1in Equation (6), the required end condition is
3
2 ε + 17 2 ph
3y
1− (4ε + 10ph
3)y
2+
5
2 ε + 5ph
3y
3−
1
2 ε + ph
3y
4− h
3− 10
3 f
0+ 17
2 f
1− 10f
2+ 5f
3− f
4−
− 1
2 ε + 10 3 ph
3α
0− εhα
2+ O(h
5) = 0 . (7)
Similarly, the end condition for system (2) is
16ε + 103 12 ph
3y
1− (19ε + 10ph
3)y
2+ (10ε + 5ph
3)y
3− (2ε + ph
3)y
4− h
3− 43
12 f
0+ 103
12 f
1− 10f
2+ 5f
3− f
4−
5ε + 43 12 ph
3α
0+ εh
2α
3+ O(h
5) = 0 . (8)
3 Convergence of the method
The system of Equations (5) and (7) provides the required quartic spline solution of bvp ( 1), which can be written in the matrix form
AY − h
3DF = C , (9)
where
A =
(
32ε +
172ph
3) −(4ε + 10ph
3) (
52ε + 5ph
3) −(
12ε + ph
3)
11ph
3− 72ε δ
2δ
1ph
3+ 24ε 11ph
3− 72ε δ
2δ
1. .. . .. . ..
ph
3+ 24ε 11ph
3− 72ε δ
2δ
1ph
3+ 24ε 11ph
3− 72ε δ
2
,
3 Convergence of the method E50
with δ
1= ph
3− 24ε , δ
2= 11ph
3+ 72ε ,
D =
17
2
−10 5 −1
11 11 1
1 11 11 1
. .. ... ... ...
1 11 11 1
1 11 11
,
C = (c
1, c
2, . . . , c
N−1)
T, Y = (y
1, y
2, . . . , y
N−1)
Tand F = (f
1, f
2, . . . , f
N−1)
T. Also
c
1=
− 1
2 ε + 10 3 ph
3α
0− 10
3 h
3f
0− εhy
(1)0, c
2= (−24ε − ph
3)α
0+ h
3f
0,
c
i= 0 , i = 3, 4, . . . , N − 2 , c
N−1= (24ε − ph
3)α
1+ h
3f
N.
The exact solution is defined as ¯ Y = [y(x
1), y(x
2), . . . , y(x
N−1)], then Equa- tion (9) is rewritten as
A ¯ Y − h
3DF = T + C , (10)
where T = [t
1, t
2, . . . , t
N−1]
Twith t
1= − 181
120 h
5y
(5)(ζ
1), x
0< ζ
1< x
4,
t
i= 2εh
5y
(5)(ζ
i), x
i−2< ζ
i< x
i+1, i = 2, 3, . . . , N − 1 . (11) Moreover,
A( ¯ Y − Y) = AE = T , (12)
E = ¯ Y − Y = (e
1, e
2, . . . , e
N−1)
T. (13)
3 Convergence of the method E51
To determine the error bound the row sums S
1, S
2, . . . , S
N−1of matrix A are calculated:
S
1= X
j
a
1,j= − 1 2 ε + 5
2 ph
3, S
2= X
j
a
2,j= −24ε + 23ph
3, S
i= X
j
a
i,j= 24ph
3, i = 2, 3, . . . , N − 3 , S
N−1= X
j
a
N−1,j= 24ε + 23ph
3. (14)
Since the matrix A is irreducible and monotone, A
−1exists and its elements are nonnegative. Hence, from Equation (12),
E = A
−1T . (15)
Also, from the theory of matrices
A
−1A = I
(N−1)×(N−1), where
A =
a
1,1a
1,2. . . a
1,n−1a
2,1a
2,2. . . a
2,n−1.. . .. . . .. .. . a
n−1,1a
n−1,2. . . a
n−1,n−1
,
A
−1=
a
−11,1a
−11,2. . . a
−11,n−1a
−12,1a
−12,2. . . a
−12,n−1.. . .. . . .. .. . a
−1n−1,1a
−1n−1,2. . . a
−1n−1,n−1
and I =
1 0 . . . 0 0 1 . . . 0 .. . .. . . .. ...
0 0 . . . 1
.
3 Convergence of the method E52
Since each row sum of matrix I
(N−1)×(N−1)= 1 and A
−1A = I
(N−1)×(N−1), therefore each row sum of A
−1A equals 1; that is,
a
−11,1(a
1,1+ a
1,2+ · · · + a
1,n−1) + a
−11,2(a
2,1+ a
2,2+ a
2,n−1) + · · · + a
−11,n−1(a
n−1,1+ a
n−1,2+ · · · + a
n−1,n−1) = 1 ,
a
−11,1S
1+ a
−11,2S
2+ · · · + a
−11,n−1S
N−1= 1 , which is written in compact form as
N−1
X
i=1
a
−1k,iS
i= 1 , k = 1, 2, . . . , N − 1 . (16) Defining S
j= min S
i, then from Equation (16),
1 > S
ja
−1k,1+ a
−1k,2+ · · · + a
−1k,N−1.
It follows that
N−1
X
i=1
a
−1k,i6 1/S
j= 1/(h
3B
i0), (17) where
B
i0= (1/h
3)S
j> 0 , 1 6 i
06 N − 1 . From Equation (12), the error terms can be written as
e
j=
N−1
X
i=1