Download Solving Two-Point Second Order Boundary Value Problems Using

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 4 (2016) pp 2407-2410
© Research India Publications. http://www.ripublication.com
Solving Two-Point Second Order Boundary Value Problems Using
Two-Step Block Method with Starting and Non-Starting Values
Zurni Omar
Mathematics Department, School of Quantitative Sciences,
Universiti Utara, Malaysia, Sintok, Kedah, Malaysia.
Oluwaseun Adeyeye
Mathematics Department, School of Quantitative Sciences,
Universiti Utara, Malaysia, Sintok, Kedah, Malaysia.
This was the concept Ha (2001) considered where the
shooting technique was implemented together with the fourth
order Runge-Kutta to solve the two point boundary value
problem.
Islam, Aziz and Sarler (2010) also solved the two-point
second order boundary value problem using collocation
method with Haar wavelets while Liang and Jeffrey (2010)
adopted the homotopy analysis method to solve the two point
second order boundary value problem. Hasni, Majid and Senu
(2013), Jator and Li (2009), Sagir (2013) and See, Majid and
Suleiman (2014) developed different block methods using
numerical integration and interpolation methods to directly
solve the two-point second order boundary value problems.
However, these authors only presented the solution of the
boundary value problems but emphasis was not made on how
the method was implemented. Neither was it discussed if the
accuracy of the method can be improved by the introduction
of starting values. Hence, this paper focuses on the direct
solution of the two-point second order boundary value
problems with investigation on the effect of implementing the
block method with or without starting values.
The organization of this paper is as follows. Section Two
shows the derivation of the two-step block method from
Taylor series while Section Three shows the analysis of the
basic properties of the block method. Numerical examples are
solved in Section Four and a conclusion of the paper is given
in Section Five.
Abstract
Second order boundary value problems can be solved directly
using block methods. This solution can be implemented with
or without the use of starting values. However, a comparison
of the effects of starting values on the results gotten from
solving a second order boundary value problem has not been
investigated into. Therefore, in this work a two-step block
method for the numerical solution of a two-point second order
boundary value problem is described. The derivation of the
method from Taylor series expansions leads to a family of
block matrix equations which are used to solve the boundary
value problems. The method is implemented with starting and
non-starting values and the results are compared with
previously existing methods. The results show an
improvement in the accuracy of the method when
implemented with starting values.
Keywords: Boundary Value Problems, Block Method, Taylor
series, Starting and Non-Starting Values.
Introduction
This paper is aimed at finding the numerical solution to the
second order boundary value problem of the form
y ''  f  x, y, y ' , y  a    , y  b   
where
(1)
a, b,  ,  are given constants.
Derivation of the Two-Step Block Method
The discrete scheme for the two-step block method is
constructed from the general form of the linear multistep
method of the form
Just like initial value problems, boundary value problems
often have multiple solutions or the solution may even fail to
exist. Conventionally, boundary value problems are solved
numerically by transforming it into a system of first order
equations and then solved by any initial value method
suitable. However, the additional initial conditions need to be
assumed before the first order system can be solved. An
iterative process known as the shooting method is adopted to
vary the assumed initial conditions on one boundary until the
condition on the other boundary is satisfied (Hoffman, 2001).
(2)
Expanding individual terms in (2) using Taylor series
expansion and substituting the expansions back in (2) gives
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 4 (2016) pp 2407-2410
© Research India Publications. http://www.ripublication.com
(3)
Rewriting (3) in matrix form and equating coefficients of
y
( m)
Adopting matrix inverse method,
xn yields
yn 1 , yn  2 , yn' 1 , yn'  2
are determined as
(11)
which is equivalent to the following block,
Adopting
0 , 1 ,
matrix inverse method, the values
0 , 1 and  2 are obtained as below
of
(12)
(4)
Analysis of the Basic Properties of the Block Method
Substituting (4) in (2) gives the discrete scheme
Order and Error Constant
Associated with any linear multistep method are the
polynomials
(5)
Taking the first derivative of (2) at
xn , we expect the method
of the form
(13)
known as the first and second characteristic polynomials
respectively.
The minimum property a linear multistep method should
possess in order to guarantee convergence is identified as
stated in the theorem below.
(6)
Adopting the same approach of the discrete scheme by taking
the Taylor series expansion of (6), substituting the expansions
back in (6) and then solving the corresponding matrix
expression. The coefficients
01 , 11 , 01 , 11
and
 21
Theorem 3.1 (Fatunla, 1988)
A linear multistep method is convergent iff it is consistent and
zero-stable.
are
found to be
Definition 3.1 (Butcher, 2008)
The linear operator associated with equation (2) is defined as
(7)
Substituting (7) in (6) gives
In the same vein, the derivatives at
xn 1 and xn  2
(8)
are gotten
On expanding
(14)
yn  j and f n  j to obtain
to be
The
(9)
method
C0  C1 
is
said
 Cq  Cq 1 
to
be
of
order
q
 Cq  m1  0, Cq 2  0
if
and Cq  2 is the error constant, where m is the order of the
(10)
Combining equations (5), (8), (9) and (10) together gives the
following equation which can be written in a matrix form
differential equation under consideration.
The order of the two-step block method is gotten from
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 4 (2016) pp 2407-2410
© Research India Publications. http://www.ripublication.com
Plotting the roots of the stability polynomial in boundary
locus approach displays the region of absolute stability as
shown below.
This implies that the block method has order 3 with error
T
constant C   1 , 2 
5


45 45


Definition 3.2 (Jator and Li, 2009)
A linear multistep method is consistent if it has order
p  1.
Figure 1: Region of Absolute Stability for (12)
Adopting the approach in Jator (2007) to analyze the two-step
block method for zero stability, the block method (12) is
normalized to give the first characteristic polynomial
  R
Numerical Problems
The block method is implemented with and without starting
values and the results are displayed for each boundary value
problem considered. Two boundary value problems were
considered and for each example, the results are compared
with the results of previously existing methods. This is to
show the accuracy of the self-starting approach in comparison
to the implementation starting with Taylor series.
The following notations are used in the tables:
2SBMSS: Two-Step Block Method implemented with nonstarting values
2SBMTSS: Two-Step Block Method implemented with
starting values
JL2009: Block Method from Jator and Li (2009)
H2013: Block Method from Hasni, Majid and Senu (2013)
as
where A is the identity matrix of dimension 2 and
matrix of dimension 2 given by
0
The roots of
  R  0
A1 is a
satisfy R j  1, j  1, 2 . Hence,
the block is said to be zero-stable. Hence, the block method is
convergent since it is both consistent and zero-stable.
The region of absolute stability is determined by obtaining the
stability polynomial from
Problem 1 (Jator and Li, 2009)
(15)
where z   h and m is the order of the differential equation.
Hence, the stability polynomial for the block method is gotten
as
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 4 (2016) pp 2407-2410
© Research India Publications. http://www.ripublication.com
Table 1: Comparison of results for solving Problem 1 with
stepsize h = 0.1
Conclusion
This paper has presented the implementation of a two-step
block method with the introduction of starting values. The
starting value adopted is the conventional Taylor series
expansion. It is observed from both Tables 1 and 2 that the
results of the two-step block method in comparison to the
results of Jator and Li (2009) yielded the same results. This
shows the consistency in self-starting implementation of block
methods, however, when the block method was implemented
with Taylor series starting values, the accuracy was given a
boost and also seen to give better accuracy in Table 3.
References
[1]
Problem 2 (Jator and Li, 2009)
[2]
[3]
[4]
Table 2: Comparison of results for solving Problem 2 with
stepsize h = 0.1
[5]
[6]
[7]
Problem 3 (Hasni, Majid and Senu, 2013)
[8]
[9]
Table 3: Comparison of results for solving Problem 3 with
stepsize h = 0.01
[10]
[11]
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