Babangida B^{*} and Musa H
Department of Mathematics and Computer Sciences, Umaru Musa Yar'adua University, Katsina, Nigeria
Received Date: September 13, 2016; Accepted Date: September 27, 2016; Published Date: October 03, 2016
Citation: Babangida B, Musa H (2016) Diagonally Implicit Super Class of Block Backward Differentiation Formula with OffStep Points for Solving Stiff Initial Value Problems. J Appl Computat Math 5: 324. doi: 10.4172/21689679.1000324
Copyright: © 2016 Babangida B, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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A new formula called 2point diagonally implicit super class of BBDF with two offstep points (2ODISBBDF) for solving stiff IVPs is formulated. The method approximates two solutions with two offstep points simultaneously at each iteration. By varying a parameter ρ ∈ (–1,1) in the formula, different sets of formulae can be generated. A specific choice of ρ =3/4 is made and it was shown that the method is both zero and Astable. A comparison between the new method and the existing 2point block backward differentiation formula with offstep points (2OBBDF) is made. The results show that the new method outperformed existing 2OBBDF method in terms of accuracy.
Offstep; Diagonally implicit super class of block backward differentiation formula; Stiff IVPs; Implicit block method; Astability
Consider a system of first order stiff initial value problems (IVPs) of the form:
With in the interval a ≤ x ≤ b where and .
System (1) is said to be stiff if it contains widely varying time scales, i.e., some components of the solution decay much more rapidly than others. Most realistic stiff systems do not have analytical solutions so that a numerical procedure must be used. Stiff ODEs occur in many fields of engineering and physical sciences such as electrical circuits, vibrations, chemical reactions, kinetics etc.
Developing methods for solving stiff problems remains a challenge in modern numerical analysis. Curtiss and Hirschfelder [1] discover Backward Differentiation Formula (BDF). Since then most of the improvements in the class of linear multistep methods have been based on BDF because of its special properties. Ibrahim [2] introduced rpoint block BDF (BBDF). Super class of block BDF, which is both zero and Astable, was developed by Suleiman [3,4]. The method is derived from 2point block BDF and outperformed both 2BBDF and 1BDF in terms of accuracy.
In order to gain an efficient numerical approximation in terms of accuracy and computational time, a super class of diagonally implicit BBDF method can be considered. The study of diagonally implicit for multistep attracted some researchers such Ababneh [5], Alexander [6], Musa [7] and Zawawi [8]. Abasi [4] developed a 2point Block BDF Method with offstep points for solving Stiff ODEs which differs from all the methods above because it calculates two solution values with offstep points simultaneously at each iteration. The motivation of this research is to develop a new method that would be called diagonally implicit superclass of BBDF with offstep points.
Derivation
In this work, two solution values, y_{n+1} and y_{n+2} with step size h, and two offstep points and which are chosen at the points where the step size is halved, are formulated in a block simultaneously. The formulae are computed using two back values y_{n1} and y_{n} with step size h. The formula is derived with the aid of this diagram below (Figure 1).
Definition 2.1: The 2point diagonally implicit super class of block backward differentiation formula with offstep points is defined as:
Where, represents the first point, k = i = 1 represents the second point, represents the third point and k = i = 2 represents the fourth point. The formula (2) is derived from Taylor’s series expansion as follows:
Definition 2.2: Linear operator L_{i} associated with first, second, third and fourth point DI2SBBDF with offstep point is defined by:
First point: To derive the first point let and define the operator as
Expanding (7) as Taylor series about x_{n} and collecting like terms gives
Where,
The coefficient of is normalized to 1. Solving the simultaneously equation (9) for the values of and gives the formula for as
Similar procedure is applied as in the derivation of first point to obtain the second, third and fourth points as:
For absolute stability of the method, ρ is Chosen to be in the interval (–1,1) as in Suleiman [3]. By choosing in equation (10), (11), (12) and (13) to obtain the 2–point diagonally implicit super class of BBDF with offstep points as follows:
In matrix form, equation (14) can be written as
Definition 2.3: Method (15) is diagonally implicit if the matrix in its left hand side is an upper triangular.
This section derives the order of the method corresponding to the equations in (14). It can be written in the following form:
Equation (16) can be written as in matrix form as:
Definition 3.1: The order of the block method (14) and its associated linear operator L given by:
Expanding the function and its derivative as Taylor series around x gives
Substituting (19) and (20) into (18) represents
The difference operator (21) and the associated method (14) is considered of order p if E_{0} = E_{1} = E_{2} = … = E_{p} = 0 and E_{p+1} ≠ 0
In this case
Therefore, the method (14) is of order 2, with error constant
The stability properties of method (14) are discussed here. We begin by defining zero and Astability taken from Suleiman [3].
Definition 4.1: A LMM is said to be zero stable if no root of the first characteristics polynomial has modulus greater than one and that any root with modulus one is simple.
Definition 4.2: A LMM is said to be Astable if its stability region covers the entire negative halfplane
The method (14) can be rewritten in matrix form as follows:
Definition 4.3: Let Y_{m} and F_{m} be vectors defined by
Y_{m} = [y_{n+1}, y_{n+1},…, y_{n+r}]^{T}, F_{m} = [f_{n+1},f_{n+2},…,f_{n+r}]^{T} r = 2, and n = 2m (See Suleiman [3]).
Method (14) can be written in matrix form as follows:
A_{0}Y_{m} = A_{1}Y_{m–1}+h(B_{0}F_{m–1}+B_{1}F_{m}). (27)
Where
Substituting scalar test equation y′ = λy(λ<0, λ complex) into (27) and using gives
The stability polynomial of (14) is given by
i.e.,
For zero stability, we set in (30) to obtain:
Solving equation (31) for t gives the following roots:
t = 0, t = 0, t = 0.350014 and t = 1. (32)
From the definition 4.1, method (14) is zerostable. The stability region of method (14) is determined by substituting t = e^{iθ} and the graph is shown below (Figure 2):
From the definition 4.2, method (14) is Astable.
Newton’s iteration is used in implementing the method. The procedure is described as follows. We begin by defining the error.
Definition 5.1
Let y_{i} and y(x_{i}) be the approximate solution of (1)
Then the absolute error is given by:
(error_{i})t = (yi)t –(y(xi))t. (33)
The maximum error is defined by
Where, T is the total number of steps and N is the number of equations.
Define from (14)
Where and are the back values.
Let denote the (i + 1)th iterative values of y_{n+j} and define
Newton’s iteration for the 2point SBBDF with offstep point method takes the form:
This can be written as
and in matrix form, equation (38) is equivalent to
To validate the efficiency of the methods developed, the following stiff IVPs are solved:
Exact solutions: y_{1}(x) = e^{–39x}+ e^{–x}
y_{2}(x) = e^{–39x}+ e^{–x}
Eigen values: 1 and 39
Source: Musa [9]
Exact solution: y_{1}(x) = e^{–x}
y_{2}(x) = e^{–x}
Eigen values: 1 and 200
Source: Ibrahim [2].
Exact Solution: y(x) = sin x + e^{–20x}.
Source: Abasi [4]
Exact Solution: y(x) = e^{–100x} + x
Source: Abasi [4]
The numerical results for the test problems given in section 6 are tabulated. The problems are solved with 2OBBDF and 2ODISBBDF methods. The number of step taken to complete integration and maximum error for the different methods is presented and compared in the tables below. In addition, the graph of Log_{10} (MAXE) against h for each problem is plotted (Figures 26) in order to give the visual impact on the performance of the method. The notations used in the tables are listed below:
The following abbreviations are used in the tables:
2ODISBBDF = 2point super class BBDF with offstep points
2OBBDF = 2point block BDF method with offstep points of order 5
h = step size
NS = total number of steps
MAXE = maximum error
Time = computational time in seconds (Table 1).
H  Method  NS  MAXE  Time  

1.  10^{–2} 10^{–4} 10^{–6} 
2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 
1000 1000 100000 100000 10000000 10000000 



2.  10^{–2} 10^{–4} 10^{–6} 
2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 
500 500 50000 50000 5000000 5000000 
7.17251e003 1.03577e004 7.35564e005 1.12034e008 7.35775e007 1.96752e010 
1.40432e003 1.68700e001 1.41352e001 1.13470e+000 2.41100e+000 9.01800e+001 

3.  10^{–2} 10^{–4} 10^{–6} 
2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 
100 100 10000 10000 100000 100000 
8.05923e002 1.86882e002 1.46355e003 4.39784e006 1.47126e005 4.48628e010 
5.90201e004 1.18580e001 2.01000e002 5.03090e001 2.98923e+000 3.76200e+001 

4.  10^{–2} 10^{–4} 10^{–6} 
2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 2OBBDF 2ODISBBDF 
500 500 50000 50000 5000000 5000000 
1.95750e002 2.62911e002 7.16455e003 1.03577e004 7.35564e005 1.12034e008 
3.08300e003 1.99800e001 5.92900e002 1.38300e+000 9.91000e+000 1.15600e+002 
Table 1: Numerical results for problems 1, 2, 3 and 4.
To give the visual impact on the performance of the method, the graphs of Graph of Log_{10} (MAXE) against h for the problems tested are plotted. Given below are the graphs of the scaled maximum error arranged problem by problem.
From the table 1 above it can be seen that 2ODISBBDF method outperformed 2OBBDF method in terms of accuracy. The graphs also show that the scaled errors for the 2ODISBBDF method are smaller when compared with that in 2OBBDF method.
A new method of order 2 that is suitable for solving stiff initial value problems has been developed. The stability analysis has shown that the method is both zero and Astable. Accuracy and the execution time of the derived method are compared with the existing 2point block backward differentiation formula with offstep points (2OBBDF). This comparison shows that the new method outperformed the existing 2OBBDF method in terms of accuracy. The computation time for the new method is seen to be competitive. The graphs also show that the scaled errors for the 2ODISBBDF method are smaller when compared with that in 2OBBDF method.