Numerical Solution of First Order Initial Value Problems Using a Self-Starting Implicit Two-Step Obrechkoff-Type Block Method

Corresponding Author: Oluwaseun Adeyeye Department of Mathematics, School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia Email: adeyeye_oluwaseun@ahsgs.uum.edu.my Abstract: The conventional two-step implicit Obrechkoff method is a discrete scheme that requires additional starting values when implemented for the numerical solution of first order initial value problems. This paper therefore presents a two-step implicit Obrechkoff-type block method which is self-starting for solving first order initial value problems, hence bypassing the rigour of developing and implementing new starting values for the method. Numerical examples are considered to show the new method performing better when com-pared with previously existing methods in literature.


Introduction
In a bid to model real-life problems in areas of engineering, biological sciences, physical sciences, electronics and many others, initial value problems are most times encountered (Shokri and Shokri, 2013).A sample first order initial value problem takes the form given below: ( ) ( ) , , y f x f y a α ′ = = (1) However in most cases, these initial value problems cannot be solved analytically and hence the need for numerical methods.These numerical methods are adopted to obtain an approximate solution to the initial value problem under consideration (James et al., 2013).Numerous numerical approaches have been proposed by scholars for the numerical approximation of initial value problems.These methods range from discrete schemes (Lambert, 1973;Butcher, 2008;Fatunla, 1988), to predictor-corrector methods (Kayode and Adeyeye, 2011;Adesanya et al., 2008;Awoyemi and Idowu, 2005) and then block methods (Omar and Kuboye, 2015;Hasni et al., 2013;Areo and Adeniyi, 2013).
One distinct family of methods for the numerical approximation of (1) above is the Obrechkoff methods.This family of methods is regarded to be distinct due to the presence of higher derivatives in the method.The general form of the k-step Obrechkoff method with l derivatives of y is given by Lambert (1973) as: with the implicit k = 2; l = 2 Adams-type method given as: (3) However, adopting the method presented in Equation 3 above requires the introduction of additional one-step numerical methods such as Euler or Trapezoidal method.This approach most times does not provide optimal accuracy and hence the need to adopt block methods.Block methods have been found to give better approximation as seen in the studies by (Jator and Li, 2012;Omar, 2004;Adesanya et al., 2014) amongst many others.Hence, this work presents a self-starting block method that will bypass the rigour of developing separate starting values and also with expected better accuracy when compared to previously existing methods in literature.
The second section of this article will show how the block method is derived and also discuss some basic properties of the block method while the next section will present some numerical examples and results.The fourth section of this article concludes this work.
This method is obtained by considering the following two-step method given as: Following the same steps adopted for deriving the implicit k = 2; l = 2 Nyström-type method given in Equation 7, the additional method is derived to give: Hence, Equation 7and 9 present the desired block method for the solution of (1).
However, to check for the order and the convergence of this block method, the following theorem and definitions are adopted.
Theorem 2.1 (Fatunla, 1988) A linear multistep method is convergent iff it is consistent and zero-stable.
Definition 2.1 (Jator and Li, 2012) A linear multistep method is consistent if it has order p≥1.Definition 2.2 (Butcher, 2008) The linear operator associated with Equation 2 is defined as: ( ) Expanding y n+j and ( ) i n j y + using Taylor series to obtain: The method is said to be of order q if C 0 = C 1 =... = C q = 0, C q+1 ≠ = 0 and C q+1 is the error constant.
Adopting this approach to the block methods to test for zero stability, the block method is normalized to give the first characteristic polynomial as: The roots of ρ(R) = 0 satisfy |R j | ≤ 1, j = 1, 2. Hence, the block is said to be zero-stable.
Therefore, since the block method is consistent and zero-stable, it is likewise convergent.
The region of absolute stability is determined by obtaining the stability polynomial from: Hence, the stability polynomial for the block method is gotten as: Plotting the roots of the stability polynomial in boundary locus approach displays the region of absolute stability as shown Fig. 1.

Numerical Examples
This section presents some numerical examples and corresponding results of the new two-Step Obrechkoff-Type block method in comparison to the two-step method in Equation 3 and other existing methods in recent literature.
The following notations are used in the tables of results:

Tested Problem 3.1 (Mixture Model)
In an oil refinery, a storage tank contains 2000 gal of gasoline that initially has 100 lb of an additive dissolved in it.In the preparation for winter weather, gasoline containing 2 lb of additive per gallon is pumped into the tank at a rate of 40 gal min −1 .The well-mixed solution is pumped out at a rate of 45 gal min −1 .Using a numerical integrator, how much of the additive is in the tank 0.1, 0.5 and 1 min after the pumping process begins?
Let y be the amount (in pounds) of additive in the tank at time t.We know that y = 100 when t = 0. Thus, the Initial Value Problem (IVP) modeling the mixture process is: The numerical result is shown in Table 1a and 1b.

Tested Problem 3.2 (SIR Model)
The SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time.The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people S(t), number of people infected I(t) and the number of people who have recovered R(t).This is a good and simple model for many infectious diseases including measles, mumps and rubella.It is given by the following three coupled equations: ( ) where, µ, γ and β are positive parameters.Define y to be: and adding Equations 13-15, we obtain the following evolution equation for y: ( )