A NEW HIGHER ORDER SECOND DERIVATIVE BLENDED BLOCK LINEAR MULTISTEP METHODS FOR THE SOLUTIONS STIFF INITIAL VALUE PROBLEMS

This paper is concerned with the accuracy and efficiency of a higher order second derivative blended block linear multistep method for the approximate solution of stiff initial value problems. The main methods were derived by blending of two linear multistep methods using continuous collocation approach. These methods are of uniform order ten. The stability analysis of the block methods indicates that the methods are A–stable, consistent and zero stable hence convergent. Numerical results obtained using the proposed new block methods were compared with those obtained by the well known ODE solver ODE 15s to illustrate the accuracy and effectiveness. The proposed block method is found to be efficient and accurate hence recommended for the solution of stiff initial value problems.


INTRODUCTION
Mathematical modeling of many problems in real life, Science, Medicine, Engineering and the like gave rise to systems of linear and non linear Differential Equations. In some cases, the differential equations could be solved analytically while in other case like the Holling Tanner equations and the Van Der Pol equations they are too complicated to be solved by analytical methods. Thus solving such problems becomes an uphill task hence the application of numerical methods for approximate solutions to these differential equations.
In this paper, the application of the nine step order ten blended block linear multistep method for the numerical solutions of stiff initial value problems (1) was considered. A potentially good numerical method for the solution of stiff system of ordinary differential equations (ODEs) must have good accuracy and some wide region of absolute stability. One of the first and most important stability requirements for linear multistep methods is A-stability as proposed by Enright (1974). The nine step blended block linear multistep methods is of a high order and A stable hence the application of the method here which makes it suitable for the solution of non linear ODEs. The solution of stiff system of ODEs has been considered by Chollom Kumleng (2012) and so on.

THE NINE STEP BLENDED LINEAR MULTISTEP METHOD
The nine step blended linear multistep method is constructed based on the continuous finite difference approximation approach using the interpolation and collocation criteria described by Lie and Norsett (1981) called multistep collocation (MC) and block multistep methods by Onumanyi et al. (1994Onumanyi et al. ( ,1999. We define based on the interpolation and collocation methods the continuous form of the k-step 2nd derivative new method as '' n n n n n n n n n n n n n x f x f x f x f x f y x x y h h x y x f x f x f x   2677τ 348107τ 71957τ  τ-+  -+  2  3  4  1260h 2240h 580608h 362880h  6  7  8  9  36929τ  6613τ  547τ  41τ  y(τ+x ) =y + -+  -n n+8  5  7  6  8  829440h  774144h  967680h  870912h  10  11  53τ  53τ  127588h  ---9 10 467775 29030400h 29030400h ae ö ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç è ø % f n      are less than or equal to 1. Since the block method is consistent and zero-stable, the method is convergent (Henrici 1962). These new methods are consistent since their orders are 11, it is also zero-stable, above all, there are A -stable as can be seen in figure 1. The new ten step discrete methods that constitute the block method have the following orders and error constants as shown below. The nine step blended block multistep methods has uniform order of and error constants of

REGIONS OF ABSOLUTE STABILITY OF THE METHODS
The absolute stability regions of the newly constructed blended block linear multistep methods (8) and (12)

NUMERICAL EXAMPLES
We report here a numerical example on stiff problem taken from the literature using the solution curve. In comparison, we also report the performance of the new blended block linear multistep methods and the well-known Matlab stiff ODE solver ODE15S on the same problems and on the same axes.

Problem1 Oregonator (Chemical Reaction) Problem
The oregonator chemical reaction model is a theoretical model of autocatalytic reaction. It is a chemical dynamics of the oscillatory reaction. It is a reaction between     934 293 514 352  645 788 801  ,  ,  ,  ,  , , , , 100000 1000000 1000000 100000 1000000 10000000 1000000 1000000 100000 Problem 2 is a model of the relationship between Lidocaine and Irregular Heart beat. Lidocaine belong to a group of drugs known as anti-arrhythmic which work by preventing sodium from being pumped out on the cells of the heart to help the heart beat normally. From our solution curves, it was observed that normalcy in the heart beat can be attained with the use of Lidocaine within the correct dosage. Our solution curves coincide with the solutions of ODE 15s. The numerical results from figures 2 and 3 reveal the accuracy of the newly constructed higher order blended block linear multistep methods (BBLMM) for step numbers 9. It can be seen clearly from the curve that our new methods perform favourably better than the well known ODE15S for the problems solved in problem 1and 2. It was also observed that the new methods have better stability regions than the conventional Adams Moulton method for step number 9.

RECOMMENDATIONS
This method is recommended for the solution of stiff system of ODEs since they are A-stable which implies a wider range of stability for effective performance.