CONVERGENCE TEST FOR THE EXTENDED 3 - POINT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR INTEGRATING STIFF IVP

In this work, a new scheme is generated from the extended 3–point super class of block backward differentiation formula for integrating stiff IVP and the proposed method is subjected to convergence test. The proposed scheme is found to be zero stable, consistent and of order 5. Thus, possess all the required criteria for convergence. The scheme can approximate the values of three points at a time per integration step. The scheme maintained the same technique of co-opting a stability control parameter ( ⍴ ) in the formula and by adjusting its value within the interval (−1,1) , more A-stabled schemes can be generated. However, this research considers ⍴ = − 1 9 and arrived at zero and A– Stabled method, capable of solving any stiff IVPs. Hence, the proposed convergent scheme can be used for integrating stiff IVPs and archives accuracy of scale error and less executional time.


INTRODUCTION
Block numerical method can generate more approximate solution values at a time per integration step, this phenomena makes it an easier to converge faster than single step scheme, which generate only one solution value per iteration. Backward differentiation formula was first discovered by Curtiss & Hirschfield (1952), and then it is extended by (Cash, 1980). The implicit block BDF method was proposed by (Ibrahim et al., 2007), Super class aspect of BBDF formula by (Sueiman et al.,2013), subsequent development of super class BBDF by (Musa & Unwala, 2019), the diagonally implicit aspect of BBDF formula (Zawawi et al, 2012), other extension and developments can be found in (Abdullahi et al., 2023), (Sagir & Abdullahi, 2023a), (Fatokun et al.,2005). These schemes among others possessed various degree of accuracy of the scaled error in one way or the other when it comes to solutions of stiff IVPs. A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small (Sulaiman et al., 2013). Due to the preferences of seeking approximate solutions to most of the stiff problems, numerical schemes are been developed continuously with various capacities to handle current realities of stiff IVPs. Most of the methods stated are zero stable, A-stable or both, and recorded an effective accuracy of the scaled errors and of executional time. This research aimed at using an implicit super class formula, extended 3 -point super class of backward differentiation formula for solving first order stiff IVPs developed by (Musa & Unwala, 2019) to generate another scheme with different value of the free parameter, ρ = − 1 9 and go further to test a convergence criteria for the proposed scheme.

MATERIALS AND METHODS
In this section, we consider the extended 3 -point super class of backward differentiation formula for solving first order stiff IVPs developed by ( Musa & Unwala, 2019) of the form From (1) we consider ρ = − 1 9 throughout the work. Theorem (1): Henrici (1962) stated the following conditions for convergence of Linear Multi-Step Method (LMM): i. Necessary condition for convergence of the Linear Multi-step Method (1) is that the modulus of none of the root of the associated polynomial ( ) exceeds one, and that the roots of modulus one is simple. The condition, thus imposed on ( ) is called the condition of zero stability. ii.
A necessary condition for convergence of the Linear Multi-step Method (1) is that the order of the associated difference operator be at least one. The condition that the order ≥ 1, is called the condition of consistency. To investigate the convergence of the method (1), the method need to meet conditions I and II in the stated theorem.

Condition of Zero Stability
The stability of the method (1) can be obtains by applying the standard test equation of the form ′ = ʎ ʎ ,  (3) The stability polynomial of (1) is obtained by evaluating from (3) using the relation   = 0 (6) Solving (6) for gives the roots as t = 1, t = 0.455458367 and t = −0.027686857 Therefore by definitions (1), the method is zero Stable.

Condition of Consistency
A necessary condition for convergence of the Linear Multi-step Method (1) is that the order of the associated difference operator be at least one. The condition that the order ≥ 1, is called the condition of consistency.

Order of the Method
In this section, we derive the order of the methods (1). Now, define the method (1) in the general matrix form as follows ∑ * Where 0 * , 1 * , 0 * and 1 * are square matrices defined by  (9) where p is unique integer such that E q = 0, q = 0,1, … p and E p+1 ≠ 0, where the E q are constant matrice (Suleiman et al., 2013). Now using the definition (1)