MAPLE SIMULATION CODES FOR STABILITY ANALYSIS OF VARIABLE STEP SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR INTEGRATING A SYSTEM OF FIRST ORDER STIFF IVPS

Strength of numerical scheme is rated by the properties it possessed and in turn the kind of problems it can handle. Zero stable method can effectively handle ODEs problem. While, an A – stable method can solve stiff ODEs problem. Analyzing stability of block methods are been carried out using various software. This work aimed at using simplified Maple simulation code to critically analyze avariable step size multi-block backward differentiation formula for the solution stiff initial value problems of ordinary differential equations. The Graphical comparisons of the simulated result obtained is made using Matlab to depict the performing schemes.


INTRODUCTION
A Numerical method is a differential equation involving a number of consecutive approximations from which it will be possible to compute the solutions, sequentially. Backward Differentiation formula (BDF) is a family of implicit method for the numerical integration of ordinary differential equations. Stiff ordinary differential equations are equations where certain implicit methods, in particular block backward differentiation formulas (BBDF), perform better, usually better than explicit ones (Curtiss & Hirschfelder, 1952). The formula undergoes different development and modifications.
The following scholar contribute tremendously with regard to BDF and BBDF (Cash, 1980); (Ibrahim et al.,2007); (Sulaiman et al.,2013a(Sulaiman et al., & 2013b; (Musa & Unwala, 2019), (Sagir & Abdullahi, 2022), ( Soomro et al., 2022), (Nasarudin et al.,2020), (Abdullahi et al, , & 2023. A system of stiff ordinary differential equations represent a couple of physical systems varying with very different times scales: That is they are systems having some components varying much more rapidly than others. Most of the methods stated are zero stable, A-stable or both, and displays different degree of accuracy of the scale error and executional time.

MATERIAL AND METHODS
In this section, we are considering Maple code for the critical analysis of the steps adopted in achieving zero and A-stable criteria of a 2-point multi -block super class of BBDF developed by Abdullahi et al (2023)  Maple Code for Analyzing Zero -Stability of the Methods Definition 1 (Zero Stability): A linear multistep method is said to be zero stable if no root of the first characteristics polynomial has modulus higher than 1 and that any root with modulus 1 is simple. (Sulaiman et al,2013) In the method (1) and if = 1.The constant coefficient matrix can be found as FUDMA Journal of Sciences (FJS) ISSN online: 2616-1370ISSN print: 2645-2944Vol. 7 No. 4, August, 2023, pp 113 -121 DOI: https://doi.org/10.33003/fjs-2023-0704-1905  To find the first characteristic polynomial, using the coefficients matrices, we use
If r = 2 the constant coefficient matrix is given as Step Step 3: (3)

CONCLUSION
A new 2 point multiblock super class of BDF for integrating system of first order stiff IVPs is considered in this work for a critical stability analysis. A simplified Maple algorithm is adopted to analyzes how to achieve zero and Astability criteria, which remained necessary properties for optimal performance of a numerical scheme, particularly in handling stiff system of IVPs are studied with simplified code. The scheme considered in the work is variable step size, which has a variable in the formula that can have different step sizes ratios. In this work, = 1 , = 2 & = 1 2 are adapted in generating the methods and all its stability criteria.