Results
This aspect shows the computed effects of block Adams family yielded through Mathematica pseudocodes (MP) for the aim of demonstrating proficiency, high level of accuracy and skillfulness in avoiding wasted time and effort.
Two practical illustrations were considered and enforced employing MP at different convergency limits of \(10^{-2}\), \(10^{-3}\),\(10^{-4}\), \(10^{-5}\),\(\ 10^{-8}\), \(10^{-9}\ \)and\(10^{-13}\). See [11, 17, 18, 25, 26, 27] for details. The Mathematica pseudocodes of block Adams Family is coded utilizing Mathematica Kernel in concert with an algorithmic rule. This Mathematica pseudocodes is executed side by side as developed by the block Adams family. See appendix.
Time-tested problem 1. Consider the inhomogeneous IVP:
\(y^{{}^{\prime}}\left(x\right)=-10{(y-1)}^{2}\),\(\text{\ y}\left(0\right)=2\),
having the exact result \(y\left(x\right)=1+\frac{1}{1+10x}\). This time-tested problem 1 has been worked-out and published by [25, 26, 27] displaying that many predictor-corrector method, block method and block hybrid method suffers setbacks in the course of implementation such as lack of stability, fixity or firmness. Again, large number of predictor-corrector method [25, 26, 27] ends up as predicting-correcting technique without any concern for determining a suitable step size thereby result to developing the convergency limits for the loop. This time-tested problem 1 has been incorporated in [0,0.1] assuming length \(h=0.01\) and published in [25, 26, 27]. Table 1 is comprising of method employed, max-errors and convergency limits at various levels that displays the technique introduced in this study. MP executed more better as a results of the special quality of desire suitable step size leading to the discovery of the convergency limits of the loop.
Time-tested problem 2. Consider the initial value ODE:
\(y^{{}^{\prime}}=-10xy,\) \(y\left(0\right)=1\)
of which the exact result is
\(y\left(x\right)=exp(-5x^{2})\)
and has shown up in [11, 17, 18] employing the time-interval\([0,10]\) having various lengths. Table 2 presents the method employed, max-errors and convergency limits of [11, 17, 18] and MP as the technique of this study as slated in (18).
The language applied are enlisted beneath. Table I and Table II displays the computed results of time-tested problems 1 and 2 employing MP equated with subsisting techniques. The typifies make reference to on Table I and Table II are showed beneath.