Introduction
According to [1, 28], Mathematica is the universe solely
incorporated environs for proficient computation. Since issued in 1988,
it has had unfathomed outcome on the manner electronic computers are
applied in various proficiencies and other areas.
Technically, Mathematica is seen to a great degree viewed as a principal
achievement of software system technology. It is observed as the most
prominent individual applications of all time formulated. Thus, it
comprises of a huge lay out of new algorithmics and prominent
proficiency inventions. Amongst its most essential inventions are its
interlinked algorithmic noesis cornerstone and its conceptions of
symbolical computer programming and of document-centered interfaces. See
[1, 24, 28].
Mathematica as a math software package is split up into two component
parts: the Mathematica forepart (notebook) and Mathematica kernel
(core). The forepart takes stimulus and manifests the end product. The
forepart is the most significant that grants the user to act together
with the schemes for the objective of performing infinitesimal calculus
and to uphold them recycle or for point of reference. The Mathematica
kernel is the unseeable component part of the ciphering computer program
that put in effect completely the calculations. This aspect is very
essential to this research study because all codes written on MPIBAF and
executed is implemented on this platform. See [1, 12, 24, 28].
Prompted by the need to design a Mathematica pseudocodes of block Adams
family under the Mathematica kernel platform to estimate engineering
science problems, biologic and natural phenomenon. A number of bookmen
started to invent efficient and real accurate techniques to solve ODEs
(1). Among them admits; [13] developed hybrid block predictor-hybrid
block corrector for the solution of first-order ordinary differential
equations. This approach involves tedious manual computation and fixed
step size strategy. [29] devised the predictor-corrector block
iteration method for solving ODEs. This technic is coded in C language
and executed via variable step size strategy. [17] implemented a
seventh-order linear multistep method in a predictor-corrector mode or
block mode: which is more efficient for the general second order initial
value problem. Again, the implementation is a combination carried out on
a composed cipher in PERL computer programming terminology and
Mathematica as well. This combination is manually executed on fasten
step size strategy. [2] considers a new block-predictor corrector
algorithm for the solution of\(y^{{}^{\prime\prime\prime}}=f\left(x,y,y^{{}^{\prime}},y^{{}^{\prime\prime}}\right)\) and whose execution takes
a difficult task with fixed step size scheme. All of these techniques
and implementations involves utilizing different fixed length to arrive
at the any max-errors. This eventually leads to increase human effort
and large space size used. Hence, the need to design firstly,
Mathematica pseudocodes for implementing block Adams family as discussed
in the abstract. Secondly, a suitable length that will ensure
efficiency, accelerate the mode of convergence and stabilize the loop
via the introduction of the convergency limits.
Consider the first-order ordinary differential equation of the form
\(y^{{}^{\prime}}=f\left(x,y\right),\) \(y(x_{0})=y_{0},\) for\(\text{xϵ}\left[a,\ \ b\right]\), (1)
where \(f\) meets the Lipschitz precondition as acknowledged in [7, 8,
10, 17].
Hence, the primary objective of this study is geared towards the
development of Mathematica pseudocodes for implementing block Adams
family. This build up exist in literatures as discussed in [19-23].
Other superior qualities of (1) necessitate varying the step size,
determining worth step size and checking max ciphered computer errors.
Scientifically, for computation involving the use of Mathematica
pseudocodes to demonstrate the superiority of the study and engender a
new breed of computational mathematics. See [1, 12, 24, 28].
The motivation behind this study stem from the fact that, computational
mathematics without the use of Mathematical packages carries a difficult
task during looping which leads to low accuracy and large global errors
inherent due to approximation via manual computation. Thus, there is a
need to build an iterative code for looping with better accuracy and low
computer error.
The remain of this study is examined as succeeds: Section 2 Materials
and Methods. Section 3 Results. Section 4 Discussions as stated in [3,
4, 20].