Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The interpretation – and compilation of predictive equations to represent the general
trend – of collected data is aided immensely by its graphical representation. Whilst,
by and large, predictive equations are more accurate and convenient for use in applications
than graphs, the latter is often preferable since it visually illustrates deviations
in the data, thereby giving an indication of reliability and the range of validity of the
equation. Combination of these two tools – a graph for demonstration and an equation
for use – is desirable to ensure optimal understanding. Often, however, the functional
dependencies of the dependent variable are only known for large and small values
of the independent variable; solutions for intermediate quantities being obscure for
various reasons (e.g. narrow band within which the transition from one regime to
the other occurs, inadequate knowledge of the physics in this area, etc.). The limiting
solutions may be regarded as asymptotic and the powered addition to a power,
s, of such asymptotes, f0 and f¥ , leads to a single correlating equation that is applicable
over the entire domain of the dependent variable. This procedure circumvents
the introduction of ad hoc curve fitting measures for the different regions and subsequent,
unwanted jumps in piecewise fitted correlative equations for the dependent
variable(s). Approaches to successfully implement the technique for different combinations
of asymptotic conditions are discussed. The aforementioned method of powered
addition is applied to experimental data and the semblances and discrepancies
with literature and analytical models are discussed; the underlying motivation being
the aspiration towards establishing a sound modelling framework for analytical and
computational predictive measures. The purported procedure is revealed to be highly
useful in the summarising and interpretation of experimental data in an elegant and
simplistic manner. / AFRIKAANSE OPSOMMING: Die interpretasie – en samestelling van vergelykings om die algemene tendens voor te
stel – van versamelde data word onoorsienbaar bygestaan deur die grafiese voorstelling
daarvan. Ten spyte daarvan dat vergelykings meer akkuraat en geskik is vir
die gebruik in toepassings as grafieke, is laasgenoemde dikwels verskieslik aangesien
dit afwykings in die data visueel illustreer en sodoende ’n aanduiding van die betroubaarheid
en omvang van geldigheid van die vergelyking bied. ’n Kombinasie van
hierdie twee instrumente – ’n grafiek vir demonstrasie en ’n vergelyking vir aanwending
– is wenslik om optimale begrip te verseker. Die funksionele afhanklikheid van
die afhanklike veranderlike is egter dikwels slegs bekend vir groot en klein waardes
van die onafhanklike veranderlike; die oplossings by intermediêre hoeveelhede onduidelik
as gevolg van verskeie redes (waaronder, bv. ’n smal band van waardes
waarbinne die oorgang tussen prosesse plaasvind, onvoldoende kennis van die fisika
in hierdie area, ens.). Beperkende oplossings / vergelykings kan as asimptote beskou
word en magsaddisie tot ’n mag, s, van sodanige asimptote, f0 en f¥, lei tot ’n enkel,
saamgestelde oplossing wat toepaslik is oor die algehele domein van die onafhanklike
veranderlike. Dié prosedure voorkom die instelling van ad hoc passingstegnieke
vir die verskillende gebiede en die gevolglike ongewensde spronge in stuksgewyspassende
vergelykings van die afhankilke veranderlike(s). Na aanleiding van die
moontlike kombinasies van asimptotiese toestande word verskillende benaderings
vir die suksesvolle toepassing van hierdie tegniek bespreek. Die bogemelde metode
van magsaddisie word toegepas op eksperimentele data en die ooreenkomste en verskille
met literatuur en analitiese modelle bespreek; die onderliggend motivering ’n
strewe na die daarstelling van ’n modellerings-raamwerk vir analitiese- en rekenaarvoorspellingsmaatreëls.
Die voorgestelde prosedure word aangetoon om, op ’n elegante
en eenvoudige wyse, hoogs bruikbaar te wees vir die lesing en interpretasie van
eksperimentele data.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4166 |
| Date | 03 1900 |
| Creators | De Wet, Pierre |
| Contributors | Du Plessis, J. P., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 117 p. : ill. |
| Rights | University of Stellenbosch |
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