Abstract:
In this dissertation various models with variational forms similar to
that of the wave equation are considered, i.e. second order hyperbolic
type partial differential equations. These models include several linear
vibration problems and heat conduction models taking phase-lag into
account.
Clearly numerical methods need to be used to solve these problems
and the Finite Element Method (FEM) is used in this study. Before
applying such a method, existence of a solution needs to be established.
Therefore, a review of the work by Van Rensburg and Van der Merwe
(2002) on general second order hyperbolic type problems was done.
The results were not only presented, but additional remarks and a
discussion which assists in applying the theory were also included. To
obtain convergence results and error estimates when FEM is applied
to the various models, general convergence results were presented. For
this the article by Basson and Van Rensburg (2013) was used.
The first model considered consists of two serially connected Timoshenko
beams. One of the beams was modelled as embedded in an
elastic material, while the other beam is either free or subjected to
a prescribed external load. This model can be adapted for a single
beam with di fferent loads on separate parts. To apply the convergence
theory though it was necessary to use the double beam model, while a
single beam model can be used when FEM is applied. This was demonstrated
when these models were used to model a plant with a tap root
system. In this biological application various things were investigated,
including different forms of FEM, a comparison of the results for the
static double beam and static single beam, and the dynamics of the
beam. These experiments indicated that the two models compare well
and gave insight into how the parameter modelling the resistance of
the soil in
influences key aspects of how the plant reacts due to external
forces.
Models for rigid bodies attached to beams were also investigated. The
equations used to describe the dynamics of a beam with a tip body
were derived, with special attention given to the interface conditions.
Consequently, a model problem for an intermediate rigid body between
two Timoshenko beams was investigated.
Hyperbolic heat conduction models were also considered and the application
to bio-heat transfer in skin was discussed. Specifically, a model
from the work by Dekka and Dutta (2019) was investigated. Their
approach to existence of solutions was scrutinized and it was found
that their application of existence results from the 2002 article by Van
Rensburg and Van der Merwe is incomplete. Due to this the exposition
of the theory is improved in the dissertation.
For all the mentioned models, the existence and uniqueness of a solution
were obtained by defining the relevant function spaces and proving the
required properties. Convergence was also established from the general
convergence results and the systems of ordinary differential equations
were obtained which can be used to obtain numerical approximations.
