Evaluation of applying Crum-based transformation in solving two point boundary value problems
Date
2016
Authors
Jogiat, Aasif
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Abstract
The aim of this research project is evaluating the application of the Crum-based transformation
in solving engineering systems modelled as two-point boundary value problems. The boundary
value problems were subjected to the various combinations of Dirichlet, Non-Dirichlet and Affine
boundary conditions. The engineering systems that were modelled were in the elds of electrostatics,
heat conduction and longitudinal vibrations. Other methods such as the Z-transforms and iterative
methods have been discussed. An attractive property of the Crum-based transformation is that
it can be applied to cases where the eigenparameters (function of eigenvalues) generated in the
discrete case are negative and was therefore chosen to be explored further in this dissertation. An
alternative matrix method was proposed and used instead of the algebraic method in the Crum-
based transformation. The matrix method was tested against the algebraic method using three unit
intervals. The analysis revealed, that as the number of unit intervals increase, there is a general
increase in the accuracy of the approximated continuous-case eigenvalues generated for the discrete
case. The other observed general trend was that the accuracy of the approximated continuous-
case eigenvalues decrease as one ascends the continuous-case eigenvalue spectrum. Three cases:
(Affine, Dirichlet), (Affine, Non-Dirichlet) and (Affine, Affine) generated negative eigenparameters.
The approximated continuous-case eigenvalues, derived from the negative eigenparameters, were
shown not to represent true physical natural frequencies since the discrete eigenvalues, derived from
negative eigenparameters, do not satisfy the condition for purely oscillatory behaviour. The research
has also shown that the Crum-based transformation method was useful in approximating the shifted
eigenvalues of the continuous case, in cases where the generated eigenparameters were negative:
since, as the number of unit intervals increase, the post-transformed approximated eigenvalues
improved in accuracy. The accuracy was also found to be better in the post-transformed case than
in the pre-transformed case. Furthermore, the approximated non-shifted and shifted continuous-
case eigenvalues (except the approximated continuous-case eigenvalues generated from negative
eigenparameters) satis ed the condition for purely oscillatory behaviour.
Description
A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the
Witwatersrand, in ful llment of the requirements for the degree of Master of Science in Engineering, Johannesburg, 2016
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Citation
Jogiat, Aasif (2016) Evaluation of applying Crum-based trans-formation in solving two point boundary value problems, University of the Witwatersrand, <http://hdl.handle.net/10539/22660>