Abstract
This dissertation presents a high-order compact finite difference method (CFDM) for solving ordinary
differential subjected to Robin boundary conditions. Compact finite difference schemes (CFDS) have
recently drawn the attention of many researchers due to their remarkably high accuracy. Their primary
benefit is that they can achieve it accuracy on very small grids. As far as we know, no CFDS have been
derived for Robin boundary conditions. According to the literature, researchers typically utilized a first-order
explicit finite difference method to approximate Neumann and Robin boundary conditions. However, this
compromises the overall accuracy of the scheme. As a result, we developed newhigher-order finite difference
schemes for approximating Robin boundary conditions in this work, which are sixth-order accurate.
We test the applicability and performance of sixth-order CFDS in a wide range of ordinary differential
equations (ODEs). The method is utilized to solve general ODEs subjected to Robin boundary conditions.
We also used CFDS to solve Lane-Emden equations with several types of boundary conditions arising in
various real-world problems. Lastly, the method is used to solve singular boundary value problems subjected
to Neumann and Robin boundary conditions. Linear and nonlinear equations are considered. The nonlinear
equations are dealt with by the quasilinearisation technique. The results are compared against published
results from other methods, some results are also compared to exact solutions. We also carry out a theoretical
convergence analysis of the CFDS, which is found to be consistent with the numerical rate of convergence.
CFDS yields highly accurate results, presented as graphs and tables.