Dr. Paul Muir2025-12-302025-12-30https://dspace.academy.edu.ly/handle/123456789/1871In this thesis, we discussed methods for solving BVODEs that involve defect control. We reviewed some of the background literature for standard CMIRK schemes and Hermite- Birkho interpolants. Then, numerical experiments were presented with plots of the absolute scaled defect to observe the location of the maximum defect for each subinterval for standard CMIRK schemes and Hermite-Birkho (H-B) interpolants on BVODEs. These results showed that standard CMIRK schemes, for orders 4, 5 and 6, do not, in general, lead to defects for which the location of the maximum defect on each subinterval and for each problem is the same.Mono-Implicit Runge-Kutta (MIRK) methods and continuous MIRK (CMIRK) methods, are used in the numerical solution of boundary value ordinary di erential equations (ODEs). One way of assessing the quality of the numerical solution is to estimate its maximum defect, which is the amount by which the solution fails to satisfy the ODE. The standard approach is to perform two point sampling of the defect on each subinterval of a mesh which partitions the problem domain to estimate the maximum defect.Continuous Runge-Kutta Methods