DEPARTMENT OF MATHEMATICS

This project work will introduced the reader to the concept of metrics (a class of functions which is regarded as generalization of the notion of distance) and metric spaces with examples. It emphasises the notion of vector space which generalizes
the concept of addition and scalar multiplication. The notion of inner product allows us to generalize the notion of the dot product of vectors. It also allows us to talk about angle between vectors, and their norms. We can discuss the notion of distance between vectors. Also the combination of inner product with a vector gives a scalar. An inner product space is a special type of vector space that has mechanism for computing a version of dot product that can be defined in real or complex vector space, as long as it satisfies some conditions. The properties arising from a metric in an inner product space is important example of a metric space

Real life models that are characterized by randomness are best described using Stochastic Ordinary Differential Equations (SODEs). Most SODEs that satisfy existence and uniqueness theorem are often insoluble via the use of analytic methods. Numericals solution are derived and the complexity of generating approximate solutions to SODEs are heightened by the presence of the phenomenon called stiffness. Hence, A-stable numerical methods are desired. This
is a strigent requirement that can only be met by implicit methods.
Two families of A-stable numerical methods for numerical approximation of SODEs are derived using Ito Taylor, Taylor’s series and undetermined coefficients methods. The stability analysis of both families of methods are established using the Boundary locus method.
Families of methods developed are A-stable for 𝑘 ≤ 12. Mean-square stable and strong stable for order 𝑝 = 1. The Numerical implementation generated on the standard test problems in the literature shows that the numerical solution to methods developed are in most cases better when compared to numerical solution generated by existing methods in the literature designed for stiff SODEs. The numerical solutions are also compared with exact solution where they are
available. The numerical solution mimic the exact solution, hence the proposed methods are well suited for the treatment of stiff SODEs.

This project focuses on exploring iterative techniques for solving linear systems. The goal is to examine the efficiency and accuracy of iterative methods such as Jacobi and Gauss-seidel method used in solving linear systems commonly found in scientific applications