Master of Science ( Mathematics ) UKM: September 2023

SEMESTER 1( 16 CREDITS )

STPD6024 Research Methodology

This course provides guidance in planning, implementing and succeed in scientific research. Students are introduced to the philosophy of science and ethics necessary to be adopted by researchers. Students are given exposure to determine and manage risks in scientific research. Apart from that, issues and rules related to research such as intellectual property, copyright, plagiarism etc. will be discussed. Subsequently, students are guided to plan their research and prepare a research proposal. For this, students are trained with techniques in information search both manually and on-line, identifying issues and research objectives, planning research and experimental design within their period of study. Students are required to prepare their research proposal according to the format and free from plagiarism. Students are given the opportunity to present and defend their proposal in a seminar. Students will be evaluated based on written and oral presentation of the research proposal, and final examination

STQM6114 Topology

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. This course concerns with properties that are preserved under continuous deformations of objects that emerges through the development of concepts from geometry and set theory. The most basic and traditional division of topology namely point set topology will be considere

STQM6124 Algebra

This course begins by reviewing back one main algebraic structures that is groups and rings together with some concepts related to both. This includes subgroup/subring, ideal, quotient group/ring and mapping of group/ring. Then the extension to polynomial ring and field is introduced. Various extensions of field to domain such as integral domain, Euclidean domain and unique factorisation domain. The embeddedment of domain into field leads to the construction of Galois group. This course ends with reviewing several selected articles on algebraic structures.

STQM6134 Functional Analysis

This course generalizes the study of linear algebra, in particular on finite and infinite dimensional vector spaces. This study is supported by various limit-related structures such as metric, inner product, norm and topology. Then it is added together with linear operators acting upon these spaces. The combination of algebraic and limit-related from new spaces namely Banach and Hilbert space. Thus, this course is basically the study of the properties of these spaces.

CORE COURSE - PURE MATHEMATICS

SEMESTER 2 ( 24 CREDITS )

STQM6224 Complex Analysi

This course gives a view of basic analytic functions such as power series representations, Cauchy-Goursat’s theorem with various versions, maximum modulus theorem with various versions, conformal mappings, Riemann mapping theorem and Phragmen-Lindelof’s theorem. This course also introduces a theorem in analytic function space and shows the application of Runge’s theorem and Mittag-Leffler’s theorem. Harmonic functions including solutions to Dirichlet’s problem, singularities, Picard’s theorem and special functions such as the gamma functions, zeta functions and important theorems for entire functions are also introduced.

STQM6988 Research Project

Research project is a compulsory course, which is either a practical training, an industrial training, a literature review or a research. Every student does this project under the supervision of a supervisor. Each student must choose a suitable topic within his/her programme module and it must be approved by the supervisor. The student must complete a report, which is either a critical review to the selected topic, a new theory or a new model in its own way

**STQM6074 Numerical Analysis

This course covers numerical methods for solving ordinary/partial differential equations (ODEs/PDEs). The problems considered include initial value problems and boundary value problems for ODEs. Numerical methods discussed include one-step and multi-step methods with fixed or variable stepsize for stiff and non-stiff as well as chaotic equations/system of equations. Further, the topics covered include stability and error analysis. Introduction to numerical methods for PDEs such as finite difference/element methods. Analysis of hyperbolic and elliptic equations. Convergence, consistency, order and stability of methods. Applications to certain problems in engineering/science.

**STQM6324 Numerical Methods for Heat Transfer and Fluid Flow

This course will present heat transfer and fluid flow models and their numerical solutions. The course begins with heat transfer and fluid flow model formulations. Steady and unsteady heat conduction up to three dimensions will be discussed. Next, the course discusses Crank-Nicholson method, steady and unsteady convection and diffusion up to three dimensions, and their numerical solution schemes include hybrid and power laws. Flow regimes and numerical solution methods will also be presented.

**STQM6414 Dynamical Systems

The course aims to introduce basic concepts in discrete-time and continuous-time dynamical systems. These include discussion on some topics such as locally property, stability comprises structural stability, hyperbolic and homoclinic point, strange attractor, Lyapunov exponent etc. Some other concepts such as bifurcation, chaos and fractal will also be explored.

CORE COURSE - APPLIED MATHEMATICS

SEMESTER 1( 16 CREDITS )

STPD6024 Research Methodology

STQM6414 Dynamical Systems

STQM6074 Numerical Analysis

STQM6534 Fluid Mechanics

The aim of this course is to show how the ideal and viscous fluids can be modelled mathematically, and further, to investigate the behaviour of the fluids analytically and numerically, especially towards the Navier-Stokes equation. This course starts with general introduction to fluid and the principle of fluid static and kinematic. Discussion on ideal fluid includes continuity, Euler and Bernoulli equations. Potential flow and incompressible flow will also be discussed. Most parts of this course discuss viscous fluid, which leads to Navier-Stokes equation, its derivations and exact solutions, as well as steady and unsteady flows. Basic flows, Stokes flow, laminar and turbulent flows, dimensional analysis, similarity method as well as Reynolds number and its importance will also be discussed. In addition, boundary layer theory and fluid instabilities will also be discussed in detail

CORE COURSE - APPLIED MATHEMATICS

SEMESTER 1( 24 CREDITS )

STQM6988 Research Project

STQM6064 Mathematical Modeling and Methods

Mathematical modeling is a process of building mathematical formulation for a physical phenomenon to gain better insights about it. The course intends to train the students in building, analyzing and solving mathematical models for certain complex problems (especially deterministic models of physics). Fundamental concepts of mathematical modeling will be explained. Dimensional procedures, approximation and dimensional analysis will be discussed first. Models introduced are linear and nonlinear. Analytical solution methods discussed include some of the followings: perturbation expansion technique, asymptotic method, transformations, special functions, Fourier series, calculus of variations and integral methods. The usage of computer algebra systems like Maple/Mathematica will be emphasized.

**STQM6154 Network Science

This course introduces mathematical theories in network science. Network science is a multidiscipline field which investigate problems that can be understood through network approach. Among the aims of network science are to find cross-network equations and increase understanding of systems which are represented by networks through data analysis. The use of network science can be found in mathematics, social networks, biological systems and transportations.

**STQM6524 Non-Deterministic Linear Dynamical Systems Modeling

This course is designed to exhibit the capability to model the dynamical system with non-deterministic condition as a stochastic process which fulfills the linear stochastic differential equations. It, furthermore can lead to the stochastic integral. This includes various Newtonian dynamical systems with noise, planning of monitoring system, management and screening of information. From this model, the definition of the concept of stochastic process is exhibited and in addition the analytical and numerical Ito’s Stochastic Calculus is constructed to solve the mentioned model. The relationship between the stochastic differential equations and the diffusion process is discusses to the research boundary.

**STQM6324 Numerical Methods for Heat Transfer and Fluid Flow

**- Elective Course

Source: http://www.ukm.my/siswazahfst/mathematics-programme-cw/