This is an advance course in Mathematical Methods to prepare students for a higher level of mathematics. The topics to be covered in this course include Calculus of variation; Lagrangeâ€™s function and its associated density; Necessary condition for a weak relative extremum; Lagrangeâ€™s equation and geodesic problems; Du-Bios-Raymond equation and corner conditions; variable end points and related theorems; sufficient conditions for a minimum; isoperimetric problems; variational integral transforms â€“ Laplace, Fourier and Hankel transforms; convolution theorems.
This course focuses on the followings: Navier-Stokes and energy equations for viscous incompressible fluids. Dynamical similarity and Reynolds number. Steady one-dimensional flow of viscous fluid. Two-dimensional flow and small disturbance theory. Radial flow between plane walls, axi-symmetric jets. Boundary layer theory. Inviscid compressible flow, energy equation and compressibility effect. Unsteady one-dimensional flow. Equations of motion for some specific types of flow and ensuring solutions.
This course is compulsory for applied mathematicians who wish to spenalise in the areas of operations research, multi-stage system, simulation, modelling, discrete and continuous optimization, The course synopsis is as follows: Single variable optimization, multivariable techniques and gradient method; Linear programming models; The Simplex: formulation and theory; Integer programming; transportation problem; non-linear programming;Quadratic programming; Kuhn-Tucker methods; Optimality criteria; Two-person zero-sum games.
This course is compulsory for both pure and applied mathematicians,particularly those who wish to further their studies in real or complex analysis.The course synopsis is spelt out as follows: Definitions of open, closed sets and neighbourhood; topological spaces, coarser and finer topologies; bases and subbases; separation axioms; compactnesss; local compactness; and connectedness. Construction of new topologies from given ones; subspaces and quotient spaces.Continuous functions; homeomorphisms, topological invariants, spaces of continuous functions. Pointwise and uniform convergence.
Theory and solutions of first-order equations, second order linear equations: classification, characteristics, canonical forms, cauchy problems, elliptic equations, Laplaceâ€™s and poissonâ€™s formula; property of harmonic equations. Hyperbolic equations: wave equations, Retarded potential transmissions line, Reiman method. Parabolic equations: Diffusion equation: Singularity function, boundary and initial-value problem. Separation of variables , heat equations, wave equations, laplace equation in two dimension Laplaceâ€™s equation for a circular disk mean-value theorem, poissonâ€™s integral formular. Method of characteristics for linear and quasilinear wave equations. The Dâ€™Alembert solution for the wave equation. The single first order differential equation. The Quasilinear first order Pde.