Algebra: The set R of real numbers, Relation of order in R, Principle of Mathematical Induction, Complex numbers; Analysis: Functions: Odd, Even and Periodic functions Hyperbola functions and their graphs; Co-ordinate geometry: Conic sections in rectangular co-ordinates, parabola, ellipse and hyperbola; Parametric equations: Plane polar coordinates, polar curves; Differentiation:

Relies theonim and the mean-value theorems, Taviors theorem, Repeated Differentiation, Applications for Differentiation, Indeterminate form; Vector algebra and its application.

This course emphasizes programming in a scientific environment using either FORTRAN 90 or C++ Syntax and semantics. It will broaden students’ on how to program using the either of the programming Languages and prepare students with the necessary programming background to proceed with C++/Fortran 90. Topics will include algorithm development, data types, control structures, functions, inpu t/ output, arrays, strings, data structures, bits, pointers, dynamic memory allocation, library functions and the preprocessor. The following features are also introduced: Inline functions, references, and default argu ments, function overloading and function ternplates.

Emphasis of theoretical basis should be emphasized. It is strongly recommended that the course be taught with at least one of the following mathematical software; MATLAB, MATHEMATICA, MAPLE. MATHCAD or any other mathematical package that the instructor may be familiar with. Students should not only be able to use the inbuilt furictionalities of the adopted software but also should be able to write their own programs in the adopted mathematical software. Nearly singular systems. Block Direct Methods.E.g. Block LU decomposition. Some direct methods of solving linear system of equations. Some iterative methods of solving linear system of equations.


GMRES Iteration
Alternating-Directed Implicit (ADI) Iterative Methods. Block Iterative Methods. Using iterative Methods to solve large linear sparse systems.

1. Methods of solving constrained problems in Rⁿ, n > I

• Linear programming methods, Duality. Theorem of

complementary Slackness principle

• Transportation problem
  
• Assignment problem
  
• Integral orogramming problem
  

2. Unconstraint Non-linear problems in Rⁿ, n >1.

• Steepest descent
  
• Newton’s and Conjugate direction methods
  
• Nelder and Mead

1.0 Classical Control Theory

1.1 Review of Laplace/Fourier Transforms and their use in the algebraic representation of linear systems: Transfer functions

1.2 Transient Response of linear system

1.3 Frequency Response of linear system

1.4 Feedback control: Route-Hurwitz, Nyquist Stability criteria. Root loci.

1.5 Statistical method of systems Identification: Elements of Time series Analysis

1.6 Sample data process

1.7 Feedback Controls System Design: Proportional, Derivative and Integral control and their combinations

General introduction to integral equations: Fredholm and Volterra equations. Application of integral equations to partial differential equations: Integral Representaions of the laplace and Poisson equations. The Helmhiz equation Symmetric Kernels: the complex Hilbert space. The Fredholm operator and its properties. Orthonormal systems of functions. Fundamental properties of eigenvalues and eigenfunctions for symmetric kernels. The bilinear form of symmetric kernels. The Hilbert Schmidt theorem, Its application in the solution of symmetric integral equations.


The Rayleigh-Ritz method for finding the first eigenvalue

Singular Integral Equations: The Abel Integral Equation

Integral Transformations: The Fourier transform: Laplace Transform merbods.

Application to Volterra Equation with convolution-type kernels.

Introduction
Equations of first order, first degree and higher (variable separable, exact equations, variation of parameters and understanding coefficients series solutions)

Fourier and Laplace Transforms-Application to linear system of differential equations

Systems of Linear Differential Equations

Autonomous and norn-autonomous equations. Fundamental of solutions and application.

System of non-linear differential equations

• Equilibrium Stationary, periodic solution, integral manifold
  

• The phase diagram, trajectories
  

• Critical points
  

introduction to the theory of stability

• Stability by linearization, direct method
  

• Poincare, Lyapunov, asymptotic stability
  

Introduction to perturbation theory

• Centre manifold

• Bifurcation of equilibrium solutions

• Hopf bifurcation

1. The simple linear model

2. The General Linear Model (Matrix formulation). The multiple Linear Regression and the Polynomial regression models as examples

3. Least squares estimation for the full rank General Linear models. Statistical inference in GLM.

4. Decomposition of the Total, Regression and Error sum of squares. Uses of the various decompositions.

Review of Probability Theory, Regularity of Stochastic Processes, Convergence of Random Walks to Brownian Motion. Brownian Motion and its Martingales, Diffusion Processes, Stochastic integrals, Ito’s Formula, Stochastic Differential Equations. Application in Industry and finance.

Techniques of Banach Spaces and Hilbert Spaces, Linear functionals, Dual Spaces, Orthogonal Projections and Ries-Representation Theorem, Reflexive Spaces, Banach-Steinhaus Theorem and Applications, Hann-Banach Theorem, Open Mapping Theorem, Closed Graph Theorem, Weak Convergence, Adjoint operators.