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  • Applied Linear Algebra

  • Nonlinear Optimization

  • Numerical Methods

  • Vector and Complex Analysis

  • Differential Equations and Applications

  • Approximation Techniques

 

Sl. No

Topic

Lectures

1.

Solution of Systems of Linear Equations.

  • Introduction
  • Basic Ideas of Applied Linear Algebra
  • Systems of Linear Equations
  • Square Non-Singular Systems
  • Ill-Conditioned and Ill-Posed Systems

1-5

2.

The Algebraic Eigenvalue Problem.

  • The Algebraic Eigenvalue Problem 
  • Canonical Forms, Symmetric Matrices 
  • Methods of Plane Rotations 
  • Householder Method, Tridiagonal Matrices 
  • QR Decomposition, General Matrices

6-10

3.

Selected Topics in Linear Algebra and Calculus

  • Singular Value Decomposition 
  • Vector Space: Concepts 
  • Multivariate Calculus 
  • Vector Calculus in Geometry 
  • Vector Calculus in Physics 

11-15

4.

An Introductory Outline of Optimization Techniques.

  • Solution of Equations 
  • Introdcution to Optimization 
  • Multivariate Optimization 
  • Constrained Optimization: Optimality Criteria 
  • Constrained Optimization: Further Issues

16-20

5.

Selected Topics in Numerical Analysis

  • Interpolation
  • Numerical Integration 
  • Numerical Solution of ODE's as IVP 
  • Boundary Value Problems, Question of Stability in IVP Solution 
  • Stiff Differential Equations, Existence and Uniqueness Theory 

21-25

6.

Ordinary Differential Equations

  • Theory of First Order ODE's
  • Linear Second Order ODE's
  • Methods of Linear ODE's
  • ODE Systems 
  • Stability of Dynamic Systems 

26-30

7.

Application of ODE's in Approximation Theory

  • Series Solutions and Special Functions 
  • Sturm-Liouville Theory 
  • Approximation Theory and Fourier Series
  • Fourier Integral to Fourier Transform, Minimax Approximation

30-34

8.

Overviews: PDE's, Complex Analysis and Variational Calculus

  • Separation of Variables in PDE's, Hyperbolic Equations
  • Parabolic and Elliptic Equations, Membrane Equation 
  • Analytic Functions 
  • Integration of Complex Functions 
  • Singularities and Residues 
  • Calculus of Variations

35-40

  • Exposure to undergraduate mathematics.

  • Applied Mathematical Methods by B. Dasgupta (Pearson Education).

Applied Mathematical Methods: http://home.iitk.ac.in/~dasgupta/MathBook/

  1. Advanced Engineering Mathematics by E. Kreyszig (Wiley)

  2. Scientific Comuting by M. T. Heath (McGraw-Hill)



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