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Mathematical Methods in Engineering and Science

Mathematical Methods in Engineering and Science. Instructor: Dr. Bhaskar Dasgupta, Department of Mechanical Engineering, IIT Kanpur. The aim of this course is to develop a firm mathematical background necessary for advanced studies and research in the fields of engineering and science. Solution of linear systems. The algebraic eigenvalue problem. Selected topics in linear algebra and calculus. An introductory outline of optimization techniques. Selected topics in numerical analysis. Ordinary differential equations. Application of ODEs in approximation theory. Partial differential equations. Complex analysis and variational calculus. (from nptel.ac.in)

Lecture 24 - Boundary Value Problems, Question of Stability in IVP Solution

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Module I. Solution of Linear Systems

Lecture 01 - Introduction

Lecture 02 - Basic Ideas of Applied Linear Algebra

Lecture 03 - Systems of Linear Equations

Lecture 04 - Square Non-singular Systems

Lecture 05 - Ill-conditioned and Ill-posed Systems

Module II. The Algebraic Eigenvalue Problem

Lecture 06 - The Algebraic Eigenvalue Problem

Lecture 07 - Canonical Forms, Symmetric Matrices

Lecture 08 - Methods of Plane Rotations

Lecture 09 - Householder Method, Tridiagonal Matrices

Lecture 10 - QR Decomposition, General Matrices

Module III. Selected Topics in Linear Algebra and Calculus

Lecture 11 - Singular Value Decomposition

Lecture 12 - Vector Space: Concepts

Lecture 13 - Multivariate Calculus

Lecture 14 - Vector Calculus in Geometry

Lecture 15 - Vector Calculus in Physics

Module IV. An Introductory Outline of Optimization Techniques

Lecture 16 - Solution of Equations

Lecture 17 - Introduction to Optimization

Lecture 18 - Multivariate Optimization

Lecture 19 - Constrained Optimization: Optimality Criteria

Lecture 20 - Constrained Optimization: Further Issues

Module V. Selected Topics in Numerical Analysis

Lecture 21 - Interpolation

Lecture 22 - Numerical Integration

Lecture 23 - Numerical Solution of ODEs as IVP

Lecture 24 - Boundary Value Problems, Question of Stability in IVP Solution

Lecture 25 - Stiff Differential Equations, Existence and Uniqueness Theory

Module VI. Ordinary Differential Equations

Lecture 26 - Theory of First Order ODEs

Lecture 27 - Linear Second Order ODEs

Lecture 28 - Methods of Linear ODEs

Lecture 29 - ODE Systems

Lecture 30 - Stability of Dynamic Systems

Module VII. Application of ODEs in Approximation Theory

Lecture 31 - Series Solutions and Special Functions

Lecture 32 - Sturm-Liouville Theory

Lecture 33 - Approximation Theory and Fourier Series

Lecture 34 - Fourier Integral to Fourier Transform, Minimax Approximation

Module VIII. Overviews: PDEs, Complex Analysis and Variational Calculus

Lecture 35 - Separation of Variables in PDEs, Hyperbolic Equations

Lecture 36 - Parabolic and Elliptic Equations, Membrane Equation

Lecture 37 - Analytic Functions

Lecture 38 - Integration of Complex Functions

Lecture 39 - Singularities and Residues

Lecture 40 - Calculus Variations