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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)

Introduction


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

References
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.