Advanced Numerical Analysis

Course Description

This is an advanced course on Numerical Analysis by Prof. Sachin C. Patwardhan, Department of Chemical Engineering, IIT Bombay. It has been designed with the following learning objectives in mind:
- Clearly bring out role of approximation theory in the process of developing a numerical recipe for solving an engineering problem
- Introduce geometric ideas associated with the development of numerical schemes
- Familiarize the student with ideas of convergence analysis of numerical methods and other analytical aspects associated with numerical computation

It is shown that majority of problems can be converted to computable forms (discretized) using three fundamental ideas in the approximation theory, namely Taylor series expansion, polynomial interpolation and least square approximation. In addition, the student is expected to clearly understand role of the following four fundamental tools:
- Linear Algebraic Equation
- Nonlinear Algebraic Equations
- Ordinary Differential Equations- Initial Value Problem
- Optimization

Advanced Numerical Analysis
Prof. Patwardhan on Lecture 12: Solving ODE - BVPs Using Firute Difference Method
Not yet rated

Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/syllabus/syllabus.php?subjectId=103101111

# Lecture Play Lecture
I. Equation Forms in Process Modeling
1 Introduction to Numerical Analysis and Overview (1:33:47) Play Video
2 Fundamentals of Vector Spaces (46:53) Play Video
3 Basic Dimension and Sub-space of a Vector Space (48:29) Play Video
4 Introduction to Normed Vector Spaces (47:34) Play Video
II. Fundamentals of Vector Spaces
5 Examples of Norms, Cauchy Sequence and Convergence, Introduction to Banach Spaces (37:02) Play Video
6 Introduction to Inner Product Spaces (53:54) Play Video
7 Cauchy Schwaz Inequality and Orthogonal Sets (43:42) Play Video
8 Gram-Schmidt Process and Generation of Orthogonal Sets (28:30) Play Video
9 Problem Discretization Using Appropriation Theory (52:51) Play Video
10 Weierstrass Theorem and Polynomial Approximation (37:09) Play Video
11 Taylor Series Approximation and Newton's Method (47:50) Play Video
III. Problem Discretization Using Approximation Theory
12 Solving ODE - BVPs Using Firute Difference Method (46:32) Play Video
13 Solving ODE - BVPs and PDEs Using Finite Difference Method (48:22) Play Video
14 Finite Difference Method (contd.) and Polynomial Interpolations (48:34) Play Video
15 Polynomial and Function Interpolations, Orthogonal Collocations Method for Solving (34:06) Play Video
16 Orthogonal Collocations Method for Solving ODE - BVPs and PDEs (1:03:48) Play Video
17 Least Square Approximations, Necessary and Sufficient Conditions (50:31) Play Video
18 Least Square Approximations: Necessary and Sufficient Conditions (55:24) Play Video
19 Linear Least Square Estimation and Geometric Interpretation (48:54) Play Video
20 Geometric Interpretation of the Least Square Solution (Contd.) and Projection (48:14) Play Video
21 Projection Theorem in a Hilbert Spaces (Contd.) and Approximation (53:18) Play Video
22 Discretization of ODE-BVP using Least Square Approximation (52:55) Play Video
23 Discretization of ODE-BVP using Least Square Approximation and Gelarkin Method (49:11) Play Video
24 Model Parameter Estimation using Gauss-Newton Method (52:08) Play Video
25 Solving Linear Algebraic Equations and Methods of Sparse Linear Systems (53:12) Play Video
26 Methods of Sparse Linear Systems (Contd.) and Iterative Methods for Solving (48:43) Play Video
IV. Solving Linear Algebraic Equations
27 Iterative Methods for Solving Linear Algebraic Equations (52:41) Play Video
28 Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis (56:18) Play Video
29 Iterative Methods for Solving Linear Algebraic Equations: (58:01) Play Video
30 Iterative Methods for Solving Linear Algebraic Equations: Convergence (46:19) Play Video
31 Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis (46:32) Play Video
32 Optimization Based Methods for Solving Linear Algebraic Equations: Gradient Method (48:13) Play Video
33 Conjugate Gradient Method, Matrix Conditioning and Solutions (40:17) Play Video
34 Matrix Conditioning and Solutions and Linear Algebraic Equations (Contd.) (54:53) Play Video
35 Matrix Conditioning (Contd.) and Solving Nonlinear Algebraic Equations (53:00) Play Video
V. Solving Nonlinear Algebraic Equations
36 Solving Nonlinear Algebraic Equations: Wegstein Method and Variants of Newton's Method (35:02) Play Video
37 Solving Nonlinear Algebraic Equations: Optimization Based Methods (48:47) Play Video
38 Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis (56:35) Play Video
VI. Solving Ordinary Differential Equations – Initial Value Problems (ODE-IVPs)
39 Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis (Contd.) (56:33) Play Video
40 Solving Ordinary Differential Equations - Initial Value Problems (ODE-IVPs) (57:10) Play Video
41 Solving Ordinary Differential Equations - Initial Value Problems (59:55) Play Video
42 Solving ODE-IVPs : Runge Kutta Methods (contd.) and Multi-step Methods (55:50) Play Video
43 Solving ODE-IVPs : Generalized Formulation of Multi-step Methods (55:06) Play Video
44 Solving ODE-IVPs : Multi-step Methods (contd.) and Orthogonal Collocations Method (52:19) Play Video
45 Solving ODE-IVPs: Selection of Integration Interval and Convergence Analysis (51:26) Play Video
46 Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.) (47:18) Play Video
47 Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.) (57:52) Play Video
48 Methods for Solving System of Differential Algebraic Equations (48:43) Play Video
49 Methods for Solving System of Differential Algebraic Equations (56:12) Play Video

Comments

There are no comments. Be the first to post one.
  Post comment as a guest user.
Click to login or register:
Your name:
Your email:
(will not appear)
Your comment:
(max. 1000 characters)
Are you human? (Sorry)
 
Disclaimer:
CosmoLearning is promoting these materials solely for nonprofit educational purposes, and to recognize contributions made by Indian Institute of Technology, Bombay (IIT Bombay) to online education. We do not host or upload any copyrighted materials, including videos hosted on video websites like YouTube*, unless with explicit permission from the author(s). All intellectual property rights are reserved to IIT Bombay and involved parties. CosmoLearning is not endorsed by IIT Bombay, and we are not affiliated with them, unless otherwise specified. Any questions, claims or concerns regarding this content should be directed to their creator(s).

*If any embedded videos constitute copyright infringement, we strictly recommend contacting the website hosts directly to have such videos taken down. In such an event, these videos will no longer be playable on CosmoLearning or other websites.