# Numerical Methods of Ordinary and Partial Differential Equations

### Course Description

Course Outline: Ordinary Differential Equations: Initial Value Problems (IVP) and existence theorem. Truncation error, deriving finite difference equations. Single step methods for I order IVP- Taylor series method, Euler method, Picard’s method of successive approximation, Runge Kutta Methods. Stability of single step methods.
Multi step methods for I order IVP - Predictor-Corrector method, Euler PC method, Milne and Adams Moulton PC method. System of first order ODE, higher order IVPs. Stability of multi step methods, root condition. Linear Boundary Value Problems (BVP), finite difference methods, shooting methods, stability, error and convergence analysis. Non linear BVP, higher order BVP. (24 Lectures)

Partial Differential Equations: Classification of PDEs, Finite difference approximations to partial derivatives. Solution of one dimensional heat conduction equation by Explicit and Implicit schemes (Schmidt and Crank Nicolson methods ), stability and convergence criteria.
Laplace equation using standard five point formula and diagonal five point formula, Iterative methods for solving the linear systems. Hyperbolic equation, explicit / implicit schemes, method of characteristics. Solution of wave equation. Solution of I order Hyperbolic equation. Von Neumann stability. (16 Lectures)

Lecture 9: Multi-Step Methods (Explicit)
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### Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/syllabus/111105038/

# Lecture Play Lecture
I. Ordinary Differential Equations
1 Motivation with few Examples (53:54) Play Video
2 Single-Step Methods for IVPs (55:02) Play Video
3 Analysis of Single-Step Methods (55:51) Play Video
4 Runge-Kutta Methods for IVPs (56:09) Play Video
5 Higher Order Methods/Equations (55:38) Play Video
6 Error, Stability, and Convergence of Single-Step Methods (58:04) Play Video
7 Tutorial I (58:36) Play Video
8 Tutorial II (57:24) Play Video
9 Multi-Step Methods (Explicit) (55:09) Play Video
10 Multi-Step Methods (Implicit) (53:49) Play Video
11 Convergence and Stability of Multi-Step Methods (59:21) Play Video
12 General Methods for Absolute Stability (56:45) Play Video
13 Stability Analysis of Multi-Step Method (54:37) Play Video
14 Predictor-Corrector Methods (57:26) Play Video
15 Some Comments on Multi-Step Methods (55:24) Play Video
16 Finite Difference Methods: Linear BVPs (58:24) Play Video
17 Linear/Non-Linear Second Order BVPs (56:34) Play Video
18 BVPS - Derivative Boundary Conditions (55:29) Play Video
19 Higher Order BVPs (58:04) Play Video
20 Shooting Method BVPs (59:23) Play Video
21 Tutorial III (57:45) Play Video
II. Partial Differential Equations
22 Introduction to First Order PDE (53:03) Play Video
23 Introduction to Second Order PDE (57:11) Play Video
24 Finite Difference Approximations to Parabolic PDEs (56:58) Play Video
25 Implicit Methods for Parabolic PDEs (55:17) Play Video
26 Consistency, Stability and Convergence (54:48) Play Video
27 Other Numerical Methods for Parabolic PDEs (57:55) Play Video
28 Tutorial IV (55:28) Play Video
29 Matrix Stability Analysis of Finite Difference Scheme (53:50) Play Video
30 Fourier Series Stability Analysis of Finite Difference Scheme (53:01) Play Video
31 Finite Difference Approximations to Elliptic PDEs I (53:16) Play Video
32 Finite Difference Approximations to Elliptic PDEs II (58:09) Play Video
33 Finite Difference Approximations to Elliptic PDEs III (57:48) Play Video
34 Finite Difference Approximations to Elliptic PDEs IV (57:26) Play Video
35 Finite Difference Approximations to Hyperbolic PDEs I (57:46) Play Video
36 Finite Difference Approximations to Hyperbolic PDEs II (56:37) Play Video
37 Method of characteristics for Hyperbolic PDEs I (55:07) Play Video
38 Method of characterisitcs of Hyperbolic PDEs II (59:28) Play Video
39 Finite Difference Approximations to 1st order Hyperbolic PDEs (54:41) Play Video
40 Summary, Appendices, Remarks (55:28) Play Video