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Numerical Methods of Ordinary and Partial Differential Equations

Numerical Methods of Ordinary and Partial Differential Equations. Instructor: Dr. G. P. Raja Sekhar, Department of Mathematics, IIT Kharagpur.

Ordinary Differential Equations: Initial Value Problems (IVP) and existence theorem. Truncation error, deriving finite difference equations. Single step methods for IVPs - Taylor series method, Euler's method, Runge Kutta Methods. Stability of single step methods. Multi step methods for IVPs - Predictor-Corrector method, Euler PC method, Milne and Adams Moulton PC method. System of first order ODE, higher order IVPs. Stability of multistep methods, root condition. Linear Boundary Value Problems (BVP), finite difference methods, shooting methods, stability, error and convergence analysis. Nonlinear BVP, higher order BVP.

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 first order Hyperbolic equation. (from nptel.ac.in)

Motivation with Few Examples


Ordinary Differential Equations
Lecture 01 - Motivation with Few Examples
Lecture 02 - Single Step Methods for Initial Value Problems: Taylor Series Method
Lecture 03 - Analysis of Single Step Methods: Euler's Method
Lecture 04 - Runge-Kutta Methods for Initial Value Problems
Lecture 05 - Higher Order Runge-Kutta Methods, Higher Order Equations
Lecture 06 - Error-Stability-Convergence of Single Step Methods
Lecture 07 - Tutorial I: Single Step Methods
Lecture 08 - Tutorial II: Single Step Methods
Lecture 09 - Multistep Methods (Explicit)
Lecture 10 - Multistep Methods (Implicit)
Lecture 11 - Convergence and Stability of Multistep Methods
Lecture 12 - General Methods for Absolute Stability
Lecture 13 - Stability Analysis of Multistep Methods
Lecture 14 - Predictor-Corrector Methods
Lecture 15 - Some Comments on Multistep Methods: Alternate Ideas behind Predictor-Corrector Methods
Lecture 16 - Finite Difference Methods - Linear Boundary Value Problems
Lecture 17 - Linear/Nonlinear Second Order Boundary Value Problems
Lecture 18 - Boundary Value Problems - Derivative Boundary Conditions
Lecture 19 - Higher Order Boundary Value Problems
Lecture 20 - Shooting Method - Boundary Value Problems
Lecture 21 - Tutorial III: Boundary Value Problems
Partial Differential Equations
Lecture 22 - Introduction to First Order Partial Differential Equations
Lecture 23 - Introduction to Second Order Partial Differential Equations
Lecture 24 - Finite Difference Approximations to Parabolic Partial Differential Equations
Lecture 25 - Implicit Methods for Parabolic Partial Differential Equations
Lecture 26 - Consistency, Stability and Convergence
Lecture 27 - Other Numerical Methods for Parabolic Partial Differential Equations
Lecture 28 - Tutorial IV: Explicit, Implicit and Derivative Boundary Conditions
Lecture 29 - Matrix Stability Analysis of Finite Difference Scheme
Lecture 30 - Fourier Series Stability Analysis of Finite Difference Scheme
Lecture 31 - Finite Difference Approximations to Elliptic PDEs I
Lecture 32 - Finite Difference Approximations to Elliptic PDEs II
Lecture 33 - Finite Difference Approximations to Elliptic PDEs III
Lecture 34 - Finite Difference Approximations to Elliptic PDEs IV
Lecture 35 - Finite Difference Approximations to Hyperbolic PDEs I
Lecture 36 - Finite Difference Approximations to Hyperbolic PDEs II
Lecture 37 - Method of Characteristics for Hyperbolic PDEs I
Lecture 38 - Method of Characteristics for Hyperbolic PDEs II
Lecture 39 - Finite Difference Approximations for First Order Hyperbolic PDEs
Lecture 40 - Summary

References
Numerical Methods of Ordinary and Partial Differential Equations
Numerical Methods of Ordinary and Partial Differential Equations. Instructor: Dr. G. P. Raja Sekhar, Department of Mathematics, IIT Kharagpur.