Numerical methods of Ordinary and Partial

Overview

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, Picards 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 - Stability of multi step methods, root condition - System of first order ODE, higher order IVPs - Linear Boundary Value Problems (BVP), finite difference methods, shooting methods, stability, error and convergence analysis - Non linear BVP, higher order BVP - 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