Numerical Methods of Ordinary and Partial Differential Equations

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)