Objective: The objective of this subject is to expose student to understand the importance of finite difference methods for solving ordinary and partial differential equations.
Ordinary Differential Equations: Multistep (Explicit and Implicit) Methods for Initial Value problems, Stability and convergence analysis, Linear and nonlinear boundary value problems, Quasilinearization. Shooting methods.
Finite Difference Methods : Finite difference approximations for derivatives, boundary value problems with explicit boundary conditions, Implicit boundary conditions, Error analysis, Stability analysis, Convergence analysis.
Cubic splines and their application for solving two point boundary value problems.
Partial Differential Equations: Finite difference approximations for partial derivatives and finite difference schemes for Parabolic equations : Schmidt’s two level, Multilevel explicit methods, Crank-Nicolson’s two level, Multilevel implicit methods, Dirichlet’s problem, Neumann problem, Mixed boundary value problem. Hyperbolic Equations : Explicit methods, implicit methods, One space dimension, two space dimensions, ADI methods. Elliptic equations : Laplace equation, Poisson equation, iterative schemes, Dirichlet’s problem, Neumann problem, mixed boundary value problem, ADI methods.
1. M.K.Jain : Numerical Solution of Differential Equations, Wiley Eastern, Delhi, 1983.
2. G.D.Smith : Numerical Solution of Partial Differential Equations, Oxford University Press, 1985.
1. P.Henrici : Discrete variable methods in Ordinary Differential Equations, John Wiley, 1964.
2. A.R. Mitchell: Computational Methods in Partial Differential equations, John Wiley & Sons, New York, 1996.
3. Steven C Chapra, Raymond P Canale: Numerical Methods for Engineers, Tata McGraw Hill, New Delhi, 2007.