JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Pre-requisites

Mathematics, Differential Equations, Linear Algebra

Course Objectives:

• To solve mathematical and engineering problems by numerical methods
• To improve students programming skills in solving engineering problems by numerical methods

Course Outcomes

Students will understand basics of numerical analysis. Students will be able to

find roots of polynomial equations using numerical analysis and solutions of ordinary differential equations including initial value problems, Boundary value problems and Numerical differentiation of data and functions. Students will be able to conduct numerical integration and differentiation and will be able to use numerical methods to solve engineering problems.

UNIT - I

Solution of Linear Algebraic Equations: Gaussian elimination - LU decomposition - Pivoting strategies - Operation Count - Matrix inversion- Special cases -Tridiagonal and block tridiagonal systems - Well conditioned and Ill conditioned system-Matrix and Vector norms Condition Number and its implications.

UNIT - II

Solution of Non-linear Algebraic Equations: Bisection - Newton-Raphson and Secant method. System of non-linear equations: Basics of finite difference method Discretization of spatial and time derivatives using Taylor’s series- Truncation error and order of discretization - Fourier (von Neumann) stability analysis.

UNIT - III

Solution of Ordinary Differential Equations: Initial Value problems-Euler explicit and implicit methods - Runge-Kutta method – Predictor - Corrector methods - Boundary value problem - Shooting method - Finite difference method applied to pin fin heat dissipation - Stiff problems - Meaning of stiffness - Further insights into stiffness by the application of Euler explicit and implicit method to a stiff problem - Solution of stiff problem - Example – Chemical kinetics.

UNIT - IV

Solution of Elliptic Partial Differential Equations: Physical problems governed by elliptic PDE’s - Five-point and nine-point discretization of Poisson’s equation - Iterative methods - Point Iterative methods – Jacobi, Gauss-Seidel, and SOR - Detailed theory of the convergence of iterative methods - Global Iterative methods – Steepest Descent and Conjugate Gradient.

UNIT - V

Classification of PDEs and characteristics of a PDE - Solution of Parabolic Partial Differential Equations: Physical problems governed by parabolic PDE’s Operator splitting and ADI methods.

TEXT BOOKS:

• 1. Numerical Mathematics and Computing, by Ward Cheney and David Kincaid, International Thomson Publishing Company
• 2. Applied Numerical Analysis, by Curtis Gerald and Patrick Wheatley, Addison-Wesley R19 M.Tech. Thermal Engineering

REFERENCE BOOKS:

• 1. Analysis of Numerical Methods, by E. Isaacson & H. B. Keller, John Wiley & Sons
• 2. Numerical Solution of Partial Differential Equations: Finite Difference Methods, by G. D. Smith, Oxford University Press, 1985
• 3. Matrix Computations, by G. H. Golub, Johns Hopkins University Press Numerical Recipes, by W. H. Press et al R19 M.Tech. Thermal Engineering M. TECH. I Year II Sem. (THERMAL ENGINEERING)