NUMERICAL METHODS [SH 603]
Course objective:
To introduce numerical methods used for the solution of engineering problems. The course emphasizes algorithm development and programming and application to realistic engineering problems.
 Introduction, Approximation and errors of computation (4hours)
 Introduction, Importance of Numerical Methods
 Approximation and Errors in computation
 Taylor's series
 Newton's Finite differences (forward, backward, central difference, divided difference)
 Difference operators, shift operators, differential operators
 Uses and Importance of Computer programming in Numerical Methods.
 Solutions of Nonlinear Equations (5 hours)
 Bisection Method
 Newton Raphson method (two equation solution)
 RegulaFalsi Method, Secant method
 Fixed point iteration method
 Rate of convergence and comparisons of these Methods
 Solution of system of linear algebraic equations (8 hours)
 Gauss elimination method with pivoting strategies
 GaussJordan method
 LU Factorization
 Iterative methods (Jacobi method, GaussSeidel method)
 Eigen value and Eigen vector using Power method
 Interpolation (8 hours)
 Newton's Interpolation (forward, backward)
 Central difference interpolation: Stirling's Formula, Bessel's Formula
 Lagrange interpolation
 Least square method of fitting linear and nonlinear curve for discrete data and continuous function
 Spline Interpolation (Cubic Spline)
 Numerical Differentiation and Integration (6 hours)
 Numerical Differentiation formulae
 Maxima and minima
 NewtonCote general quadrature formula
 Trapezoidal, Simpson's 1/3, 3/8 rule
 Romberg integration
 Gaussian integration (Gaussian – Legendre Formula 2 point and 3 point)
 Solution of ordinary differential equations (6 hours)
 Euler's and modified Euler's method
 RungeKutta methods for 1st and 2nd order ordinary differential equations
 Solution of boundary value problem by finite difference method and shooting method.
 Numerical solution of Partial differential Equation (8 hours)
 Classification of partial differential equation (Elliptic, parabolic, and Hyperbolic)
 Solution of Laplace equation (standard five point formula with iterative method)
 Solution of Poisson equation (finite difference approximation)
 Solution of Elliptic equation by Relaxation Method
 Solution of one dimensional Heat equation by Schmidt method
Practical:
Algorithm and program development in C programming language of following:
 Generate difference table.
 At least two from Bisection method, Newton Raphson method, Secant method
 At least one from Gauss elimination method or Gauss Jordan method. Finding largest Eigen value and corresponding vector by Power method.
 Lagrange interpolation. Curve fitting by Least square method.
 Differentiation by Newton's finite difference method. Integration using Simpson's 3/8 rule
 Solution of 1st order differential equation using RK4 method
 Partial differential equation (Laplace equation)
 Numerical solutions using Matlab.
References:
 Dr. B.S. Grewal, "Numerical Methods in Engineering and Science ", Khanna Publication.
 Robert J schilling, Sandra l harries , " Applied Numerical Methods for Engineers using MATLAB and C.", Thomson Brooks/cole.
 Richard L. Burden, J. Douglas Faires, "Numerical Analysis", Thomson / Brooks/cole
 John. H. Mathews, Kurtis Fink ,"Numerical Methods Using MATLAB" ,Prentice Hall publication
 JAAN KIUSALAAS , "Numerical Methods in Engineering with MATLAB", Cambridge Publication
Evaluation scheme:
The questions will cover all the chapters of the syllabus. The evaluation scheme will be as indicated in the table below
Unit 
Chapter 
Topics 
Marks 
1 
1 & 2 
All 
16 
2 
3 
All 
16 
3 
4 
All 
16 
4 
5 
All 
16 
6 
6.1 & 6.2 
5 
6 
6.3 
16 
7 
All 
Total 
80 
*Note: There may be minor deviation in marks distribution
