Numerical Methods

Course Objectives
To present the theory of numerical computational procedures for solving engineering problems. Solution of ordinary and partial differential equations will be included.

1.0 Solution of Nonlinear Equations: (10 hours)
1.1 Review of calculus, continuity, differentiability, intermediate value theorem, Taylor’s theorem
1.2 Absolute, relative, and round off errors, error bounds for computational errors
1.3 Bisection method, its error bounds and convergence
1.4 Newton’s method, secant method and their convergence
1.5 Fixed point iteration, its convergence properties,
1.6 Zeros of polynomials by Horner’s method

2.0 Interpolation and Approximation: (10 hours)
2.1 Taylor’s polynomial approximation, Lagrange’s interpolation
2.2 Newton’s interpolation and divided differences
2.3 Iterative interpolation
2.4 Cubic spline interpolation
2.5 Least squares method of fitting continuous and discrete data or functions

3.0 Numerical Differentiation and Integration: (5 hours)
3.1 Numerical differentiation formulas
3.2 Newton-Cote’s numerical integration formulas, composite numerical integration
3.3 Romberg integration algorithm
3.4 Gaussian integration formulas

4.0 Linear Algebraic Equations: (10 hours)
4.1 Review of the properties of matrices
4.2 Matrix form of Gaussian elimination, pivoting strategies, ill-conditioning
4.3 Cholesky’s and related algorithms for matrix factorization
4.3 Eigen values and eigen vectors and the power method

5.0 Solution of ordinary Differential Equations: (7 hours)
5.1 Euler’s method for solving ordinary differential equations of 1st order and other related methods
5.2 Runge-Kutta methods
5.3 Extension to higher order equations
5.4 Initial value problems
5.5 Boundary value problems
6.0 Solution of partial Differential Equations: (3 hours)
6.1 Introduction to the solution of partial differential equations
6.2 Engineering examples

Reference Books:
1.0 W. Cheney and D. Kincaid, “Numerical Mathematics and computing”, Edition, Brooks/Cole publishing Co.,1985.
2.0 C.F. Gerald and P. O. Wheatley, “Applied Numerical Analysis”, 4th Edition, Addison-Wesley Publishing Company, New York.
3.0 S. Yakowitz and F. Szidarovszky, “An Introduction to Numerical Computations”, 2nd Edition, Macmillan publishing Co., New York.

Reference Book for Programs in C:
1.0 W.H. press, B. P. Flannery et. al., “Numerical Recipes in C”, 1st Edition, Cambridge Press,1988.