Numerical Methods Syllabus - BCIS (PU)

Course Objectives

To be familiar with the theory and algorithms of numerical methods for solving nonlinear and set of algebraic equations, interpolation from given set of data, numerical differentiation and integration, solution of ordinary and partial differential equations.

Course Description

The numerical methods course involves solving engineering problems drawn from all fields of engineering. It includes error analysis, roots of nonlinear algebraic equations, solution of set of equations, curve fitting and interpolation, numerical integration and differentiation, solution of ordinary and partial differential equations. It also deals with the algorithm of solution of various methods that can be implemented in a digital computer.

Course Outcomes

At the end of this course, the students are expected to be able to:

• know the mathematical background for the different numerical methods
• understand the different methods for numerical solution of the nonlinear equations and solution of system of linear and nonlinear equations
• understand the different numerical methods for interpolation and data fitting
• know the numerical differentiation, integration and solution of set of ordinary and partial differential equations
• understand how numerical methods are used to solve various problems which are difficult to solve by analytical method

know how numerical method can generate solutions in a manner that can be implemented on digital computers

Course Contents

1. Introduction                                                                                                 4 hours

Importance of numerical methods, Review of calculus, Taylor’s theorem, Errors in numerical computations, Use of computer programming in numerical methods

2. Solution of Nonlinear Equations 8 hours

Nonlinear equations and their solutions, Trial and error method, Graphical method, Iterative methods: Bisection method, False position method, Secant method, Newton’s method and Fixed point iteration method, Rate of convergence of iterative methods, Newton’s method for polynomials and Horner’s rule

3. Solution of Set of Algebraic Equations                                                 9 hours

Existence of solutions for linear set of equations, Gaussian elimination method, pivoting, ill-conditioning, Gauss-Jordan method, Matrix inversion, Matrix factorization: Doolittle algorithm, Cholesky's factorization, Iterative solution using Gauss Seidel method, Eigen value and eigen vector using power method

4. Interpolation and Approximation                                                             8 hours

Lagrange interpolation, Newton’s interpolation using divided differences and difference table, Cubic spline interpolation, Least squares method of fitting linear and nonlinear function for given data

5. Numerical Differentiation and Integration                                                 6 hours

Numerical differentiation formulas, Maxima and minima of a tabulated function, Newton-Cote’s quadrature formulas: Trapezoidal, Simpson’s 1/3 and 3/8 rule, Romberg integration, Gaussian integration

6. Solution of Ordinary Differential Equations                                                 8 hours

Review of differential equations, Initial value problem, Taylor series method, Euler's method and its accuracy, Henu's method, Runge-Kutta methods, Solution of higher order equations, Solution of boundary value problems using finite difference and Shooting method

7. Solution of Partial Differential Equations                                                 5 hours

Review of partial differential equations, Deriving difference equations, Solution of Laplacian equation and Poisson's equation

Laboratory Work

The laboratory experiments will consist of program development and testing of nonlinear equations, Linear algebraic equations, Interpolation, Numerical integration and differentiation, ordinary and partial differential equations.

Some of them are listed below:

• To solve nonlinear equation using bisection method, secant method, Newton Raphson method
• To solve polynomial equation
• Solving set of linear algebraic equations using Gauss elimination method
• Finding largest Eigen value and corresponding eigenvector by Power method.
• Interpolation using Lagrange interpolation and Newton’s interpolation
• Curve fitting by Least square method.
• Numerical differentiation, numerical integration using trapezoidal and Simpson'srule
• Solution of differential equation using Euler’s method, RK-4 method
• Solution of partial differential equation
• Using MatLab for solution of numerical problems

References

1. Gerald, C.F.  & P.O. Wheatly, Applied Numerical Analysis
2. Balagurushamy, E.  Numerical Methods
3. Chency, W. & D. Kinciad, Numerical Mathematics and Computing
4. Press, W. H., B.P. Flannery et.al., "Numerical Recipes in C"