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
