Numerical Method

Syllabus

1. Introduction, Approximation and errors of computation
   1. Introduction, Importance of Numerical Methods
   2. Approximation and Errors in computation
   3. Taylor's series
   4. Newton's Finite differences (forward , Backward, central difference, divided difference)
   5. Difference operators, shift operators, differential operators
   6. Uses and Importance of Computer programming in Numerical Methods.
2. Solutions of Nonlinear Equations
   1. Bisection Method
   2. Newton Raphson method ( two equation solution)
   3. Regula-Falsi Method , Secant method
   4. Fixed point iteration method
   5. Rate of convergence and comparisons of these Methods
3. Solution of system of linear algebraic equations
   1. Gauss elimination method with pivoting strategies
   2. Gauss-Jordan method
   3. LU Factorization
   4. Iterative methods (Jacobi method, Gauss-Seidel method)
   5. Eigen value and Eigen vector using Power method
4. Interpolation
   1. Newton's Interpolation ( forward, backward)
   2. Central difference interpolation: Stirling's Formula, Bessel's Formula
   3. Lagrange interpolation
   4. Least square method of fitting linear and nonlinear curve for discrete data and continuous function
   5. Spline Interpolation (Cubic Spline)
5. Numerical Differentiation and Integration
   1. Numerical Differentiation formulae
   2. Maxima and minima
   3. Newton-Cote general quadrature formula
   4. Trapezoidal, Simpson's 1/3, 3/8 rule
   5. Romberg integration
   6. Gaussian integration ( Gaussian – Legendre Formula 2 point and 3 point)
6. Solution of ordinary differential equations
   1. Euler's and modified Euler's method
   2. Runge Kutta methods for 1st and 2nd order ordinary differential equations
   3. Solution of boundary value problem by finite difference method and shooting method.
7. Numerical solution of Partial differential Equation
   1. Classification of partial differential equation(Elliptic, parabolic, and Hyperbolic)
   2. Solution of Laplace equation ( standard five point formula with iterative method)
   3. Solution of Poisson equation (finite difference approximation)
   4. Solution of Elliptic equation by Relaxation Method
   5. Solution of one dimensional Heat equation by Schmidt method