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)
- Regula-Falsi 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
- Gauss-Jordan method
- LU Factorization
- Iterative methods (Jacobi method, Gauss-Seidel 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
- Newton-Cote 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 RK-4 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
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