Numerical Method(NM) Syllabus

Course Description: This course contains the concepts of numerical method techniques for

solving linear and nonlinear equations, interpolation and regression, differentiation and

integration, and partial differential equations.

Course Objectives: The main objective of the course is to provide the knowledge of numerical

method techniques for mathematical modeling.

1. Solution of Nonlinear Equations

teaching hours: 8 hrs

1.1 Errors in Numerical Calculations, Sources of Errors, Propagation of Errors, Review of

Taylor's Theorem

1.2 Solving Non-linear Equations by Trial and Error method, Half-Interval method and

Convergence, Newton's method and Convergence, Secant method and Convergence,

Fixed point iteration and its convergence, Newton's method for calculating multiple roots,

Horner's method

2. Interpolation and Regression

teaching hours: 8 hrs

2.1 Interpolation vs Extrapolation, Lagrange's Interpolation, Newton's Interpolation using

divided differences, forward differences and backward differences, Cubic spline

interpolation

2.2 Introduction of Regression, Regression vs Interpolation, Least squares method, Linear

Regression, Non-linear Regression by fitting Exponential and Polynomial

3. Numerical Differentiation and Integration

teaching hours: 8 hrs

3.1 Differentiating Continuous Functions (Two-Point and Three-Point Formula),

Differentiating Tabulated Functions by using Newton’s Differences, Maxima and minima

of Tabulated Functions

3.2 Newton-Cote's Quadrature Formulas, Trapezoidal rule, Multi-Segment Trapezoidal rule,

Simpson's 1/3 rule, Multi-Segment Simpson's 1/3 rule, Simpson's 3/8 rule, Multi-

Segment Simpson's 3/8 rule, Gaussian integration algorithm, Romberg integration

4. Solving System of Linear Equations

teaching hours: 8 hrs

4.1 Review of the existence of solutions and properties of matrices, Gaussian elimination

method, pivoting, Gauss-Jordan method, Inverse of matrix using Gauss-Jordan method

4.2 Matrix factorization and Solving System of Linear Equations by using Dolittle and

Cholesky's algorithm

4.3 Iterative Solutions of System of Linear Equations, Jacobi Iteration Method, Gauss-Seidal

Method

4.4 Eigen values and eigen vectors problems, Solving eigen value problems using power

method.

5. Solution of Ordinary Differential Equations

teaching hours: 8 hrs

5.1 Review of differential equations, Initial value problem, Taylor series method, Picard's

method, Euler's method and its accuracy, Heun's method, Runge-Kutta methods

5.2 Solving System of ordinary differential equations, Solution of the higher order equations,

Boundary value problems, Shooting method and its algorithm

6. Solution of Partial Differential Equations

teaching hours: 5 hrs

6.1 Review of partial differential equations, Classification of partial differential equation,

Deriving difference equations, Laplacian equation and Poisson's equation, engineering

examples

Laboratory Works:

The laboratory exercise should consists of program development and testing of non-linear

equations, interpolation, numerical integration and differentation, linear algebraic equations,

ordinary and partial differential equations numerical solutions using appropriate languages like

C, C++ or Matlab.

List topics to be included in Laboratory Exercises:

•  Solution of non-linear equations using Bisection Method and Secant Method
•  Solution of non-linear equations using Newton’s Raphson Method and Fixed Point

Iteration Method

•  Solution of polynomial using Newton’s Method and Horner’s Rule to evaluate

polynomial

•  Polynomial interpolation using Lagrange’s Interpolation and Newton’s Divided Difference   Interpolation, Newton’s forward and backward difference interpolation

•  Fitting of linear (straight line , y=ax + b) and non-linear (exponential y=aebx,   quadraticy=ax2+bx+c) function using least square method

•  Derivatives from divided difference table
•  Integration using Trapezoidal rule, Simpson’s 1/3 rule and Simpson’s 3/8 rule, Line and

Double Integration.

•  Solution of system of linear equations using Gauss Elimination method and Gauss Jordan

Method

•  Gauss Seidel Method, Jacobi Method and Power Method
•  Solution of ordinary differential equation using Euler’s Method, Heun’s Method and 4th   order Runge-Kutta Method
•  Boundary value problems using Shooting Method
•  Laplacian Equation, Poison’s Equation