Numerical Methods(NM) Syllabus

Course Description: This course covers different concepts of numerical techniques of solving non-linear equations, system of linear equations, integration and differentiation, and ordinary and partial differential equations.

Course Objective: The main objective of this course is to provide concepts of numerical techniques for solving different types of equations and developing algorithms for solving scientific problems.

1. Solution of Nonlinear Equations

teaching hours: 7 hrs

1.1 Errors in Numerical Calculations (True Error, Relative Error, Approximate Error, Relative Approximate Error), Sources of Errors (Round-off Error, Truncation Error), Propagation of Errors, Review of Taylor's Theorem

1.2 Concept of Linear and Non-linear Equations, Solving Non-linear Equations: Trial and Error Method, Bisection Method, Newton Raphson Method, Secant Method, Fixed Point Method, False Position Method, Newton's Method for Calculating Multiple Roots, Evaluating Polynomials with Horner's Method, Derivation, Algorithm and Example of each method

2. Interpolation and Regression

teaching hours: 8 hrs

2.1 Concept of Interpolation and Extrapolation, Lagrange's Interpolation, Newton's Interpolation using divided differences, forward differences and backward differences, Derivation, Algorithm and Example of each method

2.2 Concept of Regression, Regression vs. Interpolation, Least Squares Methods, Linear Regression, Non-linear Regression: Exponential regression by linearization, and Polynomial, Derivation, Algorithm and Example of each method 

3. Numerical Differentiation and Integration

teaching hours: 9 hrs

3.1 Concept of Differentiation, Differentiating Continuous Functions (Two-Point Forward and Backward Difference Formulae, Three-Point Formula), Differentiating Tabulated Functions by using Newton’s Differences (Divided, Forward and Backward Difference Formulae), Maxima and minima of Tabulated Functions, Derivation, Algorithm and Example of each method

3.2 Concept of Integration, Newton-Cote's Quadrature Formulae: Trapezoidal rule, Multi-Segment Trapezoidal rule, Simpson's 1/3 rule, MultiSegment Simpson's 1/3 rule, Simpson's 3/8 rule, Multi-Segment Simpson's 3/8 rule, Derivation, Algorithm and Example of each method

4. Solving System of Linear Equations

teaching hours: 8 hrs

4.1 Existence of Solutions, Properties of Matrices, Matrix Representation, Gaussian Elimination Method, Partial and Complete Pivoting with Gaussian Elimination Method, Gauss-Jordan method, Inverse of matrix using Gauss-Jordan method, Derivation, Algorithm and Example of each method

4.2 Concept of LU Decomposition, Matrix factorization and Solving System of Linear Equations by using Doolittle and Cholesky Algorithm, Derivation, Algorithm and Example of each method

4.3 Iterative Solutions of System of Linear Equations, Jacobi Iteration Method, Gauss-Seidel Method, Derivation, Algorithm and Example of each method

4.4 Eigen Values and Eigen Vectors Problems, Power Method, Algorithm and Example of the method

5. Solution of Ordinary Differential Equations

teaching hours: 8 hrs

5.1 Concept of Differential Equations, Initial Value Problem, Taylor Series Method, Euler's Method, Heun's Method, Runge-Kutta Methods, Derivation, Algorithm and Example of each method

5.2 Solving System of Ordinary Differential Equations, Solution of the Higher Order Equations, Boundary Value Problems, Shooting Method, Derivation, Algorithm and Example of each method.

6. Solution of Partial Differential Equations

teaching hours: 5 hrs

6.1 Concept of Partial Differential Equations, Classification of PDE, Deriving Difference Equations, Solving Laplacian Equation and Poisson's Equation, Derivation, Algorithm and Example of each method