Numerical Methods

Course Title: Numerical Methods

Course No: ENSH 252

Nature of the Course: Theory + Lab

Semester: 4

Full Marks: 40 + 60 + 50

Pass Marks: 16 + 24 + 20

Credit Hours: 3

Course Description

Numerical Methods equips students with thorough understanding of numerical methods, focusing on their application in obtaining approximate solutions to complex mathematical problems in science and engineering. The course covers algorithm development, programming in Python, and visualization techniques for solution of non-linear equations, linear algebraic equations, interpolation, differentiation, integration, and differential equations.

Course Objectives

The objective of this course is to equip students with a thorough understanding of numerical methods, focusing on their application in obtaining approximate solutions to complex mathematical problems commonly encountered in science and engineering. Emphasizing algorithm development, programming, and visualization techniques, the course enables students to apply computational approaches effectively, enhancing their problem-solving capabilities in real-world applications.

Course Contents

1. Solution of Non-Linear Equations

7 hrs

1.1. Errors and accuracy in numerical computations

1.2. Bisection method

1.3. Regula Falsi method and secant method

1.4. Newton Raphson method

1.5. Fixed point iteration method

1.6. Comparison of the methods (Bracketing vs open-ended methods and rates of convergence)

1.7. Solution of system of non-linear equations

Direct approach

Newton Raphson method

2. Solution of System of Linear Algebraic Equations

8 hrs

2.1. Direct methods

Gauss Jordan method

Gauss elimination method, pivoting strategies (Partial and complete)

Matrix inverse using Gauss Jordan and Gauss elimination methods

Factorization methods (Do-Little's method and Crout's method)

2.2. Iterative methods

Jacobi's method

Gauss-Seidal method

2.3. Determination of largest and smallest Eigen values and corresponding vectors using the power method

3. Interpolation

9 hrs

3.1. Polynomial Interpolation

Finite differences (Forward, backward, central and divided differences)

Interpolation with equally spaced intervals: Newton's forward and backward difference interpolation, Stirling's and Bessel's central difference interpolation

Interpolation with unequally spaced intervals: Newton's divided difference interpolation, Lagrange interpolation

3.2. Least square method of curve fitting

Linear form and forms reducible to linear form

Quadratic form and forms reducible to quadratic form

Higher degree polynomials

3.3. Cubic spline interpolation

Equally spaced interval

Unequally spaced interval

4. Numerical Differentiation and Integration

6 hrs

4.1. Numerical differentiation

Differentiation using polynomial interpolation formulae for equally spaced intervals

Local maxima and minima from equally spaced data

4.2. Numerical integration

Newton Cote's general quadrature formula

Trapezoidal rule, Simpson's 1/3 and 3/8 rules, Boole's rule, Weddle's rule

Romberg integration

Gauss-Legendre integration (up to 3-point formula)

5. Solution of Ordinary Differential Equations (ODE)

8 hrs

5.1. Initial value problems

Solution of first order equations: Taylor's series method, Euler's method, Runge-Kutta methods (Second and fourth order)

Solution of system of first order ODEs via Runge-Kutta methods

Solution of second order ODEs via Runge-Kutta methods

5.2. Two-point boundary value problems

Shooting method

Finite difference method

6. Solution of Partial Differential Equations

7 hrs

6.1. Introduction and classification

6.2. Finite difference approximations of partial derivatives

6.3. Solution of elliptic equations

Laplace equation

Poisson's equation

6.4. Solution of parabolic and hyperbolic equations

One-dimensional heat equation: Bendre-Schmidt method, Crank-Nicolson method

Solution of wave equation

Laboratory Works

Practical sessions of 45 hours using Python as the programming language. Results to be visualized graphically wherever possible. Practical report contents: Working principle, Pseudocode, Source code, Test Cases. Covers Python basics, solution of non-linear equations, system of linear algebraic equations, interpolation, numerical integration, solution of ODEs, and solution of PDEs using finite difference approach.

1.Basics of Programming in Python
2.Solution of Non-Linear Equations
3.System of Linear Algebraic Equations
4.Interpolation
5.Numerical Integration
6.Solution of Ordinary Differential Equations
7.Solution of Partial Differential Equations Using Finite Difference Approach

Reference Books

1.Chapra, S. C., Canale, R. P. (2010). Numerical Methods for Engineers (6th edition). McGraw-Hill.
2.Kiusalaas, J. (2013). Numerical Methods in Engineering with Python 3 (3rd edition). Cambridge University Press.
3.Grewal, B. S. (2017). Numerical Methods in Engineering & Science (11th edition). India: Khanna Publishers.
4.Yakowitz, S., Szidarovszky, F. (1986). An Introduction to Numerical Computations (2nd edition). Macmillan Publishing.
5.Kong, Q., Siauw T., Bayen A. (2020). Python Programming and Numerical Methods. Academic Press.

Notes:

This syllabus follows the official BEI curriculum of Tribhuwan University. In case of any doubt or revision, the university's published syllabus shall be considered authoritative. https://ioe.tu.edu.np/pages/electronics-engineering-curriculum-structure-2660

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