Pokhara University
Faculty of Science and Technology
Course Code: MTH 252
Course title: Numerical methods (2-1-2)
Full Marks: 100
Pass Marks: 45
Nature of the Course: Theory and Practical
Total Lectures: 30 hours
Level: Bachelor
Program: BE
1. Course Description
This course explains how to utilize a computer to solve issues that calculus and algebra might not be able to. It fosters the development of mathematical relationships that can be utilized to model real-world situations and the problem-solving skills necessary to study other engineering courses.
2. General Objectives
The general objectives of this course are to equip students with knowledge and tools required to solve different equations that are applicable in the fields of engineering.
3. Methods of Instructions
Lecture, Tutorial, Discussion, Readings, and Practical works
4. Contents in Detail
Specific Objectives
Contents
Solve non-linear equations by different numerical methods.
Unit 1: Solution of Non-linear equations (5 hrs)
1.1. Introduction, Importance of Numerical Methods
1.2. Approximation and Errors in computation
1.3. Bisection Method
1.4. Secant method
1.5. Newton Raphson method
1.6. Fixed point iterative method
Visualize and solve mathematical relationships of practical observations.
Unit 2: Interpolation and approximation (5hrs)
2.1. Lagrange interpolation
2.2. Finite differences (forward, backward, and divided difference)
2.3. Newton’s Interpolation (forward, backward)
2.4. Least square method of fitting linear and nonlinear curve for discrete data and continuous function
2.5. Cubic Spline Interpolation
Calculate definite integration and differentiation numerically.
Unit 3: Numerical Differentiation and Integration (4 hours)
3.1. Numerical Differentiation formulae
3.2. Trapezoidal, Simpson’s 1/3, 3/8 rule
3.3. Romberg integration
3.4. Gaussian integration (2-point and 3-point formula)
Solve the system of linear equations by different techniques.
Unit 4: Solution of system of linear algebraic equations (6 hours)
4.1. Gauss elimination method and concept of pivoting
4.2. Ill-conditioned system of linear equations
4.3. LU Factorization method (Dolittle, Crout’s, Cholesky’s)
4.4. Iterative methods (Jacobi method, Gauss‐Seidel method)
4.5. Eigen value and Eigen vector using Power method
Solve the ordinary differential equations which may exist in the field of engineering.
Unit 5: Solution of ordinary differential equations (6 hours)
5.1. Review of ordinary differential equations
5.2. Runge-Kutta methods (first, second and fourth) for first and second order differential equations
5.3. Solution of boundary value problem by shooting method
Solve numerically the partial differential equations which exist in the field of engineering.
Unit 6: Numerical solution of Partial differential Equation (4 hours)
6.1. Classification of partial differential equation (elliptic, parabolic and hyperbolic)
6.2. Solution of Laplace equation (standard 5-point formula with iterative methods)
6.3. Solution of Poisson equation (finite difference approximation method)
6.4. Solution of one-dimensional Heat equation by Schmidt method
5. List of Tutorials
The following tutorial activities of 15 hours per group of maximum 24 students should be conducted to cover all the required contents of this course.
| S.N | List of Tutorials | Duration |
|---|---|---|
| 1 | Determination of a root by all methods and their comparison. | 3 hrs |
| 2 | Finding of different interpolating polynomials, regression curve, and Cubic-spline. | 2 hrs |
| 3 | Determination of the first and second order derivatives by difference method and its comparison with the exact value. Integration by Trapezoid, Simpson’s rules, Romberg method, Gaussian method, and comparison with the exact value. | 2 hrs |
| 4 | Solution of a system of linear equations by Gauss Elimination, matrix factorization, Jacobi, Gauss-Seidel method. Finding Eigen value and Eigen vector by power method. | 4 hrs |
| 5 | Solution of first and second order differential equation by RK methods and Shooting method. | 2 hrs |
| 6 | Solution of Laplace and Poisson’s equations by the five-point formula. | 2 hrs |
6. List of Practical
| SN | List of Practicals |
|---|---|
| 1. | Solution of nonlinear equations. |
| 2. | Interpolation and regression. |
| 3. | Differentiation and Integration. |
| 4. | Linear system of equations and power method. |
| 5. | Ordinary differential equations. |
7. Evaluation System and Students’ Responsibilities
Evaluation System
The internal evaluation of a student may consist of assignments, attendance, term-exams, lab reports, and projects, etc. The tabular presentation of the internal evaluation is as follows:
| Internal Evaluation | Weight | Marks | External Evaluation | Marks |
|---|---|---|---|---|
| Theory | 30% | Semester End | 50 | |
| Attendance & Class Participation | 10% | |||
| Assignments | 20% | |||
| Presentations/Quizzes | 10% | |||
| Internal Assessment | 60% | |||
| Practical | 20% | |||
| Attendance & Class Participation | 10% | |||
| Lab Report/Project Report | 20% | |||
| Practical Exam/Project Work | 40% | |||
| Viva | 30% | |||
| Total Internal | 50 | Full Marks: 50 + 50 = 100 |
Students’ Responsibilities
Each student must secure at least 45% marks separately in internal assessment and practical evaluation with 80% attendance in the class in order to appear in the Semester End Examination. Failing to get such a score will be given NOT QUALIFIED (NQ) to appear in the Semester-End Examinations. Students are advised to attend all the classes, formal exams, tests, etc., and complete all the assignments within the specified time period. Students are required to complete all the requirements defined for the completion of the course.
8. Prescribed Books and References
Text Books
- C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis (7th edition), New York.
- B. S. Grewal, Numerical Methods in Engineering and Science, Khanna Publication (10th edition).
- S.S. Sastry, Introductory Methods of Numerical Analysis (4th edition), Prentice-Hall of India, New Delhi, 2008.
References
- Richard L. Burden, J. Douglas Faires, "Numerical Analysis 7th edition," Thomson/Brooks/Cole.
- E. Balagurusamy, Numerical methods. New Delhi; Tata McGraw Hill, 2010.
- Dr. V. N. Vedamurthy & Dr. N. Ch. S. N. Iyengar, Numerical Methods, Noida, Vikash Publication House 2009.
- Rudra Pratap, Getting Started with MATLAB, Oxford University Press 2010.