# Numerical Methods

By Mansoor P   |   MES College of Engineering Kuttippuram, Kerala
Learners enrolled: 985
In Mathematical applications, there are several instances in which no direct methods are available for solving higher degree algebraic equations or transcendental equations, ie., the equations involving circular, logarithmic or exponential functions. Such equations are solved by numerical methods. The simultaneous linear algebraic equations occur quite often in various fields of engineering and science, Generally, the matrix inversion method or Cramer’s rule have been using to solve these equations. But these methods become difficult when the system consist large number of unknown variables. In such cases, numerical techniques can be adopted to find the solutions. If the values of a function f(x) are given for some finite values of x, then using the interpolation we can find the numerical value of f(x) for any other value of x in the given interval. We are familiar with various analytical methods in order to solve many ordinary differential equations. In physical problems, there are large numbers of ordinary differential equations which cannot be solved by analytical methods.In these cases, numerical solutions can be computed using various numerical methods.

The course is developed to enable the undergraduate students to get a comprehensive understanding of numerical techniques in solving various mathematical problems. We will begin the course with the methods of finding solution of algebraic and transcendental equations using

bisection method, regula falsi method, fixed point iteration method, Newton Raphson method and Horner’s method. The direct and indirect methods of solving linear system of algebraic equations will be discussed in this course. It include Gauss elimination method, Gauss Jordan method, Factorization method, Gauss Jacobi iteration method and gauss Seidel iteration method. This course also covers interpolation of equal and unequal intervals, numerical differentiation and numerical integration. Towards the end, the method finding numerical solution of first order
ordinary differential equations will be discussed. After the successful completion of this course, the learner would be familiarized with the way of solving complicated mathematical problems numerically using the appropriate numerical technique.

Summary
 Course Status : Ongoing Course Type : Core Duration : 12 weeks Start Date : 16 Jan 2023 End Date : 15 Apr 2023 Exam Date : Enrollment Ends : 15 Mar 2023 Category : Mathematics Credit Points : 4 Level : Undergraduate

### Course layout

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1 : Bisection method
Day 2 Module 2 :  Regula-Falsi method
Day 3 Module 3 :  Fixed point iteration method
Day 4 Discussion based on the above topics
Day 5 Assignment

2 Day 1 Module 4 : Newton Raphson method- Part 1
Day 2 Module 5 : Newton Raphson method- Part 2
Day 3 Module 6 : Solution of nonlinear system of equations using Newton Raphson method.

Day 4 Discussion based on the above topics
Day 5 Assignment

3 Day 1 Module 7 : Gauss elimination method
Day 2 Module 8 : Gauss Jordan method
Day 3 Module 9 : Method of Triangularisation
Day 4 Module 10: Crout’s triagularisation method
Day 5 Discussion based on the above topics
Day 6 Assignment

4 Day 1 Module 11 : Gauss Jacobi iterative method of iteration
Day 2 Module 12 : Gauss Seidel method of iteration
Day 3 Module 13 : Forward finite differences
Day 4 Module 14: Backward finite differences
Day 5 Discussion based on the above topics
Day 6 Assignment

5 Day 1 Module 15 : Central differences
Day 2 Module 16 : Newton’s Forward Interpolation
Day 3 Module 17: Newton’s backward Interpolation
Day 4 Discussion based on the above topics
Day 5 Assignment

6 Day 1 Module 18 : Relation connecting finite differences
Day 2 Module 19 : Stirling’s Interpolation Formula
Day 3 Module 20 : Gauss’ Forward Interpolation Formula
Day 4 Module 21. Gauss’ backward Interpolation Formula
Day 5 Discussion based on the above topics
Day 6 Assignment

7 Day 1 Module 22 : Newton’s divided difference Interpolation
Day 2 Module 23 : Lagrange’s interpolation
Day 3 Module 24 : Differentiation using Newton’s forward difference formula
Day 4 Discussion based on the above topics
Day 5 Assignment

8 Day 1 Module 25 : Differentiation using Newton’s backward difference formula
Day 2 Module 26: 2Differentiation using Stirling’s formula
Day 3 Module 27 : Bessel’s formula
Day 4 Discussion based on the above topics
Day 5 Assignment

9 Day 1 Module 28 : Laplace Everett’s formula
Day 2 Module 29 : Integration using Trapezoidal rule
Day 3 Module 30: Simpson’s one third rule
Day 4 Module 31: Simpson’s 3/8 th rule
Day 5 Discussion based on the above topics
Day 6 Assignment

10 Day 1 Module 32 : Evaluation of double integrals using trapezoidal rule
Day 2 Module 33 : Evaluation of double integrals using simpson’s rule
Day 3 Module 34 : Solution of ODE using Picard’s iteration method
Day 4 Discussion based on the above topics
Day 5 Assignment

11 Day 1 Module 35 : Taylor series method
Day 2 Module 36 : Euler’s method
Day 3 Module 37:  Second order Runge-Kutta method
Day 4 Discussion based on the above topics
Day 5 Assignment

12 Day 1 Module 38 : Fourth order Runge-Kutta method
Day 2 Module 39 : Adam-Bashforth method
Day 3 Module 40 : Milne’s predictor corrector method
Day 4 Discussion based on the above topics
Day 5 Assignment

### Books and references

1. Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007.
2. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 6th Ed., New age International Publisher, India, 2007.
3. C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education, India, 2008.
4. Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, 7th Ed., PHI Learning Private Limited, 2013.
5. John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012.

### Mansoor P

MES College of Engineering Kuttippuram, Kerala
MANSOOR P
Assistant Professor, Department of Mathematics, MES
College of Engineering Kuttippuram, Trikkanapuram P.O.,
Malappuram D.T., Kerala, India
679582, easyganitham@gmail.com, 09037250791
M.Phil in Mathematics, MSc. Mathematics, BSc., Mathematics
Other Merits:
Pursuing PhD. in Mathematics with Bharathiar University, Coimbatore.
Served as Course coordinator for the MOOC on
MATHEMATICAL ANALYSIS (2021 Jan - April) under
SWAYAM platform.
Served as Course coordinator for the MOOC on Calculus
(2019 Jan-April) under SWAYAM platform.
Developed and presented a number of e-content modules
in various Mathematics topic for CEC, MHRD India.
Delivered a number of lectures in Mathematics for DTH
Swayamprabha Channel 8 of MHRD at University of Calicut.
Published research articles in reputed International
Journals.Presented papers in International and Regional Seminars.
Working as translator for translating various NPTEL courses into Malayalam regional language.

### Course certificate

30% for in course assessment and 70% for end term proctored exam.