This course is designed to teach students about the basics of scientific computing for solving differential equations using B-spline collocation techniques and learn how to match numerical methods to mathematical properties.

*Learning Outcomes*

This course gives the students the knowledge of problem classes, basic mathematical and numerical concepts, and software for the solution of engineering and scientific problems formulated as differential equations. After successful completion, students should be able to design, implement, and use numerical methods for the computer solution of scientific problems involving differential equations.

*Prerequisites: Numerical Spline Techniques-I*

*Contents*

- Classification of linear/non-linear second orders ODEs. Discuss the types of boundary conditions
- Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations.
- One-step, linear multi-step, Runge-Kutta, and Extrapolation methods; convergence,
- stability, B-spline collocation methods for the numerical solution of initial and boundary values problems,
- B-spline for solving class of singular and non-singular problems,
- Trigonometric B-spline approach for solving second order system of linear/non-linear ODEs,
- B-spline solution for nonlinear differential equation arising in general relativity,
- Bratu’s problem, Perturbation’s problem.

*Recommended Books*

- de Boor, C.:
*A Practical Guide to Splines. (*Springer Verlag, (2001)) - Farin, G.
*Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide*(Academic Press Inc., 2002) - Joan, G. Y. R.
*B-splines for initial and boundary value problems*(Ph.D. Thesis, USM, Malaysia 2013)

*Suggested Books *

- M. Prenter,
*Splines and Variational Methods*. (John Wiley & Sons, (1989)) - Bartels, R.H., Bealty, J.C. and Beatty, J.C.
*An Introduction to Spline for use in Computer Graphics and Geometric Modeling*(Morgan Kaufmann Publisher, 2006). - Smith, D. G.:
*Numerical Solution of Partial Differential Equations: Finite Difference Methods*. (Oxford Press, (1990)) - Wang, R.H.
*Multivariate Spline Functions and Their Applications (Mathematics and Its Applications)*(Science Press/ Kluwer Academic Publishers, 2005). - Recently published research papers

**RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS**

The B-spline collocation method is applied to problems that are not discussed in the course using MATLAB/Mathematica. Design algorithms for numerical solutions of differential equations using other types of Splines available in the literature.

**ASSESSMENT CRITERIA**

Sessional: 20 (Presentation / Assignment 04, Attendance 08, Result Mid-Term 04, Quiz 04

Mid-Term Exam: 30

Final-Term Exam: 50

Monday--Tuesday 9:30am-11:00am

Commencement of Classes March 02, 2020

Mid Term Examination April 27 to May 04, 2020

Final Term Examination June 22-26, 2020

Declaration of Result July 03, 2020