To explore complex systems, physicists, engineers, financiers and mathematicians require computational methods since mathematical models are only rarely solvable algebraically. Numerical methods, based upon sound computational mathematics, are the basic algorithms underpinning computer predictions in modern systems science. The course will cover the classical fundamental topics in numerical methods such as, numerical integration, solution of ordinary and partial differential equations. Linear operators, multi-step methods and solution of difference equations of different types. The viewpoint will be modern, with connections made between each topic and a variety of applications.

*Intended ** Learning Outcomes: *The student should be able to:

- Derive numerical methods for various mathematical operations and tasks, such as algorithms for numerical integration, the solution of homogeneous and inhomogeneous difference equations, and the solution of differential equations.
- Apply the numerical methods (such as Milne's predictor corrector formula) to achive higher order degree of accuracy.
- Recognize Iterative methods (for example, modified Euler, RK)
- Implement a variety of numerical algorithms using appropriate programming (for example, Maple),
- Effectively use numerical tools to solve problems in their own field of interest.

** Prerequisite:** Numerical Analysis-I

**Contents**

- Numerical Integration: Trapezoidal,
- Simpson and Quadrature formulas for evaluating integrals with error estimates.
- Difference and Differential Equation: Formulation of difference equations,
- Solution of linear (homogeneous and inhomogeneous) difference equations with constant coefficients.
- The Euler and modified Euler method,
- Runge-Kutta methods and predictor-corrector type methods for solving initial value problems along with convergence and instability criteria.
- Finite difference, collocation and variational method for boundary value problems.

*Recommended Books*

- R. L. Burden and J. D. Faires:
*Numerical Analysis*, (9th Edition, Brooks /Cole, 2010.) - Monagan,. Geddes,. Labahn, Maple 7 Programming Guide, Waterloo Maple Inc 2001.

*Suggested Books*

- J.H. Mathews,
*Numerical Methods for Mathematics*, (Latest Edition, Prentice Hall International) - S. C. Chapra and R. P. Canale:
*Numerical Methods for Engineers*, (6th edition, McGraw Hill.2010) - C.F. Gerald and P.O. Wheatley,
*Applied Numerical Analysis*, (Pearson Education, Singapore, 2005)

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

Research project/assignment based on numerical techniques and algorithms often involve features which require their solutions can be awarded to students during their course work period. Examples of features are

- Accelerating convergence of numerical methods.
- Finding numerical solution of analytical complex problems
- Improving efficiency of numerical techniques

**Assessment Criteria**

- Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
- Mid-Term Exam: 30
- Final-Term Exam: 50

*Key Dates and Time of Class Meeting*

Tuesday 09:30, 11:00

Wednesday 09:30, 11:00

Commencement of Classes September 27, 2022

Mid Term Examination 19-04-2021 (Monday) to 23-04-2021 (Friday)

Final Term Examination 21-06-2021 (Monday) to 25-06-2021 (Friday)

Declaration of Result 02-06-2021 (Wednesday)