The main objective of this course is to provide the students with a range of mathematical methods that are essential to the solutions of advanced problems encountered in the fields of applied physics and engineering. In addition this course is intended to prepare the students with mathematical tools and techniques that are required in advenced courses offered in the applied physics and engineering problems.

**Pre-Requisite:**

Calculus-III

**Intended Learning Outcomes**

Upon successful completion of this course, students should be able to

- Implement basic operations in Fourier series and Laplace transforms..
- Be able to apply mathematical and computational methods to a range of problems in science and engineering.
- Be able to apply integral transforms to solve ordinary and partial differntial quations.
- Be able to apply variational methods in solving variational problems.

**Contents**

- Fourier methods
- The Fourier transforms
- Fourier analysis of the generalized functions
- The Laplace transforms
- Hankel transforms for the solution of PDEs and their application to boundary value problems
- Green's function and Transform methods
- Expansion for Green's function
- Closed form Green's function
- Perturbation techniques: Perturbation methods for algebraic equations
- Perturbation methods for differntial equations
- Variational methods
- Euler-Lagrange's equations
- Integrand involving one, two, three and n variables
- Special cases of Euler-Lagrange equations
- Necessary condition for existance of an extremum of a functional
- Constrained maxima and minima

**Recommended Books:**

1. Powers D. L. , Boundary Value Problems and Partial Differential Equations, 5th edition, Academic Press, 2005.

2. Boyce W. E., Elementary Differential Equations, 8th edition, John Wiley and Sons, 2005.

**Suggusted Books:**

1. Krasnov M. L. Makarenko G. I. and Kiselev A. I., Problems and Exercise in the Calculus of Variations, Imported Publications, Inc., 1985.

2. J. W. Brown and R. V. Churchil, Fourier Series and Boundary Value Problems, McGraw Hill, 2006.

3. A. D. Snider, Partial Differential Equations: Sources and Solutions, Prentice Hall Inc., 1999.

**Description of system of evaluation (homework, midterms, final, assignments etc.): **

- Homework, Exercises, Attendance, Assignments: 20%
- Midterm: 30%, Final Term: 50 %

**Key Dates and Time of Class Meeting**

Monday: 9:30am-11:00am

Tuesday: 2:00am-3:30am

Commencement of Classes January 13, 2020

Mid Term Examination March 09-13, 2020

Final Term Examination May 04-08, 2020

Declaration of Result May 19, 2020