R.B.V.R.R WOMEN’S COLLEGE (AUTONOMOUS) DEPARTMENT OF M.Sc. Semester IV Paper-II Syllabus 2017-18 No. of Hrs / week: 5 No. of credits : 5 RM – 402 of Variations and

Learning Objectives: The course is aimed to Lay a broad foundation for an understanding of the problems of the calculus of variations and its many methods and techniques and to prepare students for the study of modern optimal control theory. To make the students familiar with the methods of solving Integral Equations.

Learning Outcomes : On successful completion of the course students will be able to recognize difference between Volterra and Fredholm Integral Equations, First kind and Second kind, homogeneous and inhomogeneous etc. They apply different methods to solve Integral Equations. Students will have much better and deeper understanding of the fundamental concepts of the space of admissible variations and concepts of a weak and a strong relative minimum of an integral.

Unit I

Definitions of Functionals - Strong and Weak Variations - Derivations of Euler's' - Other forms of Euler's equation - Special cases – Examples - Fundamental Lemma of Calculus of Variation

Unit II

Basic concepts - Relationship between Linear differential equations and Volterra Integral equations - Resolvent Kernel of Volterra Integral equation. Differentiation of some resolvent kernels - Solution of Integral equation by resolvent kernel - The method of successive approximations - Convolution type equations.

Unit III

Solution of Integro-differential equations with the aid of the Laplace Transformation – VIE with limits (x,α), Volterra integral equation of the first kind -VIE of the first kind of the convolution type - Euler - Beta and Gamma functions and their elementary properties - Relationship between Beta and Gamma functions. VIE of the first kind of the convolution type.

Unit IV

Fredholm integral equations of the second kind - Fundamentals - Method of Fredholm Determinants- Iterated kernels constructing the resolvent kernel with the aid of iterated kernels - Integral equations with degenerated kernels - Hammerstein type equation.Characteristic numbers and Eigen functions and its properties. Solution of homogeneous equations with degenerated kernel. Non homogeneous symmetric equations.

Prescribed Books:

1. M. Krasnov, A. Kiselev, G. Makarenko, Problems and Exercises in Integral

Equations – Mir Publishers in 1971

2. L.Elsgolts, and Calculus of Variations- Mir, Moscow 1977.

Suggested Books:

1. B Van Burnt, Calculus of Variations – Springer 2. Bernard Dacorogne, Introduction to Calculus of variations – Imperial College Press 3. I M Gelfard and S V Fomin, Calculus of Variations – Prentice Hall Press 4. Abdul Majid Wazwaz, Linear & Non linear Integral Equations- Springer 5. A D Polyanin , A V Manzhirov, Handbook of Integral Equation – Chapman & Hall / CRC