Integral Equations and Calculus of Variations , Fredholm and Voltra Integral Equations , Linear B.sc

Integral Equations and Calculus of Variations , Fredholm and Voltra Integral Equations , Linear and Non-Linear Integral Equations

Syllabus

Integral Equations and Calculus of Variations

Unit-I
Preliminary Concepts: Definition and classification of linear integral equations. Conversion of initial and boundary value problems into integral equations. Conversion of integral equations into differential equations. Integro-differential equations. Fredholm Integral Equations: Solution of integral equations with separable kernels, Eigen values and Eigen functions. Solution by the successive approximations, Neumann series and resolvent kernel. Solution of integral equations with symmetric kernels, Hilbert-Schmidt theorem, Green’s function approach.

Unit-II
Volterra Integral Equations: Successive approximations, Neumann series and resolvent kernel. Equations with convolution type kernels. Solution of integral equations by transform methods: Singular integral equations, Hilbert transform.

Unit-III
Calculus of Variations: Basic concepts of the calculus of variations such as functionals, extremum, variations, function spaces, the brachistochrone problem. Necessary condition for an extremum, Euler`s equation with the cases of one variable and several variables, Variational derivative. Invariance of Euler`sequations. Variational problem in parametric form.

Unit-IV
General Variation: Functionals dependent on one or two functions, Derivation of basic formula, Variational problems with moving boundaries, Broken extremals: Weierstrass–Erdmann conditions.

Books Recommended:
1. Abdul J. Jerry: Introduction to Integral Equations with Applications, 2nd Ed., Clarkson University Wiley Publishers, 1999.

2. G. L. Chambers: Integral Equations: A short Course, International Text Book Company Ltd., 1976.

3. R. P. Kanwal: Linear Integral Equations, 2nd Ed., Birkhauser Bosten, 1997.

4. Hochstadt Harry: Integral Equations, John Wiley & Sons, 1989.

5. I. M. Gelfand, S.V. Fomin: Calculus of Variations, Dover Books, 2000.

6. Weinstock Robert: Calculus of Variations with Applications to Physics and Engineering, Dover Publications, INC., 1974.

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