Mathematical Methods

Syllabus Details

• Unit I: (Integral transforms) Laplace transforms: Definitions, properties, Laplace transforms of some elementary functions, convolution theorem, inverse Laplace transformation, applications.
• Unit II: (Fourier transforms) Definitions, properties, Fourier transforms of some elementary functions, convolution theorems, Fourier transform as a limit of Fourier Series.
• Unit III: (Integral equations) Volterra integral equations: Basic concepts, relationship between linear differential equations and Volterra integral equations, resolvent kernel of Volterra integral equations, solution of integral equations by resolvent kernel, the method of successive approximations, convolution type equations, solution of integro-differential equations with the aid of Laplace transformation. Fredholm integral equations: Fredholm equations of the second kind, fundamentals, iterated kernels, constructing the resolvent kernel with the aid of iterated kernels, integral equations with degenerate kernels, characteristic numbers and eigenfunctions, solution of homogeneous integral equations with degenerate kernel, non-homogeneous symmetric equations, Fredholm alternative.
• Unit IV: (Calculus of variations) Extrema of functionals: The variation of a functional and its properties, Euler’s equation, field of extremals, sufficient and necessary conditions for the extremum of a functional both for weak and strong extrema; Legendre and Weierstrass theorems, Hilbert invariant integral theorem, conditional extremum, moving boundary problems, discontinuous problems, one-sided variations, Ritz method.

Course Learning Objective

1. Compute the Laplace and Fourier transforms of functions and solve linear Volterra integral equations by computing the resolvent kernel or using the Laplace transforms.
2. Convert linear Volterra/Fredholm integral equations to ordinary differential equations.
3. Examine the existence of solution of non-homogeneous Fredholm integral equations using the Fredholm alternatives.
4. Write the Euler-Lagrange equations for various types of functionals containing many dependent variables, many independent variables, higher order derivatives, etc.
5. Analyze whether the extremals are minimizers/maximizers of the given functionals.
6. Find the extremals for moving boundary problems and functionals with constraints.

Recommended Books

1. Brunt, Bruce van, The Calculus of Variations, Springer-Verlag, New York, 2004.
2. Sneddon I. N., The Use of Integral Transforms, Tata McGraw Hill, 1972.
3. Spiegel, Murray R., Schaum’s Outline of Laplace Transforms, Schaum’s Outline Series, 1965.
4. Gelfand, I. M. and Fomin, S. V., Calculus of Variations, revised English edition translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963.
5. Krasnov, M. L.; Makarenko, G. I. and Kiselev, A. I., Problems and Exercises in the Calculus of Variations, translated from the Russian by George Yankovsky, 1975.
6. Krasnov, M. L.; Makarenko, G. I. and Kiselev, A. I., Problems and Exercises in Integral Equations, translated from the Russian by George Yankovsky, 1975.
7. Kanwal, Ram P., Linear Integral Equations, second edition, Birkhäuser Boston, Inc., Boston, MA, 1997.
8. Pipkin, Allen C., A Course on Integral Equations, Texts in Applied Mathematics, 9, Springer-Verlag, New York, 1991.
9. Gibbons, M. M., A Primer on the Calculus of Variations and Optimal Control Theory, Volume 50, AMS, 2009.