Fractional Calculus and Fractional Differential Equations

Code: MA663 | L-T-P-C: 3-0-0-6

Prerequisites: MA542 or equivalent

Preamble
In a number of practical problems, integral-order derivatives and differential equations do not convey the exact picture of the situation. The use of the concept of fractional calculus shows new ways to tackle these problems. This course is aimed at introducing the methods and tools of fractional order calculus to science and engineering students of B.Tech, M.Sc., M.Tech. and Ph.D. students.

Outline
Special functions of fractional calculus: Gamma function, MIttag-Leffler function, Wright function. Fractional derivatives and integrals: Grunwald-Letnikov fractional derivatives, Riemann-Liouville fractional derivatives, geometric and physical interpretation of fractional integration and differentiation, sequential fractional derivatives, properties of fractional derivatives, Laplace, Fourier and Mellin transforms of fractional derivatives. Linear fractional differential equations: Equation of a general form, existence and uniqueness theorem as a method of solution, dependence of a solution on initial conditions, Laplace transform method, standard fractional differential equations, sequential fractional differential equations. Some methods for solving fractional order equations: Mellin transform, power series, orthogonal polynomials, numerical evaluation of fractional derivatives, approximation of fractional derivatives. Texts/References:

1. Basic Theory of Fractional Differential Equations, Y. Zhou, World Scientific, 2014.
2. Fractional Differential Equations, I. Podlubny, Academic Press, 1998.
3. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, K.B. Oldham and J. Spanier, Dover Publications, 2006.
4. An Introduction to the Fractional Calculus and Fractional Differential Equations, K.S. Miller and B. Ross, Wiley-Interscience, 1993.