Fractional Differintegrations Insight Concepts

Abstract

This chapter describes the geometric and physical interpretation of fractional integration and fractional differentiation. As a start point the Riemann-Liouville (RL) fractional integration is taken. Briefly existence of fractional differintegration is discussed along with useful tricks to obtain the fractional differintegration. The geometric interpretation is developed first for RL integration process along with concept of transformed time scales, and in-homogeneous time axis. Thereafter the RL definition is geometrically explained by convolution of the power function and the integrand, and as area under shape changing curve is demonstrated. The concept of delay is developed for Grunwald-Letnikov differintegration process and this is converted into the specific definition of short-memory principle, used for computer applications. The GL differintegration is also explained as in the classical calculus by considering infinitesimal quantities for the independent variable and the function, and explained graphically. The GL definition is expanded with binomial coefficients and its application to numerical regression. These methods are advance algorithms to get digital realization for fractional order controllers. The application to solve fractional differential equation numerically is demonstrated. Small introduction is made regarding definitions of Local Fractional Derivatives (LFD) for continuous but nowhere differentiable functions. These LFD (Kolwankar-Gangal K-G definition’s) utility is extended to measure critical point behaviors of physical system and its relation to ‘fractal’ dimension. The demonstration is made to have fractional integration and fractional differentiation, for fractal distributed quantities; thus, line, surface and volume integration can be performed when the measurable quantities are distributed in fractal form, Thereby generalizing the Gauss’s and Stroke’s law for fractal distributed quantities.