This presentation with the example of a modified capacitor theory will introduce fractional derivatives and integrals (i.e. subject of ractional calculus) that we observed in several experiments-with super-capacitors and dielectric relaxations. We revisit the concept of classical capacitor theory-and derive possible new explanations to the definition of capacitance, charge stored in capacitor. Here we will describe that charge stored at any time in a capacitor is "convolution integral" of defined capacity function of a capacitor and voltage stress across it. This approach however is different to the conventional method where we multiply the capacity and voltage functions of time to obtain charge stored as function of time. This new concept is in line with the observation of charge stored, relaxation current in form of impulse function for ideal (text-book) geometrical capacitor of constant capacity, and also for capacitor relaxing with power-law decay current that is given by universal dielectric relaxation law, when an uncharged capacitor is impressed with a constant voltage stress. This universal dielectric relaxation law gives rise to fractional derivative relating voltage stress and relaxation current that is formulation of "fractional capacitor". A "fractional capacitor" we will discuss with this new concept of redefining the charge store definition i.e. via this convolution integral approach, and obtain its loss tangent value. We will also show for a fractional capacitor by the use of time varying capacity function we can convert the fractional capacity constant that is in terms of fractional units of Farads per second to the power a fractional number, to units of Farads. From the defined capacity function, we will also derive integrated capacity of capacitor. We will also give possible physical explanation by taking example of porous and non-porous pitchers of constant volume holding water and thus, explaining the various aspects of classical capacitor and fractional capacitor that we arrive with this new formulation. The classical power law relaxation, i.e. relaxation of current with inverse of power of time for a step-voltage excitation to dielectric-as popularly known as Curie-von Schweidler law is empirically derived and is observed in several relaxation experiments on various dielectrics studies since late 19th Century. This relaxation law is also regarded as "universal-law" for dielectric relaxations; and is also termed as power law. In this deliberation, we give simple mathematical treatment to derive the distribution of relaxation rates of this Curie-von Schweidler law (a non-Debye relaxation), and show that the histogram of relaxation rate follows Zipf's power law distribution. In this deliberation, we develop possible explanation that as to why Zipfian distribution of relaxation rates appears for Curie-von Schweidler Law. In this presentation we derive appearance of fractional derivative while using Zipfian power law distribution that gives notion of scale dependent relaxation rate function for Curie-von Schweidler relaxation phenomena.
Sri Shantanu Das (BARC) graduated in Electrical Engineering and Electronics Engineering, from BITS Pilani and thereafter working as scientist at BARC since 1984-85; in the area of Nuclear Reactor Control and Safety. Several of circuits developed by him are used in control systems, safety systems and safety critical systems of Indian nuclear plants of NPCIL and Research reactors. The circuits and technology developed by him are also used in private and public sector industries. Apart from circuits & systems of control and instrumentation development he is doing development on Fractional Calculus since about more than decade and a half. This development on Fractional Calculus is done basically by him in order to understand natural laws and also for engineering advancement in area of controls (especially to aim robust and energy/fuel efficient control system), in area of signal processing and in area of systems identification. He has developed this subject on his own without any guide and has taken detailed courses on this subject for MPhil, MSc, BE, ME and PhD students of Mathematics, Science and Engineering Streams, at Jadavpur Univ, at Calcutta Univ, and short courses at Mumbai Univ, and at Pune Univ, at VNIT-Nagpur, and at other places. Since 2004, he is also working on the development of meta-material science to understand exotic properties of electro magnetism and to manipulate electromagnetic flow for usage is electronics systems, and also in the application of Microwave power for Material Processing (Joining Drilling Casting), and for development of indigenous super-capacitors & its power-packs for industrial usage and in energy conversion systems. He is been awarded as Honorary Senior Research Professor at Department of Physics Jadavpur University, Adjunct Professor at DIAT Pune, and was conferred as UGC Visiting Fellow at Department of Applied Mathematics Calcutta University. He has several publications (books journal and conference papers) and also several patents/copyrights on all these science & technological developments done by him in the above mentioned field of science mathematics and engineering. He is pioneer in growth of subject of Fractional Calculus subject in India.