Fractional Differential Equations

The science and technology of ultracapacitors are reviewed for a number of electrode materials, including carbon, mixed metal oxides, and conducting polymers. More work has been done using microporous carbons than with the other materials and most of the commercially available devices use carbon electrodes and an organic electrolytes. The energy density of these devices is 3¯5 Wh/kg with a power density of 300¯500 W/kg for high efficiency (90¯95%) charge/discharges. Projections of future developments using carbon indicate that energy densities of 10 Wh/kg or higher are likely with power densities of 1¯2 kW/kg. A key problem in the fabrication of these advanced devices is the bonding of the thin electrodes to a current collector such the contact resistance is less than 0.1 cm2. Special attention is given in the paper to comparing the power density characteristics of ultracapacitors and batteries. The comparisons should be made at the same charge/discharge efficiency.

Ultracapacitors: Why, How, and Where is the Technology

http://ieeexplore.ieee.org/iel5/5610941/5624686/05624745.pdf

Power electronics

System Identification I

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

The increasing demand of electrical energy storage has motivated the wide usage of ultracapacitors. Ultracapacitors are capable of storing and delivering energy at a rate much higher than conventional rechargeable batteries. The dynamics characteristics of ultracapacitors are better exhibited by fractional calculus as compared to the conventional calculus. The present work aims at estimating the fractional model parameters of an ultracapacitor using experimental data and further investigating the dependency of fractional parameters on the operating conditions. The parameter estimation task has been formulated as an optimization problem which aims at minimizing the deviation between the model and experimental output using a hybrid optimization technique. The hybrid algorithm combines an improved version of seeker optimization algorithm for global exploration with a local search technique, i.e., Nelder---Mead simplex search. The results show that the hybrid algorithm is capable of identifying the model parameters efficiently in both time and frequency domain. The fractional behavior has been found to be dependent on the initial condition, magnitude and offset of the applied voltage.

A Hybrid Optimization-Based Approach for Parameter Estimation and Investigation of Fractional Dynamics in Ultracapacitors

Material removal during microwave drilling is basically due to thermal ablation of the material in the vicinity of the drilling tool. The microtip of the tool, also termed as concentrator, absorbs microwaves and ionizes the dielectric in its proximity creating a zone of plasma. The plasma takes the shape of a sphere owing to the atmospheric sphere, which acts as the source of thermal energy to be used for processing a material. This mechanism of heating, also called localized microwave heating, was used in the present study to drill holes in 1.2-mm-thick soda lime glass. The mechanism of material removal had been analyzed through simulation of the hot spot region, and the results were attempted to explain through experiment observations. It was realized that the glass being a poor conductor of heat, a low power (90 W in this case) yields better drilling results owing to more localized heat corresponding to a low-volume plasma sphere. The low application time prevents further heat transfer, and a localized concentration of heat becomes possible that primarily causes the material ablation. The plasma sphere appears sustain while the tool moves through the bulk of the glass thickness although its volume gets further shrunk. The process needs careful selection of the parameters. The simulation results show relatively low temperature in the top half (opposite to the tool tip) of the plasma sphere which eventually causes the semimolten viscous glass to collapse into the drill cavity as the tool advances into the bulk and stops the movement of the tool. The continued plasma sphere raises the tip temperature, which makes the tip to melt and gets blunt. The plasma formation ceases owing to larger diameter of the tool, and the tool gets stuck which could be verified through experimental results.

A simulation approach to material removal in microwave drilling of soda lime glass at 2.45 GHz

There are many functions which are continuous everywhere but not differentiable at some points, like in physical systems of ECG, EEG plots, and cracks pattern and for several other phenomena. Using classical calculus those functions cannot be characterized-especially at the non-differentiable points. To characterize those functions the concept of Fractional Derivative is used. From the analysis it is established that though those functions are unreachable at the non-differentiable points, in classical sense but can be characterized using Fractional derivative. In this paper we demonstrate use of modified Riemann-Liouvelli derivative by Jumarrie to calculate the fractional derivatives of the non-differentiable points of a function, which may be one step to characterize and distinguish and compare several non-differentiable points in a system or across the systems. This method we are extending to differentiate various ECG graphs by quantification of non-differentiable points; is useful method in differential diagnostic. Each steps of calculating these fractional derivatives is elaborated.

Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type

The enhancement of conductivity of a composite polymer as a dielectric material is an essential requirement for electrostatic storage devices. We have modified the microstructure of the polymer mat...

Nanofiller-Induced Ionic Conductivity Enhancement and Relaxation Property Analysis of the Blend Polymer Electrolyte Using Non-Debye Electric Field Relaxation Function

Mathematical modeling of many engineering and physics problem leads to extraordinary differential equations like Nonlinear, Delayed, and Fractional Order. An effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. A Fractional Differential Equation (FDE), where the leading differential operator is Riemann-Liouvelli (RL) type requires fractional order initial states which are sometimes hard to physically relate. Therefore, we must be able to solve these extraordinary systems, in space, time, frequency, area, volume, with physical reality conserved. Extra Ordinary Differential equation Systems and its solution, with Physical Principle, of action-reaction and equivalent mathematical decomposition method, are obtained as an aid for Physicists and Engineers to tackle the process dynamics with ease. This reactions-chain generates internal modes from zeroth mode reaction to first mode second mode and to infinite modes; instantaneously in parallel time or space-scales; and the sum of all these modes gives entire system reaction. This modal reaction as explained by physics theory exactly matches the principle of Adomian Decomposition Method (ADM). Fractional Differential Equation (FDE) with Riemann-Liouvelli formulation linear and non-linear is solved as per ADM. In this formulation of FDE by RL method it is found that there is no need to worry about the fractional initial states; instead one can use integer order initial states (the conventional ones) to arrive at solution of FDE. This new finding too is highlighted in this paper-along with several other problems to give physical insight to the solution of extraordinary differential equation systems. This way one gets insight to Physics of General Differential Equation Systems-and its solution-by Physical Principle and equivalent mathematical decomposition method. This facilitates ease in modeling.

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Shantanu Das

Papers

A Hybrid Optimization-Based Approach for Parameter Estimation and Investigation of Fractional Dynamics in Ultracapacitors

A simulation approach to material removal in microwave drilling of soda lime glass at 2.45 GHz

Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type

Nanofiller-Induced Ionic Conductivity Enhancement and Relaxation Property Analysis of the Blend Polymer Electrolyte Using Non-Debye Electric Field Relaxation Function

Solution of extraordinary differential equations with physical reasoning by obtaining modal reaction series