scholarly journals Rigorous time domain responses of polarizable media

1997 ◽  
Vol 40 (2) ◽  
Author(s):  
M. Caputo
Keyword(s):  
Fractional Order ◽  
Time Domain ◽  
Order Differential Equation ◽  
Step Function ◽  
Induced Polarization ◽  
Fractional Order Differential Equation ◽  
Polarization Phenomena ◽  
Electric Flux ◽  
The Time Domain ◽  
Derivatives Of

The scope of this note is to study a model of induced polarization which fits the usually accepted frequency dependent formula of Cole and Cole, but is more general and allows the time domain observations to retrieve the parameters describing the induced polarization phenomena of the medium. By introducing the memory mechanisms, represented by derivatives of fractional order, in the relation between the electric flux density and the electric field and considering the fractional order differential equation which follows, I solve it with mathematically rigorous and closed formulae and compute the responses to a step function, a box, a set of positive boxes and a set of alternating positive and negative boxes. I also introduce a method which retrieves the parameters describing the medium when comparing the theoretical curves with the observed ones. The responses to these signals also allow to estimate the temporary alteration of the medium when repeated positive (negative) signals are input; the response increases (decreases) in amplitude when the signals are all positive (negative), it decreases when the signals are alternatively positive and negative in agreement with the known attitude of the medium to induced polarization.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Fractional Order ◽  
Gamma Function ◽  
Order Differential Equation ◽  
Incomplete Gamma Function ◽  
Order System ◽  
Fractional Order Differential Equation ◽  
Laplace Domain ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
The Time Domain

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.

scholarly journals The application of hypergeometric functions to computing fractional order derivatives of sinusoidal functions

2016 ◽  
Vol 64 (1) ◽  
pp. 243-248 ◽  
Author(s):  
M. Włodarczyk ◽  
A. Zawadzki
Keyword(s):  
Fractional Order ◽  
Hypergeometric Functions ◽  
Order Differential Equation ◽  
Analytical Form ◽  
Single Equation ◽  
Fractional Order Differential Equation ◽  
Lommel Functions ◽  
Sinusoidal Functions ◽  
Derivatives Of ◽  
Sinusoidal Function

Abstract In the paper, the analytical forms of fractional order derivatives of sinusoidal function according to definitions of Riemann - Liouville and Caputo are presented. To determine the analytical form of the integrals appearing in definitions of derivatives of fractional order the Lommel functions from hypergeometrical functions family were applied. With the use of properties of the derivatives of fractional order - differential-integral there were presented the conception of generalized element of a single equation, which depending on the value of the derivative order, can be inductor, resistor, capacitor, or a hypothetical element of a fractional order differential equation.

scholarly journals Fractional-Derivative Approximation of Relaxation in Complex Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Kin M. Li ◽  
Mihir Sen ◽  
Arturo Pacheco-Vega
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
System Identification ◽  
Complex Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Time Dependent ◽  
Fractional Order Differential Equation

In this paper, we present a system identification (SI) procedure that enables building linear time-dependent fractional-order differential equation (FDE) models able to accurately describe time-dependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the Mittag-Leffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an error-cost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shell-and-tube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.

Improving the time domain response of fractional order digital differentiators by windowing

2015 ◽  
Vol 107 ◽  
pp. 282-289 ◽  
Author(s):  
Chengyan Peng ◽  
Xiaochuan Ma ◽  
Geping Lin ◽  
Min Wang
Keyword(s):  
Fractional Order ◽  
Time Domain ◽  
Time Domain Response ◽  
The Time Domain

scholarly journals THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040025
Author(s):  
JINGFEI JIANG ◽  
JUAN L. G. GUIRAO ◽  
TAREQ SAEED
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Existence Results ◽  
Extremal Solution ◽  
Variable Order ◽  
Fractional Order Differential Equation ◽  
Causal Operator ◽  
Linear Differential

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

scholarly journals Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Changjin Xu
Keyword(s):  
Boundary Conditions ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Order Differential Equation ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Fractional Order Differential Equation ◽  
Uniqueness Of Solutions ◽  
Integral Boundary

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

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