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.

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

2007 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
System Theory ◽  
Fractional Order ◽  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

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.

The incomplete gamma functions

2016 ◽  
Vol 100 (548) ◽  
pp. 298-306 ◽  
Author(s):  
G. J. O. Jameson
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Definition Of ◽  
Image Position ◽  
Selection Of

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:(1)(2)Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

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.

The Initialization Response of Multi-Term Linear Fractional-Order Systems With Constant History Functions

2011 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Order System ◽  
Fractional Order Systems ◽  
Fractional Order Differential Equation ◽  
Term Linear ◽  
Single Term ◽  
Negative Time ◽  
Short Times

In this paper, the distinction between an operator’s historical initial condition function, the consequential initialization function of the operator, and the resulting initialization response of an entire system, is discussed. The single term and two-term differential equation results with constant history functions from earlier studies are reviewed. A three-term linear fractional-order differential equation with constant history function is studied next. This system is solved by using the proper Laplace transforms for the fractional-order derivatives. The paper then presents the initialization responses for multi-term linear fractional-order systems with commensurate orders that have had arbitrarily-long constant displacements in negative time. Results for short-times and for long-times are provided. These results are obtained by using the proper Laplace transform for the fractional-order derivatives. Using the results of this paper, the initialization response of any linear, commensurate-order, fractional-order system, with arbitrarily-long constant displacements in negative time can be determined.

scholarly journals Convergent expansions of the incomplete gamma functions in terms of elementary functions

2018 ◽  
Vol 16 (03) ◽  
pp. 435-448 ◽  
Author(s):  
Blanca Bujanda ◽  
José L. López ◽  
Pedro J. Pagola
Keyword(s):  
Gamma Function ◽  
Error Bounds ◽  
Rational Functions ◽  
Incomplete Gamma Function ◽  
Elementary Functions ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Uniformly Convergent

We consider the incomplete gamma function [Formula: see text] for [Formula: see text] and [Formula: see text]. We derive several convergent expansions of [Formula: see text] in terms of exponentials and rational functions of [Formula: see text] that hold uniformly in [Formula: see text] with [Formula: see text] bounded from below. These expansions, multiplied by [Formula: see text], are expansions of [Formula: see text] uniformly convergent in [Formula: see text] with [Formula: see text] bounded from above. The expansions are accompanied by realistic error bounds.

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.

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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