Begell House Inc.
Critical Reviewsâ„˘ in Biomedical Engineering
CRB
0278-940X
36
1
2008
SPECIAL ISSUE: INSIGHTS INTO THE PHYSICS AND THE SOLUTIONS OF FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS Preface
v-ix
10.1615/CritRevBiomedEng.v36.i1.10
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
SPECIAL ISSUE
Motivation
This work was originally presented as a series of NASA reports between 1998 and 2000. The next four sections will provide an overview of each of the four papers, as well as some insight into the importance of the problems addressed. Several new books on the general topic of the fractional calculus have been written in the interim [1]-[3] and some updated information is also provided.
Insights Into the Fractional Order Initial Value Problem Via Semi-Infinite Systems
1-21
10.1615/CritRevBiomedEng.v36.i1.20
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
This paper considers various aspects of the initial value problem for fractional order differential equations. The main contribution of this paper is to use the solutions to known spatially distributed systems to demonstrate that fractional differintegral operators require an initial condition term that is time-varying due to past distributed storage of information.
A Solution to the Fundamental Linear Fractional Order Differential Equation
23-38
10.1615/CritRevBiomedEng.v36.i1.30
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
This paper provides a solution to the fundamental linear fractional order differential equation, namely, cdtqx(t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi-infinite lossy line, is used to demonstrate the theory.
Generalized Functions for the Fractional Calculus
39-55
10.1615/CritRevBiomedEng.v36.i1.40
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
R-Function Relationships for Application in the Fractional Calculus
57-78
10.1615/CritRevBiomedEng.v36.i1.50
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis.
Relationships of the R-function to the common exponential function, et, and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, et, in terms of the R-function are developed. Also, some approximations for the R-function are developed.