The fractional calculus (which admits of integrals and derivatives of non-integer order) dates back almost to the origin of the better-known ordinary (integer-order) calculus, but thus far has been treated more as a mathematical curiosity than as a scientific and engineering tool. Increasingly many physical processes are found to be best described using fractional differential equations. These processes include: viscoelasticity, rheology, electrochemistry, fractal processes, and many diffusion processes. The application of the fractional calculus to scientific and engineering problems has been inhibited by difficulties that arise from the basic definitions given heretofore for integrals and derivatives of arbitrary order. These difficulties are associated with the initialization problem, which is explained below.

Consider a function *f*(*t*), where *t *could be time or any other independent variable, and let_{ c}D_{t}^{n}*f*(*t*) denote the nth-order derivative of *f* with respect to *t* (where n is a positive real number and not necessarily a integer). One of the properties that must be preserved to make the fractional calculus compatible with the ordinary calculus is the composition or the index law, which is represented by the following equation:

The initialization problem arises as follows: One of the requirements of prior formulations of the fractional calculus is that in order to preserve composition, *f*(*t*) and all of its derivatives must be identically zero at *t* = *c*. Inasmuch as it is difficult to make all functions and derivatives of interest initially zero in many scientific and engineering problems, this requirement has effectively rendered until now the fractional calculus inapplicable in such cases.

As its name suggests, the initialized fractional calculus is formulated to address the initialization problem. In the initialized fractional calculus, it is not required that *f*(*t*) and its derivatives be zero at *t* = *c* in order to preserve composition. What enables the elimination of this requirement is the revision of definitions of integrals and derivatives of arbitrary orders to include initialization functions that carry the history of the differintegral. The initialization functions are generalizations of the constants of integration that appear in the ordinary calculus, where they are used to represent initial conditions.

Other major topics addressed in the development of the initialized fractional calculus include the following:

- Heretofore, Laplace transforms have been the primary tools for solving fractional differential equations. Therefore, the basic Laplace transforms for the initialized fractional integrals and derivatives were developed.
- The concept of a variable-structure or variable-order differintegral was introduced. This concept can be represented by the fractional differential equation
_{c}D_{t}^{q}^{(t)}*y*(*t*) =*f*(*t*) and the companion inferred integral equation_{c}D_{t}^{–q}^{(t)}*f*(*t*) =*y*(*t*), where*q*(the structure or order parameter) is a function of*t*or*y*. Some phenomena in viscoelasticity and diffusion appear to be amenable to treatment by fractional differential equations with variable order parameters.

Going beyond viscoelasticity, the initialized fractional calculus can be applied to problems that arise in a variety of scientific, engineering, and purely mathematical disciplines, including creep, percolation, material science, viscous fluid behavior, heat transfer, batteries, electromagnetics, control, communications, filtering, and chaotic systems.

*This work was done by Carl F. Lorenzo of *Glenn Research Center *and Tom T. Hartley of the University of Akron. For further information, access the Technical Support Package (TSP) *free on-line at www.nasatech.com/tsp *under the Information Sciences category.*

*Inquiries concerning rights for the commercial use of this invention should be addressed to NASA Glenn Research Center, Commercial Technology Office, Attn: Steve Fedor, Mail Stop 4—8, 21000 Brookpark Road, Cleveland, Ohio 44135. Refer to LEW-17139.*

##### NASA Tech Briefs Magazine

This article first appeared in the October, 2002 issue of *NASA Tech Briefs* Magazine.

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