Fractional calculus: A new language of complexity

February 04, 2017

By ARL Guest Author Bruce J. West

The fractional calculus, a branch of mathematics formulated alongside classical calculus, has drawn the attention of physical, biological and engineering sciences in the last decade.

Although its development started in 1695, only in recent decades has it been regarded as a tool with real-world applications rather than a concept that is purely in the mathematical realm.

The seventeenth century differential calculus of Newton and Leibniz enabled natural philosophers to quantitatively reason about the changes in physical phenomena, and laid the groundwork for the Industrial Revolution.

The eighteenth century statistics of Gauss enabled social and life scientists to quantify uncertainty in their respective domain, following the lead of physics.

Today's technical problems require even more creative techniques to circumvent or directly overcome the research challenges blocking progress in a number of disciplines.

The dynamics of complex nonlinear phenomena demands that we extend our horizons beyond analytic functions and classical analysis.

These forays into the frontiers suggest that the functions necessary to describe complex phenomena lack traditional equations of motion; whether in complex physical phenomena, where Newton's law does not describe the dynamics, or social phenomena, where Newton's law is not expected to apply.

Consequently, the replacement of ordinary differential equations by alternative descriptions are being developed and applied.

We consider complex phenomena to have multiple nonlinearly interacting components with emerging behavior entailed by, but not inferred from, the dynamics of its component parts.

Nonlinear interactions give rise to a blend of regular and erratic variability in complex phenomena, which enable them to adapt to changing environments and thereby survive.

However, to model this behavior requires use of the fractional calculus, including such Army-critical problems as: the fractional rate equation that incorporates long-term memory into the dynamics of such things as rubber and mud; the fractional diffusion equation, which incorporates spatial heterogeneity into models of reaction-diffusion effects in explosives; and fractional kinetic equations that synthesizes the effect of chaotic dynamics and probability density functions to model turbulence.

The fractional calculus offers a new way of thinking about how to overcome the research barriers preventing the solving of technical problems critical to the Army; a way of thinking made necessary by the demands of contemporary science to overcome the complexity barriers to understanding, which are present in virtually every scientific discipline.

Complexity entails the quantitative, as well as qualitative, richness of nonlinear dynamic phenomena, so that scientific understanding must encompass both.

In order to garner such understanding in a systematic way, it requires going beyond the traditional methods of modeling and simulation as the only ways of transforming data into information and subsequently into knowledge.

Consequently, the mathematical models of reality based on the differential calculus have been pushed to their limits and beyond.

Those mathematical models, which were developed and applied so successfully to the explanations for, and understanding of, physical phenomena in the nineteenth and twentieth centuries, are no longer adequate to describe the emergent phenomena of the twenty-first century in the Internet of Battle Things, or IoBT, and in the Fog of War in Cyberspace, or FoWiC.

These battlefields of the future will be composed of a substantial number of systems and devices, which will need to collaborate and communicate both with each other and with human warfighters.

In this unpredictable environment, executable and accurate information derived from data obtained on the battlefield would be vital.

Even so, converting the massive scale of information, possibly many orders of magnitude larger than what is currently being considered on the battlefield, into actionable and trustworthy data is immensely complex.

This information will include both nonlinear and nonstationary dynamic processes, but information theory, which is typically used to quantify such data, requires that the underlying processes to be ergodic.

The fractional calculus constitutes a fresh approach to formulating and solving critical scientific and engineering problems, which violate the historic assumptions that interactions in space are homogeneous and local; that interactions in time are isotropic and without memory; and that linear dynamics is a good first approximation to a system's behavior.

The new strategy made possible by the fractional calculus complements the more traditional methodologies and is entailed by complexity within the system.

The intent is to reveal how complexity forms the basis of the scientific barriers to solving problems of crucial importance to the Army, a few of which have been mentioned, and which requires this new mathematical infrastructure for their detailed formulation and resolution.

The integration of new weapon systems, new information networks and new sensor networks into the fabric of the military is part of how the United States is expected to maintain its military and social superiority throughout the twenty-first century.

Science will continue to seek understanding of the world as it is and engineering will find ways to use science to bend the world to specific tasks and carry out prescribed functions.

However, the emphasis is no longer on the single sensor; the stress is on sensor networks; the platform is replaced by networks of communicating platforms.

The concern today is how the information from new sensors is integrated into the overall informational infrastructure of the networked battlefield as in the IoBT, as well as the Internet of Things, or IoT.

This is where revolutionary mathematics enters the discussion, mathematics that entails the development of totally new strategic and tactical thinking necessitated by the effect disruptive mathematics has on existing scientific concepts.

Thus, science for the Army is not only concerned with the fielding of new devices, but also with how those devices change the networks to which they contribute, as well as how these networks are subsequently used.

We now recognize that all networks are interconnected and an advantage achieved in one sub-network does not necessarily produce an overall advantage in the host network.

In fact, the enhanced performance of one sub-network may degrade overall performance, thereby forcing an ever deeper exploration into the understanding of how complex networks interact with one another and exchange information.

Mathematics codifies the conceptual infrastructure for science, finding expression in theory used to guide technology as the engineering fulfillment of that mathematical/ theoretical reasoning.

For example, there is presently an ARL program to develop nonlocal and fractional-order methods for near-wall turbulence, large-eddy simulation and fluid-structure interaction.

Another is the use of fractional calculus to solve multi-scale physical problems as well as another to develop fractional partial differential equations for conservation laws and beyond.

In short, in this age of information and complexity, ARL scientists and engineers, in collaboration with a world-wide community of scientists and engineers, are re-examining the foundations of science, with the view of synthesizing the various scientific disciplines into a SCIENCE, without the artificial, but necessary, boundaries imposed to define a discipline.

This re-examination has suggested the recasting of physical laws into fractional forms, which may provide the missing links between complex dynamics in the physical, social and life sciences.

For further information, Dr. Bruce J. West can be contacted at

The U.S. Army Research Laboratory, currently celebrating 25 years of excellence in Army science and technology, is part of the U.S. Army Research, Development and Engineering Command, which has the mission to provide innovative research, development and engineering to produce capabilities that provide decisive overmatch to the Army against the complexities of the current and future operating environments in support of the joint warfighter and the nation. RDECOM is a major subordinate command of the U.S. Army Materiel Command.


Last Update / Reviewed: February 4, 2017