A new collection of real world applications of fractional calculus in science and engineering

doi:10.1016/j.cnsns.2018.04.019

Communications in Nonlinear Science and Numerical Simulation

Volume 64, November 2018, Pages 213-231

https://doi.org/10.1016/j.cnsns.2018.04.019 Get rights and content

Highlights

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A review of fractional calculus applications to the real world problems from science and engineering fields.
•
The real world applications of fractional calculus in different science and engineering fields are presented.
•
Fractional calculus provides better description for analyzing the dynamics of complex systems.

Abstract

Fractional calculus is at this stage an arena where many models are still to be introduced, discussed and applied to real world applications in many branches of science and engineering where nonlocality plays a crucial role. Although researchers have already reported many excellent results in several seminal monographs and review articles, there are still a large number of non-local phenomena unexplored and waiting to be discovered. Therefore, year by year, we can discover new aspects of the fractional modeling and applications. This review article aims to present some short summaries written by distinguished researchers in the field of fractional calculus. We believe this incomplete, but important, information will guide young researchers and help newcomers to see some of the main real-world applications and gain an understanding of this powerful mathematical tool. We expect this collection will also benefit our community.

Introduction

Fractional calculus (FC) is an emerging field in mathematics with deep applications in all related fields of science and engineering. Some of the results were reported in various books or related review articles [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. However, we are still at the beginning of applying this very powerful tool in many fields of research. At this moment, the fractional calculus has opened its wings even larger to cover the dynamics of complex real world and new ideas are starting to be implemented and tested on real data. In some cases, some patents were granted which make the tool of FC very promising. Though fractional calculus was introduced more than 300 years ago and applied into many fields of science and engineering, the promotion of applications is still an important task of the FC community. When we talk about FC with scientists and engineers outside of our community, two of the most frequently asked questions are about how FC has been applied and how scientists can apply it to their respective fields. Meanwhile, many FC researchers in theoretical fields are also not familiar with the application aspects. Therefore it is necessary to provide a brief introduction on successful applications of FC in science and engineering. Moreover, we should recognize that FC is not universal but has its own place in application; hence, providing some important existing successful applications of FC can offer a guide on application studies in the future.

To make this collection more comprehensive, we have invited several distinguished researchers in the application field of FC to contribute one or more application cases, and make a summary on a specific scientific/engineering area. However, there are still many experts in this field who have not been invited or contacted, due to the difficulty of email communication and the limitation of our knowledge. Furthermore, there are still many successful applications of FC which have not been included in this collection, due to the length limitation of this collection and the time limitation of submission.

This review is organized into nine sections. We begin with some important results of FC in physics, after that we briefly present some applications from the control theory and signal and image processing. The next main topics are from mechanics and dynamical systems, biology, environmental sciences, and materials. We end our review article by presenting some main results from applications of FC to multidisciplinary and other engineering fields.

Each section contains several contributions written by prestigious scientists. Each contribution contains some relevant references where the authors can find more information about the debated topics. In this review, we collected 46 contributions and we hope that the new information presented here will strongly contribute to the promotion and further development of fractional calculus and its applications.

Section snippets

Fractional Langevin equation description of viscoelastic anomalous diffusion in complex liquids

Many complex systems such as the crowded liquid inside biological cells, solutions and melts of polymeric materials, or lipid bilayer membranes are viscoelastic. Depending on the frequency with which these systems are probed, their response is more elastic or more viscous. Diffusion of tracer particles in these complex liquids is anomalous, with the mean squared displacement scaling like $〈 r^{2} (t) 〉 ≃ t^{α},$ where we speak of subdiffusion for 0 < α < 1 and superdiffusion for α > 1 [18]. Concurrently the

Ubiquitous fractional order memory system

The future states of an integer order dynamic system depend on the current one (memoryless). Nevertheless, for a fractional order system, the current state depends on the whole history (long memory). This long memory is typically a nameplate of various fractional order systems [12], [16]. Recall the first two successful applications of fractional calculus in the 1980s, i.e., fractional order viscoelasticity and fractional order quantum mechanics. Boltzmann superposition principle plays a

A study on fractional calculus applications in image processing

Fractional calculus is a fast developing mathematical discipline (that is, calculus of derivatives and integrals of any arbitrary real or complex order) has increased extensive notoriety and significance amid for more than four decades, mostly because of its applications in various apparently different and broad fields of science and engineering. It does surely give a few potentially valuable tools for solving integral, differential and integro-differential equations. Employing fractional

Long-term control for discrete fractional systems

Many engineering problems hold the feature of discrete time or space structures, for example, images, economy series, signals and so on. Some efforts have been dedicated to the applications of the continuous fractional calculus to these topics, and researchers mainly adopted the numerical discretization of the fractional calculus. But it can readily result in tedious information or numerical errors due to the memory effect. Discrete fractional calculus can avoid this and it is a straightforward

Fractional derivative models of diffusion in magnetic resonance imaging (MRI)

A common feature observed in diffusion-weighted MRI of the brain is anomalous diffusion. Hence, in white and gray matter, S(b), the signal intensity decay is often characterized by a stretched exponential $S (b) = S_{0} e x p [- {(b D_{0})}^{α}],$ where b is the degree of diffusion-weighting, D₀ is the tissue water diffusion coefficient, and 0 < α < 1 [111]. Since normal, or Gaussian diffusion decays as a single exponential $(α = 1),$ solutions to the anomalous diffusion problem proceed by neglecting important tissue

Chloride ion anomalous diffusion in concrete structures

Chloride ion erosion is one of the main reasons to affect the durability of concrete structures, and the core issue in research is the chloride ion transport mechanism analysis and modeling. As a typical porous material, concrete is uneven and anisotropic, and hence the ideal Fick’s law of diffusion is not applicable to describe the chloride ion diffusion behavior in concrete any more. In addition, due to the continuous hydration of cement binder, the geometrical, physical and chemical

Fractional derivative model for shape memory polymers

A shape-memory polymer (SMP) is a polymeric material that is capable of memorizing its original shape, and can acquire a temporary shape upon deformation and returns to its permanent shape in response to an external stimulus such as a temperature change. SMPs have been widely used from industrial to medical applications and even everyday life [137], [138].

Since the properties of SMPs are temperature dependent and often very sensitive to an external temperature change, their accurate modeling

Basic concepts of economic processes with memory

All previous investigations on the economic processes with memory were considered within the discrete-time approach. In economics the fractional differencing and integrating have been suggested in the works of Granger and Joyeux, and Hosking, using the discrete time approach only. These fractional differencing and integrating are used in economics without direct connection with the fractional calculus and the well-known finite differences of non-integer orders. We demonstrate that the

Anomalous dielectric properties

In Maxwell’s equations, which govern the propagation of electromagnetic waves, the interaction between polarization and electric fields is described by the complex susceptibility. This is an empirical law derived by matching experimental data in some mathematical model. After the simpler Debye model, more involved models have been proposed [149]. In the Havriliak–Negami (HN) model, two real powers are introduced to fit the anomalous dielectric properties observed in disordered materials and

Acknowledgments

We thank distinguished researchers and colleagues for valuable contributions on this collection. The work was supported by the National Natural Science Foundation of China (Grant Nos. 11572112, 11572111, 41628202, and 11811530069). Y. Zhang was also partially supported by the National Science Foundation grant DMS-1460319 and the University of Alabama. This paper does not necessarily reflect the view of the funding agencies.

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ReviewA new collection of real world applications of fractional calculus in science and engineering

Highlights

Abstract

Introduction

Section snippets

Fractional Langevin equation description of viscoelastic anomalous diffusion in complex liquids

Ubiquitous fractional order memory system

A study on fractional calculus applications in image processing

Long-term control for discrete fractional systems

Fractional derivative models of diffusion in magnetic resonance imaging (MRI)

Chloride ion anomalous diffusion in concrete structures

Fractional derivative model for shape memory polymers

Basic concepts of economic processes with memory

Anomalous dielectric properties

Acknowledgments

Commun Nonlinear Sci Numer Simul

Electrochim Acta

J Electroanal Chem

Bull Math Biol

Chaos Solitons Fract

Automatica

ISA Trans

IEEE/CAA J Autom Sin

Int J Control

Comput Math Appl

Nonlinear Dyn

Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

Phys A

Commun Nonlinear Sci Numer Simulat

Commun Nonlinear Sci Numer Simulat

Theor Appl Mech

Meccanica

Commun Nonlinear Sci Numer Simulat

Comput Math Appl

Fractional integrals and derivatives: Theory and applications

Theory and application of fractional differential equations

Hamiltonian chaos and fractional dynamics

Fractional calculus in bioengineering

Fractional calculus: models and numerical methods, series on complexity, nonlinearity and chaos

Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models

Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media

Fractional calculus for scientists and engineers

Fractional derivatives for physicists and engineers: Volume II. Applications

Fractals and fractional calculus in continuum mechanics

Fractional calculus view of complexity: Tomorrow’s science

Fractional thermoelasticity

Fractional kinetics in solids: Anomalous charge transport in semiconductors, dielectrics and nanosystems

Fractional-order systems and controls

Applications of fractional calculus in physics

World Scientific

Anomalous transport: Foundations and applications

Anomalous diffusion models and their properties: Non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

Phys Chem Chem Phys

In vivo anomalous diffusion and weak ergodicity breaking of lipid granules

Phys Rev Lett

Lateral motion of phospholipids and cholesterols in a lipid bilayer: anomalous diffusion and its origins

Phys Rev Lett

Anomalous diffusion and power-law relaxation in wormlike micellar solution

New J Phys

Superdiffusion dominates intracellular particle motion in the supercrowded space of pathogenic acanthamoeba castellanii

Sci Rep

On pore-fluid viscosity and the wave properties of saturated granular materials including marine sediments

J Acoust Soc Am

Connecting the viscous grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

Proceedings of the seventy-eighth EAGE conference and exhibition

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

J Acoust Soc Am

A unifying fractional wave equation for compressional and shear waves

J Acoust Soc Am

A causal and fractional all-frequency wave equation for lossy media

J Acoust Soc Am

A theoretical basis for the application of fractional calculus to viscoelasticity

J Rheology

Linking the fractional derivative and the lOmnitz creep law to non-newtonian time-varying viscosity

Phys Rev E

Tempered fractional Feynman–Kac equation: theory and examples

Phys Rev E

Review
A new collection of real world applications of fractional calculus in science and engineering