A fractional derivative for modeling real-world problems

The article “On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation” (Appl. Math. Comput. 273: 948-956, 15 January 2016), was recently named a New Hot Paper for Mathematics in Essential Science Indicators from Clarivate Analytics and is featured on our Science Research Connect blog here. The paper currently has 80 citations in the Web of Science Core Collection.


In the essay below, author Prof. Dr. Abdon Atangana talks about this paper and its implications for its field along with possible wider applications.


The concept of fractional differential operators with non-singular kernel has captured minds of several researchers in the past year due to the operators’ wider applicability to almost all fields of science, engineering, and technology. These operators have opened new windows into the modeling of complex real-world problems that could not be modeled using the Newtonian and the well-known Riemann-Liouville fractional differential operators.  These operators are the way forward in modeling real-world problems in all disciplines, as they are able to incorporate the effect of memory, the long-range dependence, into the mathematical formulation.

In this paper, I developed all the properties of the newly established concept and provided possible multidisciplinary applications. To extend the limitations of the new concept, I provided new fractional differential operators called Atangana-Baleanu fractional derivatives. The Atangana-Baleanu fractional derivatives brought new weapons into applied mathematics to model complex real-world problems more accurately because the derivatives have the following properties:

  • The derivatives have at the same time Markovian and Non-Markovian properties, while the well-known Riemann-Liouville derivative is just Markovian and the Caputo-Fabrizio derivative is non-Markovian
  • The derivative waiting time is at the same time power law, stretched exponential and Brownian motion, while Riemann-Liouville derivative is only power law and Caputo-Fabrizio only exponential decay
  • The derivative mean square displacement is a crossover from usual diffusion to sub-diffusion, while Riemann-Liouville is just power law and scale-invariant. This means the Atangana-Baleanu fractional derivative is able to describe real-world problems with different scales, or problems that change their properties during time and space—for instance, the spread of cancer, the flow of water within heterogeneous aquifers, movement of pollution within fractured aquifers, and many others. This crossover behavior is observed in many empirical systems.
  • The derivative probability distribution is at the same time Gaussian and non-Gaussian, and can cross over from Gaussian to non-Gaussian without steady state. This means that the Atangana-Baleanu fractional derivative is at the same time deterministic and stochastic while the Riemann-Liouville is only deterministic. So for instance with this crossover the Atangana-Baleanu fractional derivative is able to describe physical or biological phenomena such as a heart attack, the physiological progression from life to death, structural failure in an airplane, and many other physical occurrences with sudden change with no steady state.

To a layman, most physical problems can be expressed in terms of mathematical formulations called differential equations; the differential equation’s aim is to analyze, understand, and predict the future of a physical problem. One of the most used differential operators was that developed in the 17th century by Isaac Newton and Gottfried Leibniz, but this failed to model complex real-world problems. The concept of nonlocal operators called fractional derivatives and integrals was suggested by Bernhard Riemann and Joseph Liouville with the aim to capture more complex phenomena, but also failed to model many important real-world problems as presented in sections 1 to 4 above. Thus, those with exponential and Mittag-Leffler kernels were suggested.

After Michele Caputo and Mauro Fabrizio suggested a fractional derivative with non-singular kernel, I suggested that the properties of the Caputo-Fabrizio fractional derivative and its relations with integral also transform its application to real-world problems to help humankind use it for their research. However, some issues were raised against this new derivative as its associated integral was an average of the given function and its classical integral. It was concluded that this derivative has a local kernel and was not fractional. In mid-2016, I suggested a new version, of which the integral was an average of a given function and the Riemann-Liouville fractional integral.

The big challenge I face as an African researcher is that most papers coming from Africa are unlikely to be accepted in top journals even though they may contain novel information.  In addition, research in Africa does not receive funding—we work hard to put Africa on the map but with little or no funding.  The future of modeling will be with fractional derivatives with non-local, non-singular kernel called Atangana-Baleanu derivatives, and without any doubt, this research is leading already, given its selection as a New Hot Paper. With this new concept being developed by a young black African man, my hope is that this will encourage all blacks in the world to know that they can also achieve.


Prof. Dr. Abdon Atangana
Professor of Applied Mathematics
University of Free State
Bloemfontein South Africa