Innovation in Engineering Computational Technology
Chapter
18 J.J. Trujillo
Department of Mathematical Analysis, Faculty of Mathematics, University of La Laguna, Tenerife, Spain Keywords: fractional models, sub-diffusion, anomalous diffusion, fractional derivative, super-diffusion, non-differentiable continuous functions, fractals, generalised Weierstrass functions.
This paper is dedicated to presenting some aspects of so-called fractional differential equations. In the first part we advance some of the reasons that justify the use of fractional differential equations as an emerging tool for modelling the dynamics of numerous anomalous phenomena, both in nature and in the theory of complex systems. The main reason behind the usefulness of fractional derivatives in the close link that exists between fractional models and the "Jump" stochastic models, such as the continuous time random walk (CTRW) method, or those of the multiple trapping model type, such as Levy stable distributions and the generalisation of the central limit theorem. Since the second half of the 20th century, the CTRW method has been practically the only tool available to describe sub or super-diffusive phenomena associated with various complex systems. On the other hand, we note that fractional operators allow us to take the memory and non-locality properties of many anomalous processes into account. In the second part of this paper we briefly present some fractional operators, along with their properties, which will allow us, from a very formal standpoint, to give examples of fractional models with boundary conditions. This should prove helpful in explaining why certain fractional derivative operators are well suited to modelling the dynamics of sub and/or super-diffusive phenomena.
Specifically, consider the well-known uni-dimensional problem associated with, for
example, the diffusion of heat in an isolated bar of unit length whose extremities are
kept at zero, with a diffusivity constant
We know the solution
with
Said solution
However, neither of them can model super-diffusive phenomena since both
exponentials stop behaving as a negative exponent when Lastly, we use some examples based on Weierstrass functions to demonstrate the possibilities fractional models offer for simulating the dynamics of anomalous processes, which classical ordinary models do not.
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