Fractional calculus
In this context, the term powers refers to iterative application of a linear operator
Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695.
[citation needed] Fractional calculus was introduced in one of Niels Henrik Abel's early papers[3] where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and the unified notation for differentiation and integration of arbitrary real order.
[5][6][7] Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.
Interchanging α and β shows that the order in which the J operator is applied is irrelevant and completes the proof.
Unfortunately, the comparable process for the derivative operator D is significantly more complex, but it can be shown that D is neither commutative nor additive in general.
which has the advantage that is zero when f(t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative.
where ϕ(ν) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.
In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function
[18] In 2016, Atangana and Baleanu suggested differential operators based on the generalized Mittag-Leffler function
is the cumulative distribution function of a probability measure on the positive real numbers.
[25] Classical fractional derivatives include: New fractional derivatives include: The Coimbra derivative is used for physical modeling:[34] A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,[35][36][37][38][39][40][41] as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors[42][43][44][45] where the lower limit
Note that the second (summation) term on the right side of the definition above can be expressed as so to keep the denominator on the positive branch of the Gamma (
As described by Wheatcraft and Meerschaert (2008),[51] a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear.
Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form):
whose solution C(x,s) contains a one-half power dependence on s. Taking the derivative of C(x,s) and then the inverse Laplace transform yields the following relationship:
[53] In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.
[54][55] In these works, the classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head.
This equation[clarification needed] has been shown useful for modeling contaminant flow in heterogenous porous media.
In their work, the hydrodynamic dispersion equation was generalized using the concept of a variational order derivative.
[11] Generalizing PID controllers to use fractional orders can increase their degree of freedom.
The new equation relating the control variable u(t) in terms of a measured error value e(t) can be written as
[64] The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law.
This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media.
This link is further described in Näsholm & Holm (2011b)[66] and in the survey paper,[67] as well as the Acoustic attenuation article.
See Holm & Nasholm (2013)[68] for a paper which compares fractional wave equations which model power-law attenuation.
[69] Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.
[71] Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.
where the solution of the equation is the wavefunction ψ(r, t) – the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t, and ħ is the reduced Planck constant.
