http://wgstrom.gsenergy.io/city-of-dreams.php The meaning of fractional calculus for physics has been hard to pin down.
Podlubny said it was akin to looking for shadows on the walls from focusing on the geometrical interpretation. That being said, Bruce J. West has written extensively on the subject. In his fantastic book, The Physics of Fractal Operators , he demonstrates the deep connection between fractional derivatives and fractal geometry.
He argues that when modeling chaotic thermodynamic systems, it is necessary to use fractal operators because the separation of time-scales of classical physics is no longer valid. In systems theory, fractional operators allow us to model the memory of control systems through formalisms such as that of the Volterra Series.
A lot of restrictions of classical calculus is pointed out by Blas M. If these two cases are not obvious, then one has to look towards a new paradigm and revisit the origins of classical one. One can see the same history behind the fractional order modeling and control over classical integer order. From book: Bandyopadhyay, Bijnan, and Shyam Kamal.
Springer International Publishing, Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What is the physical meaning of fractional calculus? Ask Question. Asked 6 years, 4 months ago. Active 1 year, 9 months ago. Viewed times. What is the applications of the fractional boundary value problem? Abel published the solution of a problem presented by Hyugens in The tautochrone problem.
Abel gave his solution in the form of an integral equation that is considered the first application of fractional calculus. The integral he worked with is. The first integral equation in history had been solved. Two facts may be observed: the regard for the sum of the orders, and that unlike in classical calculus, the derivative of a constant is not zero [ 8 , 22 ]. Liouville : In , Liouville made the first great study of fractional calculus. The first formula he obtained was the derivative of a function:.
A second definition was achieved by Liouville from the defined integral:. Liouville also tackled the tautochrone problem and proposed differential equations of arbitrary order.
Liouville expanded the coefficients in Eq. And inserted those equations in Eq. Riemann : Riemann developed his Fractional Calculus theory when he was preparing his Ph. Thus, he generalizes this definition to non-integer powers and demands that. Riemann then derived Eq. Sonin and Letnikov : The Russian mathematicians N.
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Sonin and A. Letnikov — [ 29 ] made contributions taking as basis the formula for the n th derivative of the Cauchy integral formula given by. They worked using the contour integral method, with the contribution of Laurent , they achieved the definition:. Assigning c values in Eq.
This extension of the classical derivative to fractional order is important because it lets us apply it in numerical approximations. They started with the definition of derivative as a limit given by Cauchy :. In the twentieth and twenty-first centuries, more definitions will rise, but they will be given in terms within the Riemann-Liouville fractional integral and will be part of the Modern Fractional Calculus Theory, in all their fundamental definitions [ 22 ]. We will now present the assorted definitions and notations of fractional derivatives that will be used throughout this work.
It is worth pointing out that this is necessary because such notation is currently standardized [ 18 , 19 ]. As shown in Refs. It is also possible to prove that the semigroup propriety about the order of integral operators i. On the other hand, if the order of the operators is inverted, it will have. All these properties can be used in the phenomena modeling and its solution; such models have shown to improve usual approaches.
However, when using equations with Riemann-Liouville type fractional derivatives, the initial conditions cannot be interpreted physically; a clear example is that the derivative Riemann-Liouville of a constant is not zero, contrary to the impression that the derivatives gives a notion about the change that the function experiences when advancing in the time or to modify its position.
This was the motivation for another definition that is better coupled with physical interpretations; this is the derivative of Caputo type.
Michele Caputo [ 11 ] published a book in which he introduced a new derivative, which had been independently discovered by Gerasimov This derivative is quite important, because it allows for understanding initial conditions, and is used to model fractional time. In some texts, it is known as the Gerasimov-Caputo derivative. The left and right Caputo derivatives are defined as.
In his early articles and several after that, Caputo used a Laplace transformed of the Caputo fractional derivative, which is given by. The phenomenon of anomalous diffusion is mathematically modeled by a fractional partial differential equation. The parameters of this equation are uniquely determined by the fractal dimension of the underlying object.
There are some results that show the relationship between fractals and fractional operators [ 24 ]; two of the most important that motivated the particular study of the equations to determine the pressure deficit in oil wells are highlighted below. In , Nigmatullin [ 12 ] presents one of the most distinguished contributions to the search of the concrete relationship between the fractal dimension of a porous medium and the order of the fractional derivative to model a phenomena through such a medium; in this, he achieves the evolution of a physical system of a Cantor set type.
In his research, Nigmatullin proposes a relationship between the fractal dimension of a Cantor type set and the order of a fractional integral of the Riemann-Liouville type. Where the distribution to apply see Refs. The initial results were strongly questioned by different authors, including Roman Rutman see Refs. In this section, the Equation Continuity which follows from the law of conservation of mass is established.
We obtain a system formed by three partial differential equations, one for each fluid. This multiphase system must be solved considering the relevant boundary and initial conditions [ 30 ].
In the particular case of naturally fractured reservoirs see Refs. General fluid transfer equation results combining the formulas in Eq.
This differential equation contains two dependent variables, namely the humidity content and fluid pressure, but they are related. For this reason, the saturation S p is defined so that. The porous media is considered to be formed by three porous media: the matrix, fractured media, and vuggy media. The total volume of the porous media V T is equal to the sum of the total volume of the matrix V m , of the total volume of the fractured medium V F and of the total volume of the vuggy media V G.
In other words.
The porous medium as everything contains solids and voids, with the following relations:. It follows that.
The relation between the total volumetric flow of the fluid per unit area in the porous medium q , the volumetric flow per unit area in the matrix q M , the volumetric flow per unit area in the fractured medium q F , and the volumetric flow per unit area in the vuggy media q G is analogous to Eq. The continuity equations for the matrix, the fractured medium, and the vuggy media considering Eq. The equation of continuity of the porous medium, Eq.
The contributions of fluid in each porous medium are modeled with the following relations:. In the case of the monophasic flow saturated in triadic means, the continuity equations in each porous medium can be written as follows:. The porosity of each medium has been defined as the volume of the space occupied by the medium. However, the porosity can be defined as the volume of empty space in each medium with respect to the volume of the total space occupied by the porous medium as a whole. These new porosities will be denoted with subscripts in lowercase letters and clearly have. The nest system by Eqs.
The substitution of Eqs. In polar coordinates, the system reduces to. Now we will give a process of dimensionlessness to better manage the variables.