scholarly journals The mathematical description of the bulk fluid flow and that of the content impurity dispersion, obtained by replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective

2021 ◽  
Vol 13 (4) ◽  
pp. 3-16
Author(s):  
Agneta M. BALINT ◽  
Stefan BALINT
Keyword(s):  
Fluid Flow ◽  
Time Evolution ◽  
Fractional Order ◽  
Mathematical Description ◽  
Unconfined Aquifer ◽  
Bulk Fluid ◽  
Integer Order ◽  
Measured Time ◽  
2D Flow ◽  
Impurity Dispersion

In the field of fractional calculus applications, there is a tendency to admit that “integer-order derivatives cannot simply be replaced by fractional-order derivatives to develop fractional-order theories”. There are different arguments for that: initialization problem, inconsistency, use of nonsingular or singular kernels, loss of objectivity. In this paper it is shown that the mathematical description of the bulk fluid flow and that of the content impurity spread replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. More precisely, it is proven that, the mathematical description of the bulk fluid 2D flow and that of the content impurity spread, in a horizontal unconfined aquifer, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. It is also proven that, the mathematical description of a Newtonian, incompressible, viscous bulk fluid 3D flow and that of the contained impurity dispersion, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. The obtained results show the compatibility of the general temporal Caputo and general temporal Riemann-Liouville fractional order derivatives with the understanding of the “measured time” evolution. In the same time these results reveal that, the objectivity violation, when integer order temporal derivatives are replaced by classic temporal Caputo or classic temporal Riemann-Liouville fractional order derivatives, is originated in the incompatibility of the classic fractional order derivatives, with the understanding of the “measured time” evolution.

scholarly journals Mathematical description of the bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order partial derivatives is nonobjective

2020 ◽  
Vol 12 (3) ◽  
pp. 17-31
Author(s):  
Agneta M. BALINT ◽  
Stefan BALINT
Keyword(s):  
Fluid Flow ◽  
Integral Representation ◽  
Fractional Order ◽  
Mathematical Description ◽  
Reference Frames ◽  
Finite Interval ◽  
Bulk Fluid ◽  
Fractional Order Derivative ◽  
Partial Derivatives ◽  
Impurity Dispersion

In this paper it is shown that the mathematical description of a Newtonian, incompressible, viscous bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order derivative, having integral representation on finite interval, is nonobjective. This means that, two different observers describing the flow or the contained impurity dispersion with these tools obtain two different results which cannot be reconciled i.e. transformed into each other using only formulas that link the coordinates of a point in two fixed orthogonal reference frames and formulas that link the numbers representing a moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which of the obtained results is correct?

scholarly journals Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective

2020 ◽  
Vol 4 (3) ◽  
pp. 36 ◽  
Author(s):  
Agneta M. Balint ◽  
Stefan Balint
Keyword(s):  
Fluid Flow ◽  
Integral Representation ◽  
Groundwater Flow ◽  
Fractional Order ◽  
Mathematical Description ◽  
Finite Interval ◽  
Unconfined Aquifer ◽  
Time Measurement ◽  
Bulk Fluid ◽  
Partial Derivatives

In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct.

Mathematical Description of Elastic Phenomena which Uses Caputo or Riemann-Liouville Fractional Order Partial Derivatives is Nonobjective

2020 ◽  
Author(s):  
Agneta M. Balint ◽  
Stefan Balint ◽  
Silviu Birauas
Keyword(s):  
Integral Representation ◽  
Fractional Order ◽  
Mathematical Description ◽  
Reference Frames ◽  
Finite Interval ◽  
Constitutive Law ◽  
Partial Derivatives ◽  
Integer Order ◽  
Direct Replacement

In this paper it is shown that mathematical description of strain, constitutive law and dynamics obtained by direct replacement of integer order derivatives with Caputo or Riemann-Liouville fractional order partial derivatives, having integral representation on finite interval, in case of a guitar string, is nonobjective. The basic idea is that different observers, using this type of descriptions, obtain different results which cannot be reconciled, i.e. transformed into each other using only formulas that link the coordinates of the same point in two fixed orthogonal reference frames and formulas that link the numbers representing the same moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which one of the obtained results is correct?

scholarly journals Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1544
Author(s):  
Chunpeng Wang ◽  
Hongling Gao ◽  
Meihong Yang ◽  
Jian Li ◽  
Bin Ma ◽  
...  
Keyword(s):  
Image Reconstruction ◽  
Fractional Order ◽  
Continuous Functions ◽  
Kernel Functions ◽  
Superior Performance ◽  
Image Description ◽  
Orthogonal Moments ◽  
Integer Order ◽  
Geometric Invariance ◽  
Order Continuous

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

Exploiting Fractional Order PID Controller Methods in Improving the Performance of Integer Order PID Controllers: A GA Based Approach

2009 ◽  
Author(s):  
Bijoy K. Mukherjee ◽  
Santanu Metia ◽  
Sio-Iong Ao ◽  
Alan Hoi-Shou Chan ◽  
Hideki Katagiri ◽  
...  
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
Pid Controllers ◽  
Integer Order ◽  
Fractional Order Pid ◽  
Fractional Order Pid Controller

Capillary Equilibrium in Porous Materials

1965 ◽  
Vol 5 (01) ◽  
pp. 15-24 ◽  
Author(s):  
Norman R. Morrow ◽  
Colin C. Harris
Keyword(s):  
Mass Transfer ◽  
Fluid Flow ◽  
Porous Materials ◽  
Capillary Pressure ◽  
Chemical Engineering ◽  
Fluid Distribution ◽  
Bulk Fluid ◽  
Fluid Saturation ◽  
Capillary Pressure Curves ◽  
The Relationship

Abstract The experimental points which describe capillary pressure curves are determined at apparent equilibria which are observed after hydrodynamic flow has ceased. For most systems, the time required to obtain equalization of pressure throughout the discontinuous part of a phase is prohibitive. To permit experimental points to be described as equilibria, a model of capillary behavior is proposed where mass transfer is restricted to bulk fluid flow. Model capillary pressure curves follow if the path described by such points is independent of the rate at which the saturation was changed to attain a capillary pressure point. A modified suction potential technique is used to study cyclic relationships between capillary pressure and moisture content for a porous mass. The time taken to complete an experiment was greatly reduced by using small samples. Introduction Capillary retention of liquid by porous materials has been investigated in the fields of hydrology, soil science, oil reservoir engineering, chemical engineering, soil mechanics, textiles, paper making and building materials. In studies of the immiscible displacement of one fluid by another within a porous bed, drainage columns and suction potential techniques have been used to obtain relationships between pressure deficiency and saturation (Fig. 1). Except where there is no hysteresis of contact angle and the solid is of simple geometry, such as a tube of uniform cross section, there is hysteresis in the relationship between capillary pressure and saturation. The relationship which has received most attention is displacement of fluid from an initially saturated bed (Fig. 1, Curve Ro), the final condition being an irreducible minimum fluid saturation Swr. Imbibition (Fig. 1, Curve A), further desaturation (Fig. 1, Curve R), and intermediate scanning curves have been studied to a lesser but increasing extent. This paper first considers the nature of the experimental points tracing the capillary pressure curves with respect to the modes and rates of mass transfer which are operative during the course of measurement. There are clear indications that the experimental points which describe these curves are obtained at apparent equilibria which are observed when viscous fluid flow has ceased; and any further changes in the fluid distribution are the result of much slower mass transfer processes, such as diffusion. Unless stated otherwise, this discussion applies to a stable packing of equal, smooth, hydrophilic spheres supported by a suction plate with water as the wetting phase and air as the nonwetting phase. SPEJ P. 15ˆ

The multi-model approach for fractional-order systems modelling

2016 ◽  
Vol 40 (1) ◽  
pp. 331-340 ◽  
Author(s):  
Samia Talmoudi ◽  
Moufida Lahmari
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Global Model ◽  
Fractional Order Systems ◽  
New Approach ◽  
Physical Systems ◽  
Integer Order ◽  
Systems Modelling ◽  
Model Approach

Currently, fractional-order systems are attracting the attention of many researchers because they present a better representation of many physical systems in several areas, compared with integer-order models. This article contains two main contributions. In the first one, we suggest a new approach to fractional-order systems modelling. This model is represented by an explicit transfer function based on the multi-model approach. In the second contribution, a new method of computation of the validity of library models, according to the frequency [Formula: see text], is exposed. Finally, a global model is obtained by fusion of library models weighted by their respective validities. Illustrative examples are presented to show the advantages and the quality of the proposed strategy.

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