Operators of Fractional Calculus and Their Multidisciplinary Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (30 September 2022) | Viewed by 46213

Special Issue Editor


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Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
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Special Issue Information

Dear Colleagues,

Current widespread interest in various families of fractional-order integral and derivative operators, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdélyi–Kober, Liouville–Caputo, and so on, have stemmed essentially from their demonstrated applications in numerous diverse areas of the mathematical, physical, chemical, engineering, and statistical sciences. These fractional-order operators provide interesting and potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations, the fractional-calculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, applicable analysis and applied mathematics, as well as their extensions and generalizations in one, two and more variables.

In this Special Issue, we invite and welcome review, expository, and original research articles dealing with recent advances in the theory of integrals and derivatives of fractional order and their multidisciplinary applications.

Prof. Dr. Hari Mohan Srivastava
Guest Editor

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Keywords

  • operators of fractional integrals and fractional derivatives and their applications
  • chaos and dynamical systems based upon fractional calculus
  • fractional-order ODEs and PDEs
  • fractional-order differintegral and integro-differential equations
  • integrals and derivatives of fractional order associated with special functions of mathematical physics and applied mathematics
  • identities and inequalities involving fractional-order integrals and fractional-order derivatives

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Published Papers (20 papers)

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Editorial

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4 pages, 218 KiB  
Editorial
Editorial for the Special Issue “Operators of Fractional Calculus and Their Multidisciplinary Applications”
by Hari Mohan Srivastava
Fractal Fract. 2023, 7(5), 415; https://doi.org/10.3390/fractalfract7050415 - 22 May 2023
Cited by 3 | Viewed by 1341
Abstract
This Special Issue of the MDPI journal, Fractal and Fractional, on the subject area of “Operators of Fractional Calculus and Their Multidisciplinary Applications” consists of 19 peer-reviewed papers, including some invited feature articles, originating from all over the world [...] Full article

Research

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15 pages, 7305 KiB  
Article
New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term
by Mohammed Alabedalhadi, Mohammed Al-Smadi, Shrideh Al-Omari, Yeliz Karaca and Shaher Momani
Fractal Fract. 2022, 6(12), 724; https://doi.org/10.3390/fractalfract6120724 - 8 Dec 2022
Cited by 7 | Viewed by 1762
Abstract
In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system [...] Read more.
In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well. Full article
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16 pages, 694 KiB  
Article
A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy
by Mohammad Momenzadeh, Olivia Ada Obi and Evren Hincal
Fractal Fract. 2022, 6(6), 287; https://doi.org/10.3390/fractalfract6060287 - 26 May 2022
Cited by 2 | Viewed by 1601
Abstract
Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with [...] Read more.
Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with cancer. Besides, fractional differentiation remedies the situation by obtaining a meticulous model with more flexible parameters. Furthermore, a specific type of non-Newtonian calculus, bi-geometric calculus, can describe the model in terms of proportions and implies the alternative aspect of a dynamic system. Moreover, fractional operators in bi-geometric calculus are formulated in terms of Hadamard fractional operators. In this article, the development of fractional operators in non-Newtonian calculus was investigated. The model was extended in these criteria, and the existence and uniqueness of the model were considered and guaranteed in the first step by applying the Arzelà–Ascoli. The bi-geometric analogue of the numerical method provided a suitable tool to solve the model approximately. In the end, the visual graphs were obtained by using the MATLAB program. Full article
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18 pages, 319 KiB  
Article
Uniform Stability of a Class of Fractional-Order Fuzzy Complex-Valued Neural Networks in Infinite Dimensions
by Xin Liu, Lili Chen and Yanfeng Zhao
Fractal Fract. 2022, 6(5), 281; https://doi.org/10.3390/fractalfract6050281 - 23 May 2022
Cited by 2 | Viewed by 1769
Abstract
In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the [...] Read more.
In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the uniqueness of the solution of the above complex-valued neural networks is derived. Subsequently, the criteria for uniform stability of the above complex-valued neural networks are established. In comparison with related results, we do not need to construct a complex Lyapunov function, reducing the computational complexity. Finally, an example is given to show the validity of the main results. Full article
14 pages, 312 KiB  
Article
Some New Inequalities on Laplace–Stieltjes Transforms Involving Logarithmic Growth
by Hongyan Xu, Hong Li and Zuxing Xuan
Fractal Fract. 2022, 6(5), 233; https://doi.org/10.3390/fractalfract6050233 - 22 Apr 2022
Cited by 6 | Viewed by 1392
Abstract
This article is devoted to exploring the properties on the logarithmic growth of entire functions represented by Laplace–Stieltjes transforms of zero order. In order to describe the growth of Laplace–Stieltjes transforms more finely, we introduce some concepts of the logarithmic indexes of the [...] Read more.
This article is devoted to exploring the properties on the logarithmic growth of entire functions represented by Laplace–Stieltjes transforms of zero order. In order to describe the growth of Laplace–Stieltjes transforms more finely, we introduce some concepts of the logarithmic indexes of the maximum term and the center index of the maximum term of Laplace–Stieltjes transforms, and establish some new inequalities focusing on the above logarithmic indexes, the logarithmic order, the (lower) logarithmic type and the coefficients of Laplace–Stieltjes transforms. Moreover, we obtain two estimation forms on the (lower) logarithmic type of entire functions represented by Laplace–Stieltjes transform by applying these inequalities. One estimation is mainly by the center indexes of the maximum term, the other is by the logarithmic order, exponent and coefficients. Finally, we obtain the equivalence condition of entire functions with the perfectly logarithmic linear growth. This result shows that the two estimation forms can be equivalent to some extent. Full article
21 pages, 4259 KiB  
Article
Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA
by Baokun Wang, Shaohua Wang, Yibing Peng, Youguo Pi and Ying Luo
Fractal Fract. 2022, 6(4), 218; https://doi.org/10.3390/fractalfract6040218 - 12 Apr 2022
Cited by 13 | Viewed by 2271
Abstract
In this paper, the design of a fractional-order proportional integral (FOPI) controller and integer-order (IOPI) controller are compared for the permanent magnet synchronous motor (PMSM) speed regulation system. A high-precision implementation method of a fractional-order proportional integral (FOPI) controller is proposed in this [...] Read more.
In this paper, the design of a fractional-order proportional integral (FOPI) controller and integer-order (IOPI) controller are compared for the permanent magnet synchronous motor (PMSM) speed regulation system. A high-precision implementation method of a fractional-order proportional integral (FOPI) controller is proposed in this work. Three commonly used numerical implementation methods of fractional operators are investigated and compared for comprehensively evaluating the numerical implementation performance in this work. Furthermore, for the impulse response invariant implementation method, the effects of different discretization orders on the control performance of the system are compared. The high-order fractional-order controller can be implemented accurately in a control system with the field-programmable gate array (FPGA) with the capability of parallel calculation. The simulation and experimental results show that the high-precision numerical implementation method of the designed high-order FOPI controller has better performance than the ordinary precision fractional operation implementation method and traditional order integer order PI controller. Full article
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17 pages, 1394 KiB  
Article
Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels
by Manal Alqhtani and Khaled M. Saad
Fractal Fract. 2022, 6(1), 13; https://doi.org/10.3390/fractalfract6010013 - 29 Dec 2021
Cited by 23 | Viewed by 1821
Abstract
In this paper, three new models of fractal–fractional Michaelis–Menten enzymatic reaction (FFMMER) are studied. We present these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels. We construct three schema of successive approximations according to the theory of [...] Read more.
In this paper, three new models of fractal–fractional Michaelis–Menten enzymatic reaction (FFMMER) are studied. We present these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels. We construct three schema of successive approximations according to the theory of fractional calculus and with the help of Lagrange polynomials. The approximate solutions are compared with the resulting numerical solutions using the finite difference method (FDM). Because the approximate solutions in the classical case of the three models are very close to each other and almost matches, it is sufficient to compare one model, and the results were good. We investigate the effects of the fractal order and fractional order for all models. All calculations were performed using Mathematica software. Full article
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11 pages, 289 KiB  
Article
Further Integral Inequalities through Some Generalized Fractional Integral Operators
by Abd-Allah Hyder, Mohamed A. Barakat, Ashraf Fathallah and Clemente Cesarano
Fractal Fract. 2021, 5(4), 282; https://doi.org/10.3390/fractalfract5040282 - 20 Dec 2021
Cited by 12 | Viewed by 2141
Abstract
In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the [...] Read more.
In this article, we utilize recent generalized fractional operators to establish some fractional inequalities in Hermite–Hadamard and Minkowski settings. It is obvious that many previously published inequalities can be derived as particular cases from our outcomes. Moreover, we articulate some flaws in the proofs of recently affiliated formulas by revealing the weak points and introducing more rigorous proofs amending and expanding the results. Full article
13 pages, 583 KiB  
Article
Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel
by Iván Area and Juan J. Nieto
Fractal Fract. 2021, 5(4), 273; https://doi.org/10.3390/fractalfract5040273 - 14 Dec 2021
Cited by 16 | Viewed by 2897
Abstract
In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated [...] Read more.
In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series. Full article
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9 pages, 277 KiB  
Article
Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations
by Osama Moaaz, Clemente Cesarano and Sameh Askar
Fractal Fract. 2021, 5(4), 259; https://doi.org/10.3390/fractalfract5040259 - 6 Dec 2021
Cited by 2 | Viewed by 1755
Abstract
In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the [...] Read more.
In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type. Full article
18 pages, 446 KiB  
Article
Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods
by A. Torres-Hernandez and F. Brambila-Paz
Fractal Fract. 2021, 5(4), 240; https://doi.org/10.3390/fractalfract5040240 - 23 Nov 2021
Cited by 7 | Viewed by 6371
Abstract
Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a [...] Read more.
Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a simple and compact method to work the fractional calculus through the classification of fractional operators using sets. This new method of working with fractional operators, which may be called fractional calculus of sets, allows generalizing objects of conventional calculus, such as tensor operators, the Taylor series of a vector-valued function, and the fixed-point method, in several variables, which allows generating the method known as the fractional fixed-point method. Furthermore, it is also shown that each fractional fixed-point method that generates a convergent sequence has the ability to generate an uncountable family of fractional fixed-point methods that generate convergent sequences. So, it is presented a method to estimate numerically in a region Ω the mean order of convergence of any fractional fixed-point method, and it is shown how to construct a hybrid fractional iterative method to determine the critical points of a scalar function. Finally, considering that the proposed method to classify fractional operators through sets allows generalizing the existing results of the fractional calculus, some examples are shown of how to define families of fractional operators that satisfy some property to ensure the validity of the results to be generalized. Full article
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15 pages, 349 KiB  
Article
The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions
by Hari M. Srivastava, Eman S. A. AbuJarad, Fahd Jarad, Gautam Srivastava and Mohammed H. A. AbuJarad
Fractal Fract. 2021, 5(4), 210; https://doi.org/10.3390/fractalfract5040210 - 11 Nov 2021
Cited by 7 | Viewed by 2006
Abstract
The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the [...] Read more.
The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the Hadamard product of the (p,q)-extended Gauss hypergeometric function Fp,q and the Fox-Wright function rΨs(z). Some special cases of our main results are considered. Furthermore, the (p,q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p,q)-extended Bessel-Wright function is established. Full article
18 pages, 1158 KiB  
Article
A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations
by Saima Mustafa, Hajira, Hassan Khan, Rasool Shah and Saadia Masood
Fractal Fract. 2021, 5(4), 206; https://doi.org/10.3390/fractalfract5040206 - 11 Nov 2021
Cited by 7 | Viewed by 1919
Abstract
In the present note, a new modification of the Adomian decomposition method is developed for the solution of fractional-order diffusion-wave equations with initial and boundary value Problems. The derivatives are described in the Caputo sense. The generalized formulation of the present technique is [...] Read more.
In the present note, a new modification of the Adomian decomposition method is developed for the solution of fractional-order diffusion-wave equations with initial and boundary value Problems. The derivatives are described in the Caputo sense. The generalized formulation of the present technique is discussed to provide an easy way of understanding. In this context, some numerical examples of fractional-order diffusion-wave equations are solved by the suggested technique. It is investigated that the solution of fractional-order diffusion-wave equations can easily be handled by using the present technique. Moreover, a graphical representation was made for the solution of three illustrative examples. The solution-graphs are presented for integer and fractional order problems. It was found that the derived and exact results are in good agreement of integer-order problems. The convergence of fractional-order solution is the focus point of the present research work. The discussed technique is considered to be the best tool for the solution of fractional-order initial-boundary value problems in science and engineering. Full article
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19 pages, 877 KiB  
Article
(k, ψ)-Proportional Fractional Integral Pólya–Szegö- and Grüss-Type Inequalities
by Tariq A. Aljaaidi, Deepak B. Pachpatte, Mohammed S. Abdo, Thongchai Botmart, Hijaz Ahmad, Mohammed A. Almalahi and Saleh S. Redhwan
Fractal Fract. 2021, 5(4), 172; https://doi.org/10.3390/fractalfract5040172 - 18 Oct 2021
Cited by 10 | Viewed by 1701
Abstract
The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators [...] Read more.
The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators with respect to another strictly increasing continuous function ψ. Then, we state and prove some of its properties and special cases. With the help of this generalized operator, we investigate some Pólya–Szegö- and Grüss-type fractional integral inequalities. The functions used in this work are bounded by two positive functions to obtain Pólya–Szegö- and Grüss-type k-fractional integral inequalities in a new sense. Moreover, we discuss some new special cases of the Pólya–Szegö- and Grüss-type inequalities through this work. Full article
21 pages, 376 KiB  
Article
Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators
by Hari Mohan Srivastava, Artion Kashuri, Pshtiwan Othman Mohammed and Kamsing Nonlaopon
Fractal Fract. 2021, 5(4), 160; https://doi.org/10.3390/fractalfract5040160 - 9 Oct 2021
Cited by 18 | Viewed by 1671
Abstract
In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for [...] Read more.
In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results. Full article
26 pages, 391 KiB  
Article
New Estimations of Hermite–Hadamard Type Integral Inequalities for Special Functions
by Hijaz Ahmad, Muhammad Tariq, Soubhagya Kumar Sahoo, Jamel Baili and Clemente Cesarano
Fractal Fract. 2021, 5(4), 144; https://doi.org/10.3390/fractalfract5040144 - 29 Sep 2021
Cited by 22 | Viewed by 1452
Abstract
In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the [...] Read more.
In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the novel version of Hermite–Hadamard inequality. Furthermore, to improve our results, we explore two new equalities, and employing these we present some refinements of the Hermite–Hadamard-type inequality. A few remarkable cases are discussed, which can be seen as valuable applications. Applications of some of our presented results to special means are given as well. An endeavor is made to introduce an almost thorough rundown of references concerning the Mittag–Leffler functions and the Raina functions to make the readers acquainted with the current pattern of emerging research in various fields including Mittag–Leffler and Raina type functions. Results established in this paper can be viewed as a significant improvement of previously known results. Full article
13 pages, 289 KiB  
Article
Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces
by Chenkuan Li and Hari M. Srivastava
Fractal Fract. 2021, 5(3), 105; https://doi.org/10.3390/fractalfract5030105 - 31 Aug 2021
Cited by 7 | Viewed by 1709
Abstract
This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some [...] Read more.
This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems. Full article
18 pages, 302 KiB  
Article
On the Nonlinear Integro-Differential Equations
by Chenkuan Li and Joshua Beaudin
Fractal Fract. 2021, 5(3), 82; https://doi.org/10.3390/fractalfract5030082 - 30 Jul 2021
Cited by 5 | Viewed by 1642
Abstract
The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s [...] Read more.
The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function. Full article
17 pages, 355 KiB  
Article
On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus
by Gul Sana, Pshtiwan Othman Mohammed, Dong Yun Shin, Muhmmad Aslam Noor and Mohammad Salem Oudat
Fractal Fract. 2021, 5(3), 60; https://doi.org/10.3390/fractalfract5030060 - 25 Jun 2021
Cited by 15 | Viewed by 2568
Abstract
Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q [...] Read more.
Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods. Full article

Other

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33 pages, 1044 KiB  
Perspective
The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus
by Bruce J. West
Fractal Fract. 2022, 6(4), 225; https://doi.org/10.3390/fractalfract6040225 - 15 Apr 2022
Cited by 2 | Viewed by 2412
Abstract
This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively [...] Read more.
This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants. Full article
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