Special Issue "Modelling Problems Arising in Science and Engineering with Fractional Differential Operators: Beyond the Power-Law Limit"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2019).

Special Issue Editor

Prof. Dr. Jordan Hristov
Website1 Website2 Website3
Guest Editor
Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria
Interests: nonlinear transport phenomena; modeling; scaling; fractional calculus; heat and mass transfer; diffusion problems
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

You are kindly invited to contribute to the Special Issue “Modelling Problems Arising in Science and Engineering with Fractional Differential Operators: Beyond the Power-Law Limit”.

Fractional calculus has a brilliant history in the modelling of non-linear and anomalous problems in mathematics, physics, statistics and engineering, involving a variety of fractional-order integral and derivative operators, such as the ones named after Grunwald-Letnikov, Riemann-Liouville, Weyl, Caputo, Hadamard, Riesz, Erdelyi-Kober, etc., based on the power-law memory. Beyond this bright classical basis, in recent years new trends in fractional modelling involving operators with non-singular kernels have been created to model dissipative phenomena that cannot be adequately modelled by fractional differential operators based on singular kernels.

This Special Issue addresses contemporary modeling problems in science and engineering involving fractional differential operators with classical and new memory kernels. This is a call to authors involved in modeling with new and classical fractional differential operators to show their important positions in fractional modelling theory, differences in applications and how these operators should be applied. The issue offers a broad range of applied topics and multidisciplinary applications of fractional order differential operators with classical and new kernels in science and engineering.

We invite and welcome review, expository, and original research articles dealing with recent advances in the theory of fractional-order integral and derivative operators and their multidisciplinary applications. We will be glad to see your contributions with strong results demonstrating the feasibility of both the classical and the new trends in fractional calculus.

The main topics of the collections envisage some principle problems, including but not limited to:

  • Fractional modelling: new trends, new fractional operators, mathematical properties of fractional operators,
  • Memory kernels to fractional operators: identification, construction, definitions of fractional operators on their basis and relevant properties
  • Fractional-order ODEs, PDEs and integro-differential equations involving new fractional operators
  • Special functions of mathematical physics and applied mathematics associated with the new fractional operators
  • Examples beyond the classical singular kernel applications: Non-power-law relaxations involving new operators
  • Fractional modelling of the mechanics and rheology of solid materials with non-power-law relaxations
  • Anomalous diffusion models beyond the power-law behaviour
  • Fractional modelling for biomechanical and biomedical applications with new operators
  • Thermodynamic compatibility of fractional models with new kernels
  • Fractional models of heat, mass and fluid flow beyond the power-law
  • Control and signal fractional modelling problems with non-singular fractional operators
  • Dynamic and stochastic systems based on fractional calculus with non-power-law kernels
  • Fractional modelling of electrochemical and magnetic systems with non-singular operators

Prof. Dr. Jordan Hristov
Guest Editor

Manuscript Submission Information

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Keywords

  • Fractional modelling with non-power-law kernels
  • Fractional operator definitions and properties
  • Fractional-order ODEs, PDEs and integro-differential equations
  • Non-power-law relaxations involving new operators
  • Rheological modeling of fluid and solids with non-singular kernels
  • Anomalous diffusion transport beyond the power-law
  • Biomechanical and medical models with non-singular fractional operators
  • Thermodynamical model consistency when non-singular operators are involved
  • Control and identification with new fractional operators
  • Chaos and complexity with non-singular fractional operators

Published Papers (7 papers)

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Research

Open AccessArticle
Effective Method for Solving Different Types of Nonlinear Fractional Burgers’ Equations
Mathematics 2020, 8(5), 729; https://doi.org/10.3390/math8050729 - 06 May 2020
Abstract
In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to [...] Read more.
In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods. Full article
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Open AccessArticle
Certain Results Comprising the Weighted Chebyshev Function Using Pathway Fractional Integrals
Mathematics 2019, 7(10), 896; https://doi.org/10.3390/math7100896 - 25 Sep 2019
Abstract
An analogous version of Chebyshev inequality, associated with the weighted function, has been established using the pathway fractional integral operators. The result is a generalization of the Chebyshev inequality in fractional integral operators. We deduce the left sided Riemann Liouville version and the [...] Read more.
An analogous version of Chebyshev inequality, associated with the weighted function, has been established using the pathway fractional integral operators. The result is a generalization of the Chebyshev inequality in fractional integral operators. We deduce the left sided Riemann Liouville version and the Laplace version of the same identity. Our main deduction will provide noted results for an appropriate change to the Pathway fractional integral parameter and the degree of the fractional operator. Full article
Open AccessFeature PaperArticle
On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels
Mathematics 2019, 7(9), 772; https://doi.org/10.3390/math7090772 - 22 Aug 2019
Cited by 4
Abstract
We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag–Leffler kernels. The iteration process depends on [...] Read more.
We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag–Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property, and hence, the new introduced iterated fractional difference-sum operators have this semigroup property as well. Full article
Open AccessArticle
Mittag–Leffler Memory Kernel in Lévy Flights
Mathematics 2019, 7(9), 766; https://doi.org/10.3390/math7090766 - 21 Aug 2019
Cited by 2
Abstract
In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis [...] Read more.
In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems. Full article
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Open AccessArticle
On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation
Mathematics 2019, 7(8), 722; https://doi.org/10.3390/math7080722 - 08 Aug 2019
Cited by 7
Abstract
The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform [...] Read more.
The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering. Full article
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Open AccessArticle
Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay
Mathematics 2019, 7(6), 532; https://doi.org/10.3390/math7060532 - 11 Jun 2019
Cited by 5
Abstract
In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing [...] Read more.
In the present article, fractional-order partial differential equations with proportional delay, including generalized Burger equations with proportional delay are solved by using Natural transform decomposition method. Natural transform decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. Therefore, Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear Partial deferential equations particularly fractional-order partial differential equations with proportional delay. Full article
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Open AccessArticle
New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator
Mathematics 2019, 7(4), 374; https://doi.org/10.3390/math7040374 - 24 Apr 2019
Cited by 14
Abstract
In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t [...] Read more.
In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration. Full article
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