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Axioms
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Fractional Calculus, Wavelets and Fractals
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Fractional Calculus, Wavelets and Fractals

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 August 2020) | Viewed by 22355

Special Issue Editor

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Modeling, simulation, and applications of nonlinear and nonregular mathematical methods such as fractional calculus, fractals, nonlinear equations, and wavelets have recently become increasingly popular and important research subjects, playing a fundamental role in the more comprehensive description of natural phoenomena as well as in engineering and the applied sciences.

As a consequence there has been an impressive growing interest with respect to applications in mathematics, physics, engineering, economics, biology, medicine, geography, and linguistics, opening new perspectives in the formulation of axiomatic models in applied fields and developing some new and challenging fields of research. 

This Special Issue is focused on the most recent advances in axiomatic models for theoretical and applied sciences arising in all fields of science, engineering applications and other applied fields that are based on fractional calculus, fractals, wavelets, nonlinear equations and methods, nonlinear dynamical systems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integrals.

Prof. Dr. Carlo Cattani
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematics
  • physics
  • mathematical physics
  • mechanics
  • fractional calculus
  • fractal
  • fractional dynamical systems
  • fractional partial differential equations

Published Papers (10 papers)

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Research

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19 pages, 10956 KiB  
Article
Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate
by Mohammed Hamed Alshbool, Osman Isik and Ishak Hashim
Axioms 2021, 10(1), 6; https://doi.org/10.3390/axioms10010006 - 07 Jan 2021
Cited by 8 | Viewed by 2023
Abstract
In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems [...] Read more.
In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems in fractional partial differential equations. To get the FBSS, we first convert all terms in the problem to matrix forms. Then, the fundamental matrix equation is obtained and thus, the solution is obtained. Two error estimation methods based on a residual correction procedure and the consecutive approximations are incorporated to find the estimate and bound of the absolute error. The perturbation and stability analysis of the method is given. We apply the method to some illustrative examples. The numerical results are compared with the exact solutions and known second-order methods. The outcomes of the numerical examples are very encouraging and show that the FBSS is highly useful in solving fractional partial problems. The results show the accuracy and effectiveness of the method. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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14 pages, 1387 KiB  
Article
On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces
by Nallapu Vijender and Vasileios Drakopoulos
Axioms 2020, 9(4), 119; https://doi.org/10.3390/axioms9040119 - 18 Oct 2020
Cited by 4 | Viewed by 1639
Abstract
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein [...] Read more.
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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13 pages, 280 KiB  
Article
On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
by Maksim V. Kukushkin
Axioms 2020, 9(3), 81; https://doi.org/10.3390/axioms9030081 - 17 Jul 2020
Cited by 3 | Viewed by 1521
Abstract
In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives [...] Read more.
In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
14 pages, 269 KiB  
Article
On a New Generalized Integral Operator and Certain Operating Properties
by Paulo M. Guzman, Luciano M. Lugo, Juan E. Nápoles Valdés and Miguel Vivas-Cortez
Axioms 2020, 9(2), 69; https://doi.org/10.3390/axioms9020069 - 20 Jun 2020
Cited by 21 | Viewed by 2539
Abstract
In this paper, we present a general definition of a generalized integral operator which contains as particular cases, many of the well-known, fractional and integer order integrals. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
12 pages, 608 KiB  
Article
On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator
by Francesca Pitolli
Axioms 2020, 9(2), 61; https://doi.org/10.3390/axioms9020061 - 25 May 2020
Cited by 8 | Viewed by 2245
Abstract
Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative [...] Read more.
Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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12 pages, 266 KiB  
Article
Measure of Noncompactness for Hybrid Langevin Fractional Differential Equations
by Ahmed Salem and Mohammad Alnegga
Axioms 2020, 9(2), 59; https://doi.org/10.3390/axioms9020059 - 23 May 2020
Cited by 21 | Viewed by 2664
Abstract
In this research article, we introduce a new class of hybrid Langevin equation involving two distinct fractional order derivatives in the Caputo sense and Riemann–Liouville fractional integral. Supported by three-point boundary conditions, we discuss the existence of a solution to this boundary value [...] Read more.
In this research article, we introduce a new class of hybrid Langevin equation involving two distinct fractional order derivatives in the Caputo sense and Riemann–Liouville fractional integral. Supported by three-point boundary conditions, we discuss the existence of a solution to this boundary value problem. Because of the important role of the measure of noncompactness in fixed point theory, we use the technique of measure of noncompactness as an essential tool in order to get the existence result. The modern analysis technique is used by applying a generalized version of Darbo’s fixed point theorem. A numerical example is presented to clarify our outcomes. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
17 pages, 314 KiB  
Article
Existence of Solutions for Nonlinear Fractional Differential Equations and Inclusions Depending on Lower-Order Fractional Derivatives
by Subramanian Muthaiah and Dumitru Baleanu
Axioms 2020, 9(2), 44; https://doi.org/10.3390/axioms9020044 - 25 Apr 2020
Cited by 14 | Viewed by 2151
Abstract
This article deals with the solutions of the existence and uniqueness for a new class of boundary value problems (BVPs) involving nonlinear fractional differential equations (FDEs), inclusions, and boundary conditions involving the generalized fractional integral. The nonlinearity relies on the unknown function and [...] Read more.
This article deals with the solutions of the existence and uniqueness for a new class of boundary value problems (BVPs) involving nonlinear fractional differential equations (FDEs), inclusions, and boundary conditions involving the generalized fractional integral. The nonlinearity relies on the unknown function and its fractional derivatives in the lower order. We use fixed-point theorems with single-valued and multi-valued maps to obtain the desired results, through the support of illustrations, the main results are well explained. We also address some variants of the problem. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
21 pages, 367 KiB  
Article
On One Interpolation Type Fractional Boundary-Value Problem
by Kateryna Marynets
Axioms 2020, 9(1), 13; https://doi.org/10.3390/axioms9010013 - 28 Jan 2020
Cited by 4 | Viewed by 2078
Abstract
We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called “interpolation” type boundary restrictions. [...] Read more.
We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called “interpolation” type boundary restrictions. Under certain conditions, the aforementioned problem is simplified via a proper parametrization technique, and with the help of the numerical-analytic method, the approximate solutions are constructed. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
22 pages, 324 KiB  
Article
Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion
by Maksim V. Kukushkin
Axioms 2019, 8(2), 75; https://doi.org/10.3390/axioms8020075 - 23 Jun 2019
Cited by 6 | Viewed by 2602
Abstract
In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously [...] Read more.
In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)

Review

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11 pages, 915 KiB  
Review
Cantor Waves for Signorini Hyperelastic Materials with Cylindrical Symmetry
by Carlo Cattani
Axioms 2020, 9(1), 22; https://doi.org/10.3390/axioms9010022 - 13 Feb 2020
Cited by 2 | Viewed by 1914
Abstract
In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the [...] Read more.
In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the nonlinear fundamental equations of the fractional model are explicitly given. In the special case of linear approximation we explicitly compute the fractional wave profile. Full article
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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