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Fractal Fract
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Advances in Fractional and Fractal Boundary Value Problems in Applied...
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Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (10 July 2022) | Viewed by 12413

Special Issue Editors

Prof. Dr. Muhammad Bilal Riaz

E-Mail Website
Guest Editor
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland
Interests: computational mathematics fractional calculus; fluid dynamics; mathematical physics; numerical analysis
Prof. Dr. Mohamed Abdelsabour Fahmy

E-Mail Website
Guest Editor
Department of Mathematics, Umm Al-Qura University, Mecca, Saudi Arabia
Interests: boundary element algorithm; boundary value problems; fractional-order equations
Special Issues, Collections and Topics in MDPI journals
Dr. Khaled Mohammed Saad

E-Mail Website
Guest Editor
1. Department of Mathematics, College of Sciences and Arts, Najran University, Najran P.O. Box 1988, Saudi Arabia
2. Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen
Interests: numerical and analytical solutions of fractional differential equations

Special Issue Information

Dear Colleagues,

Extensive studies have been focused on fractional calculus in recent years due to its capacity to model complex phenomena more efficiently. The advantages of fractional modeling have been visualized in many engineering and scientific disciplines including biology, physics, aerodynamics, electron-analytic chemistry, ecology image processing, financial modeling, control theory for dynamical systems, disease modelling, nanotechnology, random walks, anomalous transport, anomalous diffusion, and viscoelasticity  

The fractal boundary value problems for the Fredholm and Volterra integral equations, heat conduction, and wave equations have taken much interest recently. Fractals are applied in many engineering implementations such as porous media modelling, nano fluids, fracture mechanics, and many other implementations in nanoscale. The local temperature relies on the fractal dimensions where adequate physical results can be obtained by the implementation of local fractional models and relevant solution approaches for the transport phenomena applied in fractal objects. 

We aim to combine the fractional order and fractal dimension in boundary value problems, and we would like to get new interesting solutions of the problems.  

As a result of recent enhancements in fractional and fractal calculus applications, many authors have become interested in this area. This Special Issue on “Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences” is devoted to uncovering leading investigators’ recent work in the above areas of fractional and fractal calculus. 

Prof. Dr. Muhammad Bilal Riaz
Prof. Dr. Mohamed Abdelsabour Fahmy
Dr. Khaled Mohammed Saad
Guest Editors

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. Fractal and Fractional 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 2700 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

  • fractional and fractal calculus
  • boundary value problems
  • numerical methods
  • fractional differential equations
  • fractal differential equations
  • fractal-fractional differential equations
  • mathematical modelling
  • epidemic models
  • fractional partial differential equations
  • nonlinear dynamics
  • mechatronics
  • dynamic systems
  • numerical mathematics
  • friction

Published Papers (6 papers)

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Research

14 pages, 4027 KiB  
Article
Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations
by Hakeem Ullah, Mehreen Fiza, Ilyas Khan, Nawa Alshammari, Nawaf N. Hamadneh and Saeed Islam
Fractal Fract. 2022, 6(6), 288; https://doi.org/10.3390/fractalfract6060288 - 26 May 2022
Cited by 4 | Viewed by 1481
Abstract
In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. [...] Read more.
In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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23 pages, 7908 KiB  
Article
3D Boundary Element Model for Ultrasonic Wave Propagation Fractional Order Boundary Value Problems of Functionally Graded Anisotropic Fiber-Reinforced Plates
by Mohamed Abdelsabour Fahmy
Fractal Fract. 2022, 6(5), 247; https://doi.org/10.3390/fractalfract6050247 - 29 Apr 2022
Cited by 18 | Viewed by 2451
Abstract
This paper proposes a three–dimensional (3D) local boundary element model based on meshless moving least squares (MLS) method for ultrasonic wave propagation fractional order boundary value problems of functionally graded anisotropic (FGA) fiber-reinforced plates. The problem domain is split into several circular sub-domains. [...] Read more.
This paper proposes a three–dimensional (3D) local boundary element model based on meshless moving least squares (MLS) method for ultrasonic wave propagation fractional order boundary value problems of functionally graded anisotropic (FGA) fiber-reinforced plates. The problem domain is split into several circular sub-domains. The nodal points are randomly distributed across the examined region. Each node is the focal point of a circular sub-domain that encircles it. The Laplace-transform approach is used to solve dynamic issues. In the local weak form of the governing equations for the converted quantities, a unit test function is utilized. The Gauss divergence theorem to the weak-form is used to produce local boundary-domain integral equations. A meshless approximation is achieved using the MLS method. To find time-dependent solutions, an inverse Laplace-transform approach is used. The effects of the fractional order parameter, functionally graded material, anisotropy, and the time characteristic of the laser pulse are investigated. The proposed method’s validity and performance are demonstrated for a two-dimensional problem with excellent agreement with the finite element method. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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20 pages, 1295 KiB  
Article
New Solutions of Nonlinear Dispersive Equation in Higher-Dimensional Space with Three Types of Local Derivatives
by Ali Akgül, Mir Sajjad Hashemi and Fahd Jarad
Fractal Fract. 2022, 6(4), 202; https://doi.org/10.3390/fractalfract6040202 - 4 Apr 2022
Cited by 7 | Viewed by 1534
Abstract
The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study [...] Read more.
The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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20 pages, 5639 KiB  
Article
Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique
by Ghazala Akram, Muhammad Abbas, Hira Tariq, Maasoomah Sadaf, Thabet Abdeljawad and Manar A. Alqudah
Fractal Fract. 2022, 6(3), 170; https://doi.org/10.3390/fractalfract6030170 - 19 Mar 2022
Cited by 2 | Viewed by 1835
Abstract
Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline [...] Read more.
Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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16 pages, 757 KiB  
Article
Fractional Order Mathematical Model of Serial Killing with Different Choices of Control Strategy
by Mati ur Rahman, Shabir Ahmad, Muhammad Arfan, Ali Akgül and Fahd Jarad
Fractal Fract. 2022, 6(3), 162; https://doi.org/10.3390/fractalfract6030162 - 16 Mar 2022
Cited by 16 | Viewed by 2076
Abstract
The current manuscript describes the dynamics of a fractional mathematical model of serial killing under the Mittag–Leffler kernel. Using the fixed point theory approach, we present a qualitative analysis of the problem and establish a result that ensures the existence of at least [...] Read more.
The current manuscript describes the dynamics of a fractional mathematical model of serial killing under the Mittag–Leffler kernel. Using the fixed point theory approach, we present a qualitative analysis of the problem and establish a result that ensures the existence of at least one solution. Ulam’s stability of the given model is presented by using nonlinear concepts. The iterative fractional-order Adams–Bashforth approach is being used to find the approximate solution. The suggested method is numerically simulated at various fractional orders. The simulation is carried out for various control strategies. Over time, all of the compartments demonstrate convergence and stability. Different fractional orders have produced an excellent comparison outcome, with low fractional orders achieving stability sooner. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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11 pages, 4071 KiB  
Article
A New Iterative Predictor-Corrector Algorithm for Solving a System of Nuclear Magnetic Resonance Flow Equations of Fractional Order
by Mariam Sultana, Uroosa Arshad, Muhammad Khalid, Ali Akgül, Wedad Albalawi and Heba Y. Zahran
Fractal Fract. 2022, 6(2), 91; https://doi.org/10.3390/fractalfract6020091 - 6 Feb 2022
Cited by 1 | Viewed by 1546
Abstract
Nuclear magnetic resonance flow equations, also known as the Bloch system, are said to be at the heart of both magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy. The main aim of this research was to solve fractional nuclear magnetic resonance [...] Read more.
Nuclear magnetic resonance flow equations, also known as the Bloch system, are said to be at the heart of both magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy. The main aim of this research was to solve fractional nuclear magnetic resonance flow equations (FNMRFEs) through a numerical approach that is very easy to handle. We present a New Iterative Predictor-Corrector Algorithm (NIPCA) based on the New Iterative Algorithm and Predictor-Corrector Algorithm to solve nonlinear nuclear magnetic resonance flow equations of fractional order involving Caputo derivatives. Graphical representation of the solutions with detailed error analysis shows the higher accuracy of the new technique. This New Iterative Predictor-Corrector Algorithm requires less computational time than previously published numerical methods. The results achieved in this article indicate that the algorithm is fit to use for other chaotic systems of fractional differential equations. Full article
(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)
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