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Fractal Fract
Special Issues
Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition
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Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition

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: 25 February 2025 | Viewed by 10347

Special Issue Editors

Dr. Libo Feng

E-Mail Website
Guest Editor
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia
Interests: numerical methods and analysis of fractional PDE; application of fractional mathematical models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yang Liu

E-Mail Website
Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Interests: finite element method; finite difference method; LDG methods; numerical methods for fractional PDEs
Special Issues, Collections and Topics in MDPI journals
Dr. Lin Liu

E-Mail Website
Guest Editor
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Interests: viscoelastic fluid boundary layer flow; fractional anomalous diffusion; biological heat conduction
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last few decades, the application of fractional calculus to real-world problems has grown rapidly, with the use of dynamical systems described by fractional differential equations (FDEs) as one of the ways to understand complex materials and processes. Due to the ability to model the non-locality, memory, spatial heterogeneity and anomalous diffusion inherent in many real-world problems, the application of FDEs has been attracting much attention in many fields of science and is still under development. However, generally, the fractional mathematical models from science and engineering are so complex that analytical solutions are not available. Therefore, numerical solution has been an effective tool to deal with fractional mathematical models.

This Special Issue aims to promote communication between researchers and practitioners on the application of fractional calculus, present the latest development of fractional differential equations, report state-of-the-art and in-progress numerical methods and discuss future trends and challenges. We cordially invite you to contribute by submitting original research articles or comprehensive review papers. This Special Issue will cover the following topics, but these are not exhaustive:

  1. Mathematical modeling of fractional dynamic systems;
  2. Analytical or semi-analytical solution of fractional differential equations;
  3. Numerical methods to solve fractional differential equations, e.g.,  the finite difference method, the finite element method, the finite volume method, the spectral method, etc.;
  4. Fast algorithm for the time or space fractional derivative;
  5. Mathematical analysis for fractional problems and numerical analysis for the numerical scheme;
  6. Applications of fractional calculus in physics, biology, chemistry, finance, signal and image processing, hydrology, non-Newtonian fluids, etc.

Please also feel free to read and download the published articles in our first volume:

https://www.mdpi.com/journal/fractalfract/special_issues/NSAFDE

Dr. Libo Feng
Prof. Dr. Yang Liu
Dr. Lin Liu
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

  • numerical methods
  • mathematical modeling
  • fractional calculus
  • fractional differential equations
  • numerical analysis
  • fast algorithm

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Published Papers (9 papers)

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Research

12 pages, 3493 KiB  
Article
On a Preloaded Compliance System of Fractional Order: Numerical Integration
by Marius-F. Danca
Fractal Fract. 2025, 9(2), 84; https://doi.org/10.3390/fractalfract9020084 - 26 Jan 2025
Viewed by 306
Abstract
In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, [...] Read more.
In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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16 pages, 308 KiB  
Article
Well-Posedness of the Mild Solutions for Incommensurate Systems of Delay Fractional Differential Equations
by Babak Shiri
Fractal Fract. 2025, 9(2), 60; https://doi.org/10.3390/fractalfract9020060 - 21 Jan 2025
Viewed by 488
Abstract
Systems of incommensurate delay fractional differential equations (DFDEs) with non-vanishing constant delay of retarded type are investigated. It is shown that the mild solutions are well-posed in Hadamard sense on the space of continuous functions. The analysis is local and carried out for [...] Read more.
Systems of incommensurate delay fractional differential equations (DFDEs) with non-vanishing constant delay of retarded type are investigated. It is shown that the mild solutions are well-posed in Hadamard sense on the space of continuous functions. The analysis is local and carried out for finite intervals. The strong results are obtained with weak conditions by using state-of-the-art new methods. No condition on the Lipschitz parameter is added for well-posedness results. Application of this theorem for the Hopfield neural network is carried out. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
21 pages, 337 KiB  
Article
Positive Solutions and Their Existence of a Nonlinear Hadamard Fractional-Order Differential Equation with a Singular Source Item Using Spectral Analysis
by Cheng Li and Limin Guo
Fractal Fract. 2024, 8(7), 377; https://doi.org/10.3390/fractalfract8070377 - 26 Jun 2024
Cited by 3 | Viewed by 1129
Abstract
Based on the spectral analysis method, Gelfand’s formula, and the cones fixed point theorem, some positive solutions with their existence of a nonlinear infinite-point Hadamard fractional-order differential equation is achieved on the interval [a, b] under some conditions, and particularly [...] Read more.
Based on the spectral analysis method, Gelfand’s formula, and the cones fixed point theorem, some positive solutions with their existence of a nonlinear infinite-point Hadamard fractional-order differential equation is achieved on the interval [a, b] under some conditions, and particularly the nonlinear term allows singularities for time and spatial parameters in the present study. Finally, an analysis case is carried out to reveal the principal results. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
22 pages, 3835 KiB  
Article
An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting
by Imtiaz Ahmad, Abdulrahman Obaid Alshammari, Rashid Jan, Normy Norfiza Abdul Razak and Sahar Ahmed Idris
Fractal Fract. 2024, 8(6), 364; https://doi.org/10.3390/fractalfract8060364 - 20 Jun 2024
Cited by 1 | Viewed by 1449
Abstract
The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional [...] Read more.
The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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8 pages, 4284 KiB  
Article
Dynamic Behavior and Optical Soliton for the M-Truncated Fractional Paraxial Wave Equation Arising in a Liquid Crystal Model
by Jie Luo and Zhao Li
Fractal Fract. 2024, 8(6), 348; https://doi.org/10.3390/fractalfract8060348 - 12 Jun 2024
Viewed by 918
Abstract
The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation [...] Read more.
The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation is applied to the M-truncated fractional paraxial wave equation. Moreover, a two-dimensional dynamical system and its disturbance system are obtained. The phase portraits of the two-dimensional dynamic system and Poincaré sections and a bifurcation portrait of its perturbation system are drawn. The obtained three-dimensional graphs of soliton solutions, two-dimensional graphs of soliton solutions, and contour graphs of the M-truncated fractional paraxial wave equation arising in a liquid crystal model are drawn. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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10 pages, 919 KiB  
Article
A Dynamical Analysis and New Traveling Wave Solution of the Fractional Coupled Konopelchenko–Dubrovsky Model
by Jin Wang and Zhao Li
Fractal Fract. 2024, 8(6), 341; https://doi.org/10.3390/fractalfract8060341 - 6 Jun 2024
Cited by 12 | Viewed by 1174
Abstract
The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using [...] Read more.
The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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19 pages, 1368 KiB  
Article
Dynamical Analysis of Two-Dimensional Fractional-Order-in-Time Biological Population Model Using Chebyshev Spectral Method
by Ishtiaq Ali
Fractal Fract. 2024, 8(6), 325; https://doi.org/10.3390/fractalfract8060325 - 29 May 2024
Viewed by 995
Abstract
In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, [...] Read more.
In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the L1 scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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20 pages, 8443 KiB  
Article
Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition
by Jingyu Yang, Lin Liu, Siyu Chen, Libo Feng and Chiyu Xie
Fractal Fract. 2024, 8(6), 309; https://doi.org/10.3390/fractalfract8060309 - 23 May 2024
Viewed by 1237
Abstract
The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed [...] Read more.
The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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20 pages, 372 KiB  
Article
A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations
by Jie Zhao, Shubin Dong and Zhichao Fang
Fractal Fract. 2024, 8(1), 51; https://doi.org/10.3390/fractalfract8010051 - 12 Jan 2024
Viewed by 1397
Abstract
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution [...] Read more.
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L∞(L2(Ω)) norm and for the auxiliary variable λ in the discrete L∞((L2(Ω))2) and L∞(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
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