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Fractal Fract
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Advances in Fractional Modeling and Computation, Second Edition
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Advances in Fractional Modeling and Computation, Second 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: 15 January 2027 | Viewed by 4641

Special Issue Editors

Prof. Dr. Jordan Hristov

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Guest Editor
Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria
Interests: mathematical modeling; fractional calculus; non-linear diffusion; viscoelasticity
Special Issues, Collections and Topics in MDPI journals
Dr. Slavi Georgiev

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Guest Editor
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Interests: mathematic modeling; fractional calculus; numerical methods for fractional differential equations; applications in computational and quantitative finance, mathematical and computer modeling
Special Issues, Collections and Topics in MDPI journals
Dr. Yuri Mitkov Dimitrov

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Guest Editor
Department of Mathematics, Physics and Informatics, University of Forestry, 1756 Sofia, Bulgaria
Interests: fractional calculus; numerical methods for fractional differential equations; Monte Carlo methods
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Dr. Venelin Todorov

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Guest Editor
Department of Parallel Algorithms and Machine Learning with a Laboratory in Neurotechnologies, Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Interests: applied mathematics; mathematical modeling; fractional calculus; numerical methods; stochastic and Monte Carlo methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a branch of mathematics that deals with the study of fractional order derivatives. Today, fractional calculus has many applications in various fields, including physics, engineering, finance, and biology. It can be used to model complex systems that exhibit non-local or long-range interactions, as well as to solve differential equations involving fractional derivatives. Many models of complex systems that use ordinary and partial differential equations do not have analytic solutions. There is an urgent need to develop effective computational methods for the analysis of fractional models.

The focus of this Special Issue is the development and advancement of models using fractional differential equations and processes. We welcome original and review papers on theory, computational and Monte Carlo methods, and practical applications of fractional models in physics, chemistry, biology, engineering, economics, probability, and statistics. Topics that are invited for submission include (but are not limited to) the following:

  • Fractional models in natural sciences;
  • Fractional models in economics and engineering;
  • Numerical algorithms and discretization;
  • Fractional differential systems with control theory;
  • Fractional dynamic systems;
  • Analysis of fractional models;
  • Stochastic methods for fractional models;
  • Monte Carlo methods;
  • Markov chains and processes;
  • Stochastic modeling and simulation;
  • Related fractional models.

Prof. Dr. Jordan Hristov
Dr. Slavi Georgiev
Dr. Yuri Mitkov Dimitrov
Dr. Venelin Todorov
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 250 words) can be sent to the Editorial Office for assessment.

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 models in natural sciences
  • fractional models in economics and engineering
  • numerical algorithms and discretization
  • fractional differential systems with control theory
  • fractional dynamic systems
  • analysis of fractional models
  • stochastic methods for fractional models
  • Monte Carlo methods
  • Markov chains and processes
  • stochastic modeling and simulation
  • related fractional models

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

Related Special Issue

Published Papers (4 papers)

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Research

20 pages, 4865 KB  
Article
Solitary and Cnoidal Structures in Plasmas Described by a Residual-Controlled Time-Fractional Gardner Equation
by Alvaro H. Salas, Weaam Alhejaili and Samir A. El-Tantawy
Fractal Fract. 2026, 10(4), 211; https://doi.org/10.3390/fractalfract10040211 - 24 Mar 2026
Viewed by 214
Abstract
The present work is devoted to the analysis of a time-fractional Gardner equation arising in the modeling of nonlinear plasma waves in media endowed with memory and anomalous transport effects. Building on a physically motivated soliton profile, we construct a finite-time fractional ansatz [...] Read more.
The present work is devoted to the analysis of a time-fractional Gardner equation arising in the modeling of nonlinear plasma waves in media endowed with memory and anomalous transport effects. Building on a physically motivated soliton profile, we construct a finite-time fractional ansatz in which the integer-order time variable is replaced by a fractional reparametrization that encodes the Caputo memory kernel. Within this framework, the governing evolution equation is not treated via a formal infinite expansion but rather via a finite approximation, whose quality is assessed directly via the associated residual. The Caputo fractional derivative is evaluated by a strong finite-difference formula that is second-order accurate in time and preserves the nonlocal convolution structure of the fractional operator. This combination of a finite fractional ansatz and a strong Caputo discretization allows us to compute the residual of the time analytically fractional Gardner equation and to use it as a quantitative diagnostic of accuracy and consistency. Two representative classes of nonlinear structures supported by the Gardner equation are examined in detail: a smooth solitary-wave profile and a cnoidal-wave configuration. For each example, the approximate fractional solution is generated, the corresponding residual is evaluated in space–time, and global and final-time residual norms are determined to quantify the influence of the fractional order on the wave dynamics and on the quality of the approximation. The numerical results show that the proposed residual-controlled approach yields residual magnitudes that remain one to two orders of magnitude smaller than those associated with truncated residual power-series approximations constructed from the same data, while preserving the expected qualitative features of fractional solitary and cnoidal waves in non-Markovian plasma environments. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation, Second Edition)
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18 pages, 35027 KB  
Article
A Finite Difference Method for Caputo Generalized Time Fractional Diffusion Equations
by Jun Li, Jiejing Zhang and Yingjun Jiang
Fractal Fract. 2026, 10(1), 19; https://doi.org/10.3390/fractalfract10010019 - 28 Dec 2025
Cited by 1 | Viewed by 1977
Abstract
This paper presents a finite difference method for solving the Caputo generalized time fractional diffusion equation. The method extends the L1 scheme to discretize the time fractional derivative and employs the central difference for the spatial diffusion term. Theoretical analysis demonstrates that [...] Read more.
This paper presents a finite difference method for solving the Caputo generalized time fractional diffusion equation. The method extends the L1 scheme to discretize the time fractional derivative and employs the central difference for the spatial diffusion term. Theoretical analysis demonstrates that the proposed numerical scheme achieves a convergence rate of order 2−α in time and second order in space. These theoretical findings are further validated through numerical experiments. Compared to existing methods that only achieve a temporal convergence of order 1−α, the proposed approach offers improved accuracy and efficiency, particularly when the fractional order α is close to zero. This makes the method highly suitable for simulating transport processes with memory effects, such as oil pollution dispersion and biological population dynamics. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation, Second Edition)
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27 pages, 818 KB  
Article
Second-Order L1 Schemes for Fractional Differential Equations
by Yuri Dimitrov, Slavi Georgiev, Radan Miryanov and Venelin Todorov
Fractal Fract. 2025, 9(12), 816; https://doi.org/10.3390/fractalfract9120816 - 13 Dec 2025
Viewed by 699
Abstract
Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed [...] Read more.
Difference schemes for the numerical solution of fractional differential equations rely on discretizations of the fractional derivative. In this paper, we obtain the second-order expansion formula for the L1 approximation of the Caputo fractional derivative. Second-order approximations of the fractional derivative are constructed based on the expansion formula and parameter-dependent discretizations of the second derivative. Examples illustrating the application of these approximations to the numerical solution of ordinary and partial fractional differential equations are presented, and the convergence and order of the difference schemes are proved. Numerical experiments are also provided, confirming the theoretical predictions for the accuracy of the numerical methods. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation, Second Edition)
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26 pages, 516 KB  
Article
Analysis of an ABC-Fractional Asset Flow Model for Financial Markets
by Din Prathumwan, Inthira Chaiya and Kamonchat Trachoo
Fractal Fract. 2025, 9(9), 563; https://doi.org/10.3390/fractalfract9090563 - 27 Aug 2025
Cited by 1 | Viewed by 1125
Abstract
This paper proposes a novel fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative to analyze asset price dynamics in financial markets. Compared to classical models, the proposed model incorporates a nonlocal and non-singular fractional operator, allowing for a more accurate representation [...] Read more.
This paper proposes a novel fractional-order asset flow model based on the Atangana–Baleanu–Caputo (ABC) derivative to analyze asset price dynamics in financial markets. Compared to classical models, the proposed model incorporates a nonlocal and non-singular fractional operator, allowing for a more accurate representation of investor behavior and market adjustment processes. The model captures both short-term trend-driven responses and long-term valuation-based decisions. We establish key theoretical properties of the system, including the existence and uniqueness of solutions, positivity, boundedness, and both local and global stability using Lyapunov functions. Numerical simulations under varying fractional orders demonstrate how the ABC derivative governs the convergence speed and equilibrium behavior of the system. Compared to classical integer-order models, the ABC-based approach provides smoother dynamics, greater flexibility in modeling behavioral heterogeneity, and better alignment with observed long-term financial phenomena. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation, Second Edition)
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