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Fractal Fract
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General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics
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General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics

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

Deadline for manuscript submissions: closed (25 March 2025) | Viewed by 10763

Special Issue Editors

Dr. Yi-Ying Feng

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China
Interests: fractional calculus; local fractional calculus; general fractional calculus; creep constitutive model; applied mathematics; mechanical engineering
Special Issues, Collections and Topics in MDPI journals
Dr. Jian-Gen Liu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
Interests: fractional calculus; local fractional calculus; mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus can contain different fractional operators to obtain many fractional derivatives, and the generalisation is always a key concept in mathematics. Therefore, it is of utmost importance to study the general fractional calculus that enlarges the natural limitation of various definitions for fractional derivatives.

This subject matter of this Special Issue aims at highlighting the general fractional calculus to solve problems that affect foundational mathematical research and engineering technology. Many phenomena from physics, chemistry, mechanics and electricity can be modeled using differential equations involving general fractional derivatives. In addition, the research in the field of general fractional calculus is interdisciplinary. Its development can also promote the vigorous development of several fields. Topics that are invited for submission include (but are not limited to):

  • general fractional calculus theory;
  • general fractional calculus method;
  • general fractional calculus applications;
  • fractional viscoelasticity;
  • fractional dynamical systems;
  • fractional calculus in anomalous diffusion;
  • fractional operator theory and theoretical analysis;
  • new definitions and properties of general fractional calculus;
  • memory and heritability of general fractional calculus.

Dr. Yi-Ying Feng
Dr. Jian-Gen 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

  • general fractional calculus theory
  • general fractional calculus method
  • general fractional calculus applications
  • fractional viscoelasticity
  • fractional dynamical systems
  • fractional calculus in anomalous diffusion
  • fractional operator theory and theoretical analysis
  • new definitions and properties of general fractional calculus
  • memory and heritability of general fractional calculus

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Published Papers (7 papers)

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19 pages, 363 KiB  
Article
Tempered Riemann–Liouville Fractional Operators: Stability Analysis and Their Role in Kinetic Equations
by Muhammad Umer, Muhammad Samraiz, Muath Awadalla and Meraa Arab
Fractal Fract. 2025, 9(3), 187; https://doi.org/10.3390/fractalfract9030187 - 18 Mar 2025
Viewed by 173
Abstract
Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial [...] Read more.
Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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15 pages, 310 KiB  
Article
Certain Properties and Characterizations of Two-Iterated Two-Dimensional Appell and Related Polynomials via Fractional Operators
by Mohra Zayed and Shahid Ahmad Wani
Fractal Fract. 2024, 8(7), 382; https://doi.org/10.3390/fractalfract8070382 - 28 Jun 2024
Viewed by 829
Abstract
This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for [...] Read more.
This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for these polynomials and obtains their recurrence relations. The paper also establishes corresponding results for the generalized 2-iterated 2D Bernoulli, 2-iterated 2D Euler, and 2-iterated 2D Genocchi polynomials. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
24 pages, 3382 KiB  
Article
A Two-Temperature Fractional DPL Thermoelasticity Model with an Exponential Rabotnov Kernel for a Flexible Cylinder with Changeable Properties
by Ahmed E. Abouelregal, Yazeed Alhassan, Hashem Althagafi and Faisal Alsharif
Fractal Fract. 2024, 8(4), 182; https://doi.org/10.3390/fractalfract8040182 - 22 Mar 2024
Cited by 11 | Viewed by 1762
Abstract
This article presents a new thermoelastic model that incorporates fractional-order derivatives of two-phase heat transfer as well as a two-temperature concept. The objective of this model is to improve comprehension and forecasting of heat transport processes in two-phase-lag systems by employing fractional calculus. [...] Read more.
This article presents a new thermoelastic model that incorporates fractional-order derivatives of two-phase heat transfer as well as a two-temperature concept. The objective of this model is to improve comprehension and forecasting of heat transport processes in two-phase-lag systems by employing fractional calculus. This model suggests a new generalized fractional derivative that can make different kinds of singular and non-singular fractional derivatives, depending on the kernels that are used. The non-singular kernels of the normalized sinc function and the Rabotnov fractional–exponential function are used to create the two new fractional derivatives. The thermoelastic responses of a solid cylinder with a restricted surface and exposed to a moving heat flux were examined in order to assess the correctness of the suggested model. It was considered that the cylinder’s thermal characteristics are dependent on the linear temperature change and that it is submerged in a continuous magnetic field. To solve the set of equations controlling the suggested issue, Laplace transforms were used. In addition to the reliance of thermal characteristics on temperature change, the influence of derivatives and fractional order was also studied by providing numerical values for the temperature, displacement, and stress components. This study found that the speed of the heat source and variable properties significantly impact the behavior of the variables under investigation. Meanwhile, the fractional parameter has a slight effect on non-dimensional temperature changes but plays a crucial role in altering the peak value of non-dimensional displacement and pressure. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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12 pages, 317 KiB  
Article
Investigation of the Time Fractional Higher-Dimensional Nonlinear Modified Equation of Wave Propagation
by Jian-Gen Liu and Yi-Ying Feng
Fractal Fract. 2024, 8(3), 124; https://doi.org/10.3390/fractalfract8030124 - 20 Feb 2024
Viewed by 1709
Abstract
In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time [...] Read more.
In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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20 pages, 10371 KiB  
Article
Dynamic Compressive Mechanical Property Characteristics and Fractal Dimension Applications of Coal-Bearing Mudstone at Real-Time Temperatures
by Shiru Guo, Lianying Zhang, Hai Pu, Yadong Zheng, Bing Li, Peng Wu, Peitao Qiu, Chao Ma and Yiying Feng
Fractal Fract. 2023, 7(9), 695; https://doi.org/10.3390/fractalfract7090695 - 18 Sep 2023
Cited by 5 | Viewed by 1744
Abstract
Coal-bearing rocks are inevitably exposed to high temperatures and impacts (rapid dynamic load action) during deep-earth resource extraction, necessitating the study of their mechanical properties under such conditions. This paper reports on dynamic compression tests conducted on coal-bearing mudstone specimens at real-time temperatures [...] Read more.
Coal-bearing rocks are inevitably exposed to high temperatures and impacts (rapid dynamic load action) during deep-earth resource extraction, necessitating the study of their mechanical properties under such conditions. This paper reports on dynamic compression tests conducted on coal-bearing mudstone specimens at real-time temperatures (the temperature of the rock remains constant throughout the impact process) ranging from 25 °C to 400 °C using a temperature Hopkinson (T-SHPB) test apparatus developed in-house. The objective is to analyze the relationship between mechanical properties and the fractal dimension of fractured fragments and to explore the mechanical response of coal-bearing mudstone specimens to the combined effects of temperature and impact using macroscopic fracture characteristics. The study found that the peak stress and dynamic elastic modulus initially increased and then decreased with increasing temperature, increasing in the 25–150 °C range and monotonically decreasing in the 150–400 °C range. Based on the distribution coefficients and fractal dimensions of the fractured fragments, it was found that the degree of damage of coal-bearing mudstone shows a trend of an initial decrease and then an increase with increasing temperature. In the temperature range of 25–150 °C, the expansion of clay minerals within the mudstone filled the voids between the skeletal particles, resulting in densification and decreased damage. In the temperature range of 150–400 °C, thermal stresses increased the internal fractures and reduced the overall strength of the mudstone, resulting in increased damage. Negative correlations between fractal dimensions, the modulus of elasticity, and peak stress could be used to predict rock properties in engineering. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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17 pages, 413 KiB  
Article
Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model
by Emilia Bazhlekova and Sergey Pshenichnov
Fractal Fract. 2023, 7(8), 636; https://doi.org/10.3390/fractalfract7080636 - 20 Aug 2023
Cited by 7 | Viewed by 1313
Abstract
A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions [...] Read more.
A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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11 pages, 292 KiB  
Article
Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with p-Laplacian Operator
by Tengfei Shen
Fractal Fract. 2023, 7(6), 427; https://doi.org/10.3390/fractalfract7060427 - 25 May 2023
Cited by 2 | Viewed by 1152
Abstract
The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved [...] Read more.
The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results. Full article
(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)
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