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Fractal Fract
Special Issues
Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential...
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Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations

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: 31 July 2025 | Viewed by 2626

Special Issue Editors

Prof. Dr. Xiao-Jun Yang

grade E-Mail Website
Guest Editor
1. State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep Underground Engineering and School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
2. Qin Institute of Mathematics, Shanghai Hanjing Center of Science and Technology, Shanghai 201609, China
Interests: mathematical physics; fractional calculus and applications; fractals; mechanics; number theory; integral transforms and special functions
Prof. Dr. Wenxue Zhou

E-Mail Website
Guest Editor
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
Interests: nonlinear analysis; differential equation theory and its applications; fractional differential equations; nonlinear fractional difference equations
Dr. Sunil Dutt Purohit

E-Mail Website
Guest Editor
Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India
Interests: fractional calculus; special functions; integral transforms; q-hypergeometric functions; geometric function theory; mathematical modeling
Dr. Ali Turab

E-Mail Website
Guest Editor
School of Software, Northwestern Polytechnical University, Xi'an 710072, China
Interests: mathematical modeling; applied behavior; nonlinear analysis; fractional calculus; game theory and decision theory
Dr. Ahmed Refaie Ali

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt
Interests: applied mathematics; fluid dynamics; electromagnetic theory; electrodynamics; wave propagation

Special Issue Information

Dear Colleagues,

Nonlinear and liner differential equations are used to deal with the real-world problems in natural sciences. Fractional differential equations are the generalized versions of the classical differential equations. There are still many unsolved problems in the dynamical systems theory of nonlinear PDEs of integer and fractional orders. Fractional calculus is defined by the integrals within singular kernel. General fractional calculus is defined by the integral with nonsingular kernels. The Special Issue invites submissions of original research articles, reviews, and perspectives on the latest developments in the analysis and numerical computations of nonlinear fractional and classical differential equations. Potential topics include, but are not limited to, the following:

  • Analysis of nonlinear PDEs;
  • Computational methods for mathematical models in real-world problems;
  • Integral transforms for nonlinear problems;
  • Analysis of fractional mathematical models;
  • Special functions in fractional PDEs;
  • Analysis of fractional PDEs;
  • Numeral methods for fractional PDEs;
  • Analysis of general fractional PDEs;
  • Traveling wave solutions for nonlinear PDEs;
  • Similarity solution for nonlinear PDEs;
  • Series expansion method for fractional PDEs;
  • Finite element method for fractional PDEs;
  • Finite difference method for fractional PDEs;
  • Traveling wave solutions for fractional PDEs;
  • Group analysis for fractional PDEs;
  • Reduced differential transform method for fractional PDEs;
  • Differential transform method for fractional PDEs;
  • Residual power series method for fractional PDEs;
  • Homotopy analysis method for fractional PDEs;
  • Decomposition method for fractional PDEs;
  • Homotopy perturbation method for fractional PDEs;
  • Asymptotic perturbation solution for fractional PDEs;
  • Variational iteration method for fractional PDEs;
  • Coupling methods for fractional PDEs.

Prof. Dr. Xiao-Jun Yang
Prof. Dr. Wenxue Zhou
Dr. Sunil Dutt Purohit
Dr. Ali Turab
Dr. Ahmed Refaie Ali
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

  • nonlinear PDEs
  • fractional PDEs
  • general fractional PDEs
  • fractional mathematical models
  • mathematical models in real-world problems
  • integral transforms for nonlinear problems
  • special functions
  • computational methods
  • numeral methods
  • traveling wave solutions
  • similarity solution
  • series expansion method
  • finite element method
  • finite difference method
  • group analysis
  • reduced differential transform method
  • differential transform method
  • residual power series method
  • homotopy analysis method
  • decomposition method
  • homotopy perturbation method
  • asymptotic perturbation solution
  • variational iteration method
  • coupling methods

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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.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • 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 (4 papers)

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Research

20 pages, 1657 KiB  
Article
An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials
by Youssri Hassan Youssri, Waleed Mohamed Abd-Elhameed, Amr Ahmed Elmasry and Ahmed Gamal Atta
Fractal Fract. 2025, 9(2), 78; https://doi.org/10.3390/fractalfract9020078 - 24 Jan 2025
Abstract
The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. [...] Read more.
The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. Full article
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
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23 pages, 17782 KiB  
Article
Discrete Fractional-Order Modeling of Recurrent Childhood Diseases Using the Caputo Difference Operator
by Yasir A. Madani, Zeeshan Ali, Mohammed Rabih, Amer Alsulami, Nidal H. E. Eljaneid, Khaled Aldwoah and Blgys Muflh
Fractal Fract. 2025, 9(1), 55; https://doi.org/10.3390/fractalfract9010055 - 20 Jan 2025
Viewed by 346
Abstract
This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated [...] Read more.
This paper presents a new SIRS model for recurrent childhood diseases under the Caputo fractional difference operator. The existence theory is established using Brouwer’s fixed-point theorem and the Banach contraction principle, providing a comprehensive mathematical foundation for the model. Ulam stability is demonstrated using nonlinear functional analysis. Sensitivity analysis is conducted based on the variation of each parameter, and the basic reproduction number (R0) is introduced to assess local stability at two equilibrium points. The stability analysis indicates that the disease-free equilibrium point is stable when R0<1, while the endemic equilibrium point is stable when R0>1 and otherwise unstable. Numerical simulations demonstrate the model’s effectiveness in capturing realistic scenarios, particularly the recurrent patterns observed in some childhood diseases. Full article
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
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23 pages, 397 KiB  
Article
Investigating a Nonlinear Fractional Evolution Control Model Using W-Piecewise Hybrid Derivatives: An Application of a Breast Cancer Model
by Hicham Saber, Mohammed A. Almalahi, Hussien Albala, Khaled Aldwoah, Amer Alsulami, Kamal Shah and Abdelkader Moumen
Fractal Fract. 2024, 8(12), 735; https://doi.org/10.3390/fractalfract8120735 - 13 Dec 2024
Viewed by 581
Abstract
Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness often limits their ability to accurately represent systems with discontinuous changes or abrupt [...] Read more.
Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness often limits their ability to accurately represent systems with discontinuous changes or abrupt transitions. This paper introduces a novel framework for analyzing nonlinear fractional evolution control systems using piecewise hybrid derivatives with respect to a nondecreasing function W(ι). Building upon the theoretical foundations of piecewise hybrid derivatives, we establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions, leveraging topological degree theory and functional analysis. Our results significantly improve upon existing theoretical understanding by providing less restrictive conditions for stability compared with standard fixed-point theorems. Furthermore, we demonstrate the applicability of our framework through a simulation of breast cancer disease dynamics, illustrating the impact of piecewise hybrid derivatives on the model’s behavior and highlighting advantages over traditional modeling approaches that fail to capture the multi-step nature of the disease. This research provides robust modeling and analysis tools for systems exhibiting multi-step behavior across diverse fields, including engineering, physics, and biology. Full article
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
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29 pages, 6875 KiB  
Article
Quantitative Analysis of the Fractional Fokker–Planck–Levy Equation via a Modified Physics-Informed Neural Network Architecture
by Fazl Ullah Fazal, Muhammad Sulaiman, David Bassir, Fahad Sameer Alshammari and Ghaylen Laouini
Fractal Fract. 2024, 8(11), 671; https://doi.org/10.3390/fractalfract8110671 - 18 Nov 2024
Viewed by 656
Abstract
An innovative approach is utilized in this paper to solve the fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional [...] Read more.
An innovative approach is utilized in this paper to solve the fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional Fokker–Planck–Levy (FFPL) equation numerically. Two scenarios of the FFPL equation are considered by varying the value of (i.e., 1.75, 1.85). Moreover, three cases of each scenario are numerically studied for different discretized domains with 100, 200, and 500 points in x∈[−1, 1] and t∈[0, 1]. For the FFPL equation, solutions are obtained via the FDM-APINN technique via 1000,  2000, and 5000 iterations. The errors, loss function graphs, and statistical tables are presented to validate our claim that the FDM-APINN is a better alternative intelligent technique for handling fractional-order partial differential equations with complex terms. The FDM-APINN can be extended by using nongradient-based bioinspired computing for higher-order fractional partial differential equations. Full article
(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)
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