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Fractal Fract
Special Issues
Advances in Nonlinear Differential Equations with Applications
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Advances in Nonlinear Differential Equations with Applications

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

Deadline for manuscript submissions: closed (20 February 2023) | Viewed by 7600

Special Issue Editors

Prof. Dr. Chao-Qing Dai

E-Mail Website
Guest Editor
School of Optical, Mechanical and Electrical Engineering, Zhejiang Agriculture and Forestry University, Hangzhou 311300, China
Interests: nonlinear wave; optical soliton; matter wave; integrable theory; neural network; deep learning; fiber laser
Prof. Dr. Wen-Jun Liu

E-Mail Website
Guest Editor
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Interests: nonlinear optics; fiber laser; optical solitons
Prof. Dr. Nikolay A. Kudryashov

E-Mail Website
Guest Editor
Department of Applied Mathematics, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia
Interests: nonlinear differential equation; Painleve equations; integrability and nonintegrability; exact solution; soliton; special functions; nonlinear mathematical model; point vortices; numerical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear differential equations show important and strong abilities to govern enormous physical, biological, and engineering practical problems. Abundant solutions and the related localized structures of nonlinear differential equations play important roles in many applied sciences, including condensed matter physics, nonlinear optics, biology physics, hydrodynamics, and other fields. Therefore, the continued development of more advanced, effective methods to solve nonlinear differential equations is of paramount importance.

This Special Issue, entitled “Advances in Nonlinear Differential Equations with Applications”, aims to reflect the current state-of-the-art developments in all topics relevant to nonlinear differential equations and their applications in various applied sciences. We encourage all researchers working in areas of knowledge related to nonlinear differential equations to submit the details of their research. We hope this Special Issue will provide a comprehensive background for engineers, researchers, and experts in the field. The topics to be considered in this Special Issue include, but are not limited to, the following:

  • Nonlinear differential equations and their fractional forms in physics, optical engineering, and biology;
  • Nonlinear difference-differential equations in physics, optical engineering, and biology;
  • Abundant solutions and the related localized structures of nonlinear differential equations;
  • Data-driven solutions to nonlinear differential equations via deep learning;
  • New methods to solve nonlinear differential equations;
  •  Symmetry of differential equations and its application;
  • Nonlinear differential equations and their solutions in other applied sciences.

Prof. Dr. Chao-Qing Dai
Prof. Dr. Wen-Jun Liu
Prof. Dr. Nikolay A. Kudryashov
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 differential equation
  • fractional nonlinear differential equation
  • nonlinear difference-differential equation
  • soliton
  • rogue wave
  • exact solution
  • approximate solution
  • integrable methods
  • machine learning

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

Published Papers (4 papers)

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Research

14 pages, 468 KiB  
Article
On Exact Solutions of Some Space–Time Fractional Differential Equations with M-truncated Derivative
by Ayten Özkan, Erdoĝan Mehmet Özkan and Ozgur Yildirim
Fractal Fract. 2023, 7(3), 255; https://doi.org/10.3390/fractalfract7030255 - 10 Mar 2023
Cited by 12 | Viewed by 1636
Abstract
In this study, the extended G′/G method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. [...] Read more.
In this study, the extended G′/G method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. A symbolic computation program called Maple was used to implement the method in a dependable and effective way. There are also a few graphs provided for the solutions. Using the suggested method to solve these equations, we have provided many new exact solutions that are distinct from those previously found. By offering insightful explanations of many nonlinear systems, the study’s findings add to the body of literature. The results revealed that the suggested method is a valuable mathematical tool and that using a symbolic computation program makes these tasks simpler, more dependable, and quicker. It is worth noting that it may be used for a wide range of nonlinear evolution problems in mathematical physics. The study’s findings may have an influence on how different physical problems are interpreted. Full article
(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)
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18 pages, 577 KiB  
Article
Deflection of Beams Modeled by Fractional Differential Equations
by José Villa-Morales, Luz Judith Rodríguez-Esparza and Manuel Ramírez-Aranda
Fractal Fract. 2022, 6(11), 626; https://doi.org/10.3390/fractalfract6110626 - 27 Oct 2022
Cited by 6 | Viewed by 1775
Abstract
Using the concept of a fractional derivative, in Caputo’s sense, we derive and solve a fractional differential equation that models the deflection of beams. The scheme to introduce the fractional concept can be used for different situations; in the article, we only consider [...] Read more.
Using the concept of a fractional derivative, in Caputo’s sense, we derive and solve a fractional differential equation that models the deflection of beams. The scheme to introduce the fractional concept can be used for different situations; in the article, we only consider four cases as an example of its usefulness. In addition, we establish a relationship between the fractional index and the level of stiffness (or flexibility) of the material with which the beam is made. Full article
(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)
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10 pages, 1598 KiB  
Article
Local Fractional Homotopy Perturbation Method for Solving Coupled Sine-Gordon Equations in Fractal Domain
by Liguo Chen and Quansheng Liu
Fractal Fract. 2022, 6(8), 404; https://doi.org/10.3390/fractalfract6080404 - 22 Jul 2022
Cited by 6 | Viewed by 1436
Abstract
In this paper, the coupled local fractional sine-Gordon equations are studied in the range of local fractional derivative theory. The study of exact solutions of nonlinear coupled systems is of great significance for understanding complex physical phenomena in reality. The main method used [...] Read more.
In this paper, the coupled local fractional sine-Gordon equations are studied in the range of local fractional derivative theory. The study of exact solutions of nonlinear coupled systems is of great significance for understanding complex physical phenomena in reality. The main method used in this paper is the local fractional homotopy perturbation method, which is used to analyze the exact traveling wave solutions of generalized nonlinear systems defined on the Cantor set in the fractal domain. The fractal wave with fractal dimension ε=ln2/ln3 is numerically simulated. Through numerical simulation, we find that the obtained solutions are of great significance to explain some practical physical problems. Full article
(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)
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16 pages, 2301 KiB  
Article
Second Derivative Block Hybrid Methods for the Numerical Integration of Differential Systems
by Dauda Gulibur Yakubu, Ali Shokri, Geoffrey Micah Kumleng and Daniela Marian
Fractal Fract. 2022, 6(7), 386; https://doi.org/10.3390/fractalfract6070386 - 10 Jul 2022
Cited by 5 | Viewed by 1635
Abstract
The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of [...] Read more.
The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators. Full article
(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)
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