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Mathematics
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Partial Differential Equations: Theory and Applications
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Partial Differential Equations: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Biology".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 8105

Special Issue Editor

Prof. Dr. Mihaela Negreanu

E-Mail Website
Guest Editor
Department of Applied Mathematics and Mathematical Analysis, Complutense University of Madrid, Av. Séneca, 2, 28040 Madrid, Spain
Interests: partial differential equations; mathematical biology; chemotaxis; control theory; optimization; the lubrication theory of fluid dynamics

Special Issue Information

Dear Colleagues,

The present Special Issue will cover most areas of applied mathematics in the theory of partial differential equations, with a special focus on mathematical biology. The purpose of this Special Issue is to present articles describing the latest developments in these fields, including theoretical and numerical/computational aspects.

Topics of interest to this Special Issue include, but are not limited to, numerical methods for partial differential equations, mathematical biology, chemotaxis systems, global existence, the local and global stability of solutions, the study of mixed problems, and recent developments in the field of mathematical modeling in biology.

All interested researchers are kindly invited to contribute to this Special Issue with original research articles, short communications, and review articles.

Dr. Mihaela Negreanu
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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 for partial differential equations
  • partial/ordinary differential equations
  • mathematical biology
  • chemotaxis systems
  • local and global stability
  • asymptotic stability and bifurcation
  • global existence of solutions
  • asymptotic behavior of the solution
  • conditional convergence

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Published Papers (4 papers)

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Research

18 pages, 2048 KiB  
Article
Retrieval of Optical Solitons with Anti-Cubic Nonlinearity
by Muslum Ozisik, Aydin Secer, Mustafa Bayram, Anjan Biswas, Oswaldo González-Gaxiola, Luminita Moraru, Simona Moldovanu, Catalina Iticescu, Dorin Bibicu and Abdulah A. Alghamdi
Mathematics 2023, 11(5), 1215; https://doi.org/10.3390/math11051215 - 1 Mar 2023
Cited by 8 | Viewed by 1661
Abstract
Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by [...] Read more.
Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by using the enhanced modified extended tanh expansion method and a wide range of optical soliton solutions is obtained. Second, the effects of AC parameters on soliton behavior are examined for each obtained soliton type. Methodology: In order to apply the method, the non-linear ordinary differential equation form (NLODE) of the investigated NLSE-AC is obtained by applying the defined wave transformation. Then, with the help of the proposed algorithm for the NLODE form, polynomial form, an algebraic equation system is obtained by setting the coefficients of this form to zero, and the solution of this system is also obtained. After determining the suitable solution set, the optical soliton solution of the investigated problem is obtained with the help of the serial form of the proposed method, a Riccati solution and wave transform. After checking that the solution satisfies the investigated problem, 3D and 2D graphics are obtained for the special parameter values and the necessary comments are made in the relevant sections. Findings: With the proposed method, many optical soliton solutions, such as topological, anti-peaked, combined peaked-bright, combined anti-peaked dark, singular, combined singular-anti peaked, periodic singular, composite kink anti-peaked, kink, periodic and periodic, with different amplitudes are obtained, and 3D and 2D representations have been made. Then, the effect of AC parameters on the soliton behavior in each case has been successfully studied. It has been shown that AC parameters have a significant effect on the soliton behavior, and this effect changes depending on the soliton shape and the parameters. Moreover, providing and maintaining the delicate balance between the soliton shape and the parameters and the interaction of the parameters with each other involves great difficulties. Originality: Although some soliton types of the NLSE-AC equation have been presented for the first time in this study, there is no study in the literature showing the effect of AC parameters on soliton behavior, especially for the NLSE-AC equation. Full article
(This article belongs to the Special Issue Partial Differential Equations: Theory and Applications)
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13 pages, 2246 KiB  
Communication
Solitons in Neurosciences by the Laplace–Adomian Decomposition Scheme
by Oswaldo González-Gaxiola, Anjan Biswas, Luminita Moraru and Abdulah A. Alghamdi
Mathematics 2023, 11(5), 1080; https://doi.org/10.3390/math11051080 - 21 Feb 2023
Cited by 8 | Viewed by 1324
Abstract
The paper concentrates on the solitary waves that are retrievable from the generalized Boussinesq equation. The numerical simulations are displayed in the paper that gives a visual perspective to the model studied in neurosciences. The Laplace–Adomian decomposition scheme makes this visualization of the [...] Read more.
The paper concentrates on the solitary waves that are retrievable from the generalized Boussinesq equation. The numerical simulations are displayed in the paper that gives a visual perspective to the model studied in neurosciences. The Laplace–Adomian decomposition scheme makes this visualization of the solitons possible. The numerical simulations are being reported for the first time using an elegant approach. The results would be helpful for neuroscientists and clinical studies in Medicine. The novelty lies in the modeling that is successfully conducted with an impressively small error measure. In the past, the model was integrated analytically only to recover soliton solutions and its conserved quantities. Full article
(This article belongs to the Special Issue Partial Differential Equations: Theory and Applications)
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19 pages, 318 KiB  
Article
Regularity, Asymptotic Solutions and Travelling Waves Analysis in a Porous Medium System to Model the Interaction between Invasive and Invaded Species
by José Luis Díaz Palencia, Julián Roa González, Saeed Ur Rahman and Antonio Naranjo Redondo
Mathematics 2022, 10(7), 1186; https://doi.org/10.3390/math10071186 - 5 Apr 2022
Cited by 6 | Viewed by 1665
Abstract
This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use [...] Read more.
This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the self-similar structure that permits showing the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive species predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support). Full article
(This article belongs to the Special Issue Partial Differential Equations: Theory and Applications)
9 pages, 389 KiB  
Article
Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method
by Jesús Flores, Ángel García, Mihaela Negreanu, Eduardo Salete, Francisco Ureña and Antonio M. Vargas
Mathematics 2022, 10(3), 332; https://doi.org/10.3390/math10030332 - 21 Jan 2022
Cited by 4 | Viewed by 2503
Abstract
The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method [...] Read more.
The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method. Full article
(This article belongs to the Special Issue Partial Differential Equations: Theory and Applications)
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