Special Issue "Applications of Nonlinear Diffusion Equations"

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (16 February 2020).

Special Issue Editors

Prof. Dr. Roman M. Cherniha
E-Mail Website
Guest Editor
Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs'ka Street, 01601 Kyiv, Ukraine
Interests: Non-linear PDEs: Lie and conditional symmetries, exact solutions and their properties; Application of symmetry-based methods for analytical solving nonlinear initial and boundary value problems arising in mathematical physics and mathematical biology
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear diffusion equations occur widely in the modelling of phenomena that invariably involve irreversible processes. Irreversibility may be signified by some time-monotonic function or “entropy” on the space of state functions. We welcome contributions that have some reference to real irreversible systems whose state functions involve dependence on both space and time variables or their analogues (e.g., age of individuals). Such systems may include but are not limited to heat transfer, solute transport, mixing processes, evolution of solid surfaces and crystal defects, cell migration, tumour growth, population dynamics, disease transmission, and population genetics. “Nonlinear” is a key word, but linear models may be used if the effects of nonlinear extensions are also discussed. Within this field, analysis of the properties of practical nonlinear diffusion equations and approaches to their solution remain important.

Prof. Dr. Philip Broadbridge
Prof. Dr. Roman M. Cherniha
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 papers will be 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. Entropy 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 1600 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.

Published Papers (4 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Open AccessArticle
Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells
Entropy 2020, 22(2), 248; https://doi.org/10.3390/e22020248 - 21 Feb 2020
Abstract
Dye-sensitized solar cells offer an alternative source for renewable energy by means of converting sunlight into electricity. While there are many studies concerning the development of DSSCs, comprehensive mathematical modelling of the devices is still lacking. Recent mathematical models are based on diffusion [...] Read more.
Dye-sensitized solar cells offer an alternative source for renewable energy by means of converting sunlight into electricity. While there are many studies concerning the development of DSSCs, comprehensive mathematical modelling of the devices is still lacking. Recent mathematical models are based on diffusion equations of electron density in the conduction band of the nano-porous semiconductor in dye-sensitized solar cells. Under linear diffusion and recombination, this paper provides analytical solutions to the diffusion equation. Further, Lie symmetry analysis is adopted in order to explore analytical solutions to physically relevant special cases of the nonlinear diffusion equations. While analytical solutions may not be possible, we provide numerical solutions, which are in good agreement with the results given in the literature. Full article
(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)
Open AccessArticle
Spherically Restricted Random Hyperbolic Diffusion
Entropy 2020, 22(2), 217; https://doi.org/10.3390/e22020217 - 14 Feb 2020
Abstract
This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power [...] Read more.
This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings. Full article
(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)
Show Figures

Figure 1

Open AccessArticle
Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers
Entropy 2020, 22(2), 154; https://doi.org/10.3390/e22020154 - 28 Jan 2020
Abstract
The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and [...] Read more.
The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested. Full article
(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)
Open AccessArticle
Spatiotemporal Evolution of a Landslide: A Transition to Explosive Percolation
Entropy 2020, 22(1), 67; https://doi.org/10.3390/e22010067 - 04 Jan 2020
Abstract
Patterns in motion characterize failure precursors in granular materials. Currently, a broadly accepted method to forecast granular failure from data on motion is still lacking; yet such data are being generated by remote sensing and imaging technologies at unprecedented rates and unsurpassed resolution. [...] Read more.
Patterns in motion characterize failure precursors in granular materials. Currently, a broadly accepted method to forecast granular failure from data on motion is still lacking; yet such data are being generated by remote sensing and imaging technologies at unprecedented rates and unsurpassed resolution. Methods that deliver timely and accurate forecasts on failure from such data are urgently needed. Inspired by recent developments in percolation theory, we map motion data to time-evolving graphs and study their evolution through the lens of explosive percolation. We uncover a critical transition to explosive percolation at the time of imminent failure, with the emerging connected components providing an early prediction of the location of failure. We demonstrate these findings for two types of data: (a) individual grain motions in simulations of laboratory scale tests and (b) ground motions in a real landslide. Results unveil spatiotemporal dynamics that bridge bench-to-field signature precursors of granular failure, which could help in developing tools for early warning, forecasting, and mitigation of catastrophic events like landslides. Full article
(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)
Show Figures

Figure 1

Back to TopTop