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

Deadline for manuscript submissions: closed (16 February 2020) | Viewed by 17094

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Prof. Dr. Philip Broadbridge

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Guest Editor

Department of Mathematics and Statistics, La Trobe University Melbourne, Melbourne, VIC 3086, Australia
Interests: concepts of entropy and their applications; nonlinear diffusion
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Roman M. Cherniha

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Guest Editor

Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs'ka Street, 01004 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, Collections and Topics 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 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. 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 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.

Published Papers (6 papers)

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16 pages, 311 KiB

Open AccessArticle

Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

by Lina Ji and Rui Wang

Entropy 2020, 22(8), 873; https://doi.org/10.3390/e22080873 - 08 Aug 2020

Cited by 1 | Viewed by 1758

Abstract

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry. Full article

(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)

8 pages, 263 KiB

Open AccessArticle

Stability Analysis of the Explicit Difference Scheme for Richards Equation

by Fengnan Liu, Yasuhide Fukumoto and Xiaopeng Zhao

Entropy 2020, 22(3), 352; https://doi.org/10.3390/e22030352 - 18 Mar 2020

Cited by 4 | Viewed by 3199

Abstract

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency. Full article

(This article belongs to the Special Issue Applications of Nonlinear Diffusion Equations)

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18 pages, 1585 KiB

Open AccessArticle

Exploring Nonlinear Diffusion Equations for Modelling Dye-Sensitized Solar Cells

by Benjamin Maldon, Ngamta Thamwattana and Maureen Edwards

Entropy 2020, 22(2), 248; https://doi.org/10.3390/e22020248 - 21 Feb 2020

Cited by 8 | Viewed by 3083

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 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)

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31 pages, 1612 KiB

Open AccessArticle

Spherically Restricted Random Hyperbolic Diffusion

by Philip Broadbridge, Alexander D. Kolesnik, Nikolai Leonenko, Andriy Olenko and Dareen Omari

Entropy 2020, 22(2), 217; https://doi.org/10.3390/e22020217 - 14 Feb 2020

Cited by 11 | Viewed by 2459

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)

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11 pages, 338 KiB

Open AccessArticle

Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers

by Roman Cherniha and Vasyl’ Davydovych

Entropy 2020, 22(2), 154; https://doi.org/10.3390/e22020154 - 28 Jan 2020

Cited by 4 | Viewed by 2527

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 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)

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18 pages, 2744 KiB

Open AccessArticle

Spatiotemporal Evolution of a Landslide: A Transition to Explosive Percolation

by Kushwant Singh and Antoinette Tordesillas

Entropy 2020, 22(1), 67; https://doi.org/10.3390/e22010067 - 04 Jan 2020

Cited by 7 | Viewed by 3187

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. 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)

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