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Mathematics
Special Issues
Differential Equations Applied in Fluid Dynamics
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Differential Equations Applied in Fluid Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 October 2025) | Viewed by 462

Special Issue Editor

Dr. Imre Ferenc Barna

E-Mail Website
Guest Editor
Hungarian Research Network, Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29–33, 1121 Budapest, Hungary
Interests: analytic solutions of partial differential equations of flows; laser–matter interaction
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We warmly invite you to submit your research to the Special Issue entitled “Differential Equations Applied in Fluid Dynamics”.

It is well known that most nonlinear partial differential equations (PDEs) lack a general rigorous mathematical theory. However, through various symmetry reductions, we can construct special exact solutions that reflect long-term behaviors or other global properties. Symmetry plays a crucial role in the analysis of dynamical systems, and it is often possible to express these solutions explicitly in terms of elementary or special functions, frequently appearing in highly symmetric forms. Deriving and understanding new analytic solutions remains an intellectual challenge in mathematics and mathematical physics.

These principles are particularly significant in fluid mechanics. This Special Issue seeks theoretical or numerical analyses of special solutions with symmetry assumptions for nonlinear partial (or ordinary) differential systems, such as those arising in fluid mechanics (e.g., the Navier–Stokes equations, Euler equations, Euler–Poisson equations, and their special cases). Additionally, contributions addressing solutions from general relativity (e.g., the Einstein field equations and their special cases) or other nontrivial nonlinear PDEs are welcome, provided they establish a clear connection to fluid dynamics.

We encourage you to submit your paper to the journal Mathematics and select the Special Issue “Differential Equations Applied in Fluid Dynamics” via the MDPI submission system. Papers will be published on a rolling basis, and we look forward to receiving your submission at your earliest convenience.

Dr. Imre Ferenc Barna
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

  • fluid dynamics
  • symmetry transformations
  • differential equations

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Published Papers (1 paper)

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Research

39 pages, 504 KB  
Article
Geophysical Monge–Ampère-Type Equation: Symmetries and Exact Solutions
by Andrei D. Polyanin and Alexander V. Aksenov
Mathematics 2025, 13(21), 3522; https://doi.org/10.3390/math13213522 - 3 Nov 2025
Viewed by 139
Abstract
This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for [...] Read more.
This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for the first time. An invariant point transformation is found that depends on fourteen arbitrary constants and preserves the form of the equation under consideration. One-dimensional symmetry reductions leading to self-similar and some other invariant solutions that described by single ODEs are considered. Using the methods of generalized and functional separation of variables, as well as the principle of structural analogy of solutions, a large number of new non-invariant closed-form solutions are obtained. In general, the extensive list of all exact solutions found includes more than thirty solutions that are expressed in terms of elementary functions. Most of the obtained solutions contain a number of arbitrary constants, and several solutions additionally include two arbitrary functions. Two-dimensional reductions are considered that reduce the original PDE in three independent variables to a single simpler PDE in two independent variables (including linear wave equations, the Laplace equation, the Tricomi equation, and the Guderley equation) or to a system of such PDEs. A number of specific examples demonstrate that the type of the mixed, highly nonlinear PDE under consideration, depending on the choice of its specific solutions, can be either hyperbolic or elliptic. To analyze the equation and construct exact solutions and reductions, in addition to Cartesian coordinates, polar, generalized polar, and special Lorentz coordinates are also used. In conclusion, possible promising directions for further research of the highly nonlinear PDE under consideration and related PDEs are formulated. It should be noted that the described symmetries, transformations, reductions, and solutions can be utilized to determine the error and estimate the limits of applicability of numerical and approximate analytical methods for solving complex problems of mathematical physics with highly nonlinear PDEs. Full article
(This article belongs to the Special Issue Differential Equations Applied in Fluid Dynamics)
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