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Mathematics
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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: 31 December 2026 | Viewed by 3935

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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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 (3 papers)

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Research

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38 pages, 714 KB  
Article
Reduced Integer–Fractional Dynamics of Hydrothermal Memory in Volcanic Gas and Isotope Signals
by Sebastiano Ettore Spoto
Mathematics 2026, 14(12), 2139; https://doi.org/10.3390/math14122139 - 15 Jun 2026
Cited by 2 | Viewed by 262
Abstract
Volcanic gas and isotope time series are indirect observables of coupled magmatic and hydrothermal dynamics. We formulate a reduced integer–fractional model in which ordinary differential equations describe deep recharge, pressure, gas-phase volatile inventory, and source mixing, whereas Caputo equations describe shallow hydrothermal pressure, [...] Read more.
Volcanic gas and isotope time series are indirect observables of coupled magmatic and hydrothermal dynamics. We formulate a reduced integer–fractional model in which ordinary differential equations describe deep recharge, pressure, gas-phase volatile inventory, and source mixing, whereas Caputo equations describe shallow hydrothermal pressure, thermal excess, gas pathway effectiveness, permeability, and scrubbing. Under explicit local regularity and admissibility assumptions, the mixed-order Volterra problem is locally well-posed and the physically admissible state set is positively invariant. We derive componentwise dissipative estimates and state conditions for global continuation under bounded trajectories and analyze finite-interval consistency with the integer-order limit and local stability of a frozen commensurate hydrothermal linearization. Conservative observation equations link hidden states to gas ratios, fluxes, and isotope ratios. The inverse problem is treated diagnostically; global identifiability is not claimed. Local sensitivity screening, Fisher information concepts, and scalar recovery tests are used only as preliminary local diagnostics of information content under known or misspecified forcing. Synthetic demonstrations and a reference forward solver illustrate how hydrothermal memory and sulfur scrubbing can reshape carbon dioxide/sulfur dioxide (CO2/SO2) anomalies before site-specific calibration. Full article
(This article belongs to the Special Issue Differential Equations Applied in Fluid Dynamics)
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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
Cited by 1 | Viewed by 2090
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)

Review

Jump to: Research

25 pages, 2644 KB  
Review
Compact Finite Difference Schemes: A Review of Fundamentals, Applications, and Practical Implementation
by Andrea Arroyo Ramo, J. Alberto Conejero, María Jezabel Perez-Quiles and Sergio Hoyas
Mathematics 2026, 14(11), 1958; https://doi.org/10.3390/math14111958 - 3 Jun 2026
Viewed by 729
Abstract
Compact finite difference schemes approximate spatial derivatives through implicit relations between neighboring grid points. Despite using compact stencils and relatively simple algebraic structures, these schemes achieve high-order accuracy and spectral-like resolution, reducing dispersion errors while maintaining low numerical dissipation. These properties make them [...] Read more.
Compact finite difference schemes approximate spatial derivatives through implicit relations between neighboring grid points. Despite using compact stencils and relatively simple algebraic structures, these schemes achieve high-order accuracy and spectral-like resolution, reducing dispersion errors while maintaining low numerical dissipation. These properties make them particularly attractive for problems requiring accurate spatial derivatives and computational efficiency, such as wave propagation, aeroacoustics, and turbulent flow simulations. This review presents the main ideas behind compact finite difference schemes, including their derivation from Taylor expansions and Padé approximations, their accuracy properties, and their resolution characteristics through modified wavenumber analysis. The manuscript is intended as a review and practical synthesis, rather than as the proposal of a new numerical scheme, and aims to connect the theoretical construction of compact schemes with their numerical behavior, practical implementation, and representative applications. To support reproducibility, we provide a fully documented open-source Python 3.11 notebook with a reference implementation of the schemes discussed in the paper. The examples include first- and second-order derivative calculations and representative one- and two-dimensional boundary-value problems, including Helmholtz-type equations. Finally, we survey applications across computational fluid dynamics, acoustics, geophysical flows, structural mechanics, biology, electromagnetism, and quantitative finance. Full article
(This article belongs to the Special Issue Differential Equations Applied in Fluid Dynamics)
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