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Mathematics
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Applications of Differential Equations in Sciences
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Applications of Differential Equations in Sciences

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

Deadline for manuscript submissions: 30 June 2025 | Viewed by 5417

Special Issue Editors

Prof. Dr. Angela Slavova

E-Mail Website
Guest Editor
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, Sofia 1113, Bulgaria
Interests: cellular neural networks; differential equations; modeling; numerical methods
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Nikolay K. Vitanov

E-Mail Website
Guest Editor
1. Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, Sofia 1113, Bulgaria
2. Climate, Atmosphere and Water Research Institute, Bulgarian Academy of Sciences, Blvd., Tzarigradsko Chaussee 66, 1784 Sofia, Bulgaria
Interests: nonlinear dynamics; nonlinear time series analysis; fluid mechanics; nonlinear partial differential equations; application of the methods of statistics and probability theory to natural, social and economic systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Elena V. Nikolova

E-Mail Website
Guest Editor
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, Sofia 1113, Bulgaria
Interests: mathematical modeling; theory of dynamical systems; theory of non-linear waves

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to many applications of differential equations in different fields of science. Several phenomena in nature (physics, chemistry, and biology) and society (economics) result in problems leading to the study of linear and nonlinear differential equations.

It is known that many natural phenomena are not static, but depend both on the instantaneous values of given quantities and the type of their change. Such phenomena must be described mathematically with differential equations and the mathematical models built with their help. In this way, many of the fundamental laws of physics and chemistry are defined, and in biology and economics, differential equations model the behavior of systems of great complexity. In many cases, completely different problems from unrelated scientific fields can be reduced to the same differential equations. For example, the propagation of light and sound in air and of waves on a water surface can be described by the same partial differential equation: the wave equation. Heat transfer, the theory developed by Joseph Fourier at the beginning of the 19th century, is described by another partial differential equation of the second order: heat conduction. Subsequently, it turns out that many other processes can be described with the same or similar equations, such as Brownian motion (Fokker–Planck equation) or the behavior of financial markets in the Black–Scholes model.

The main topics of this Special Issue are:

  • Applications in mathematical physics;
  • Applications in mechanics;
  • Applications in fractional calculus;
  • Applications in financial mathematics;
  • Applications in mathematical biology;
  • Applications in numerical methods and computer science.

Prof. Dr. Angela Slavova
Prof. Dr. Nikolay K. Vitanov
Prof. Dr. Elena V. Nikolova
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. 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

  • differential equations
  • mathematical physics
  • mechanics
  • financial mathematics
  • fractional calculus
  • mathematical biology
  • numerical methods
  • neuroscience

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Published Papers (5 papers)

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Research

15 pages, 997 KiB  
Article
Conservation Law Analysis in Numerical Schema for a Tumor Angiogenesis PDE System
by Pasquale De Luca and Livia Marcellino
Mathematics 2025, 13(1), 28; https://doi.org/10.3390/math13010028 - 25 Dec 2024
Viewed by 456
Abstract
Tumor angiogenesis, the formation of new blood vessels from pre-existing vasculature, is a crucial process in cancer growth and metastasis. Mathematical modeling through partial differential equations helps to understand this complex biological phenomenon. Here, we provide a conservation properties analysis in a tumor [...] Read more.
Tumor angiogenesis, the formation of new blood vessels from pre-existing vasculature, is a crucial process in cancer growth and metastasis. Mathematical modeling through partial differential equations helps to understand this complex biological phenomenon. Here, we provide a conservation properties analysis in a tumor angiogenesis model describing the evolution of endothelial cells, proteases, inhibitors, and extracellular matrix. The adopted approach introduces a numerical framework that combines spatial and time discretization techniques. Here, we focus on maintaining solution accuracy while preserving physical quantities during the simulation process. The method achieved second-order accuracy in both space and time discretizations, with conservation errors showing consistent convergence as the mesh was refined. The numerical schema demonstrates stable wave propagation patterns, in agreement with experimental observations. Numerical experiments validate the approach and demonstrate its reliability for long-term angiogenesis simulations. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
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16 pages, 1107 KiB  
Article
Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations
by Junseok Kim
Mathematics 2024, 12(23), 3796; https://doi.org/10.3390/math12233796 - 30 Nov 2024
Viewed by 539
Abstract
In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such [...] Read more.
In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
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9 pages, 14083 KiB  
Article
Erroneous Applications of Fractional Calculus: The Catenary as a Prototype
by Gerardo Becerra-Guzmán and José Villa-Morales
Mathematics 2024, 12(14), 2148; https://doi.org/10.3390/math12142148 - 9 Jul 2024
Viewed by 1507
Abstract
In this work, we study the equation of the catenary curve in the context of the Caputo derivative. We solve this equation and compare the solution with real physical models. From the experiments, we find that the best approximation is achieved in the [...] Read more.
In this work, we study the equation of the catenary curve in the context of the Caputo derivative. We solve this equation and compare the solution with real physical models. From the experiments, we find that the best approximation is achieved in the classical case. Therefore, introducing a fractional parameter arbitrarily can be detrimental. However, we observe that, when adding a certain weight to the chain, fractional calculus produces better results than classical calculus for modeling the minimum height. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
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14 pages, 300 KiB  
Article
Spatial Decay Estimates and Continuous Dependence for the Oldroyd Fluid
by Yuanfei Li
Mathematics 2024, 12(8), 1240; https://doi.org/10.3390/math12081240 - 19 Apr 2024
Viewed by 1051
Abstract
This article investigates the Oldroyd fluid, which is widely used in industrial and engineering environments. When the Oldroyd fluid passes through a three-dimensional semi-infinite cylinder, the asymptotic properties of the solutions are established. On this basis, we also studied the continuous dependence of [...] Read more.
This article investigates the Oldroyd fluid, which is widely used in industrial and engineering environments. When the Oldroyd fluid passes through a three-dimensional semi-infinite cylinder, the asymptotic properties of the solutions are established. On this basis, we also studied the continuous dependence of the viscosity coefficient. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
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24 pages, 375 KiB  
Article
Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form
by Petar Popivanov and Angela Slavova
Mathematics 2024, 12(7), 1003; https://doi.org/10.3390/math12071003 - 27 Mar 2024
Viewed by 1056
Abstract
In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the [...] Read more.
In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
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