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Mathematics
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Asymptotic Analysis and Applications
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Asymptotic Analysis and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 October 2025 | Viewed by 4668

Special Issue Editor

Dr. Zhibin Du

E-Mail Website
Guest Editor
School of Software, South China Normal University, Foshan 528225, China
Interests: graph τheory; discrete mathematics; combinatorics; matrix theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

When exact mathematical results are not available, or too unwieldy to be of much use, we have the option of modifying our original orientation to derive just concise and precise estimates of quantities. The methods in asymptotic analysis are incredibly powerful in deriving approximate solutions of problems. This Special Issue aims to collect and disseminate new advances in asymptotic analysis, especially its application in combinatorics.

Dr. Zhibin Du
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

  • analytic functions
  • asymptotic enumeration
  • recurrence relations
  • generating function asymptotics
  • asymptotic expansions
  • asymptotic approximations
  • singularities

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Published Papers (4 papers)

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Research

32 pages, 876 KiB  
Article
Physics-Informed Neural Networks and Fourier Methods for the Generalized Korteweg–de Vries Equation
by Rubén Darío Ortiz Ortiz, Ana Magnolia Marín Ramírez and Miguel Ángel Ortiz Marín
Mathematics 2025, 13(9), 1521; https://doi.org/10.3390/math13091521 - 5 May 2025
Viewed by 332
Abstract
We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg–de Vries (gKdV) equation, focusing on classical Fourier-based Crank–Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes—including the cubic case (ν=3)—and [...] Read more.
We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg–de Vries (gKdV) equation, focusing on classical Fourier-based Crank–Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes—including the cubic case (ν=3)—and diverse initial conditions such as solitons, smooth pulses, discontinuities, and noisy profiles. In addition to pure PINN and spectral models, we propose a novel hybrid PINN–spectral method incorporating a regularization term based on Fourier reference solutions, leading to improved accuracy and stability. Numerical experiments show that while spectral methods achieve superior efficiency in structured domains, PINNs provide flexible, mesh-free alternatives for data-driven and irregular setups. The hybrid model achieves lower relative L2 error and better captures soliton interactions. Our results demonstrate the complementary strengths of spectral and machine learning methods for nonlinear dispersive PDEs. Full article
(This article belongs to the Special Issue Asymptotic Analysis and Applications)
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30 pages, 424 KiB  
Article
Thermoelastic Extensible Timoshenko Beam with Symport Term: Singular Limits, Lack of Differentiability and Optimal Polynomial Decay
by Moncef Aouadi, Taoufik Moulahi and Najmeddine Attia
Mathematics 2025, 13(5), 854; https://doi.org/10.3390/math13050854 - 4 Mar 2025
Cited by 1 | Viewed by 499
Abstract
In this article, we consider the equations of the nonlinear model of a thermoelastic extensible Timoshenko beam, recently derived by Aouadi in the context of Fourier’s law. The new aspect we propose here is to introduce a second sound model in the temperatures [...] Read more.
In this article, we consider the equations of the nonlinear model of a thermoelastic extensible Timoshenko beam, recently derived by Aouadi in the context of Fourier’s law. The new aspect we propose here is to introduce a second sound model in the temperatures which turns into a Gurtin–Pipkin’s model. Thus, the derived equations are physically more realistic since they overcome the property of infinite propagation speed (Fourier’s law property). They are also characterized by the presence of a symport term. Moreover, it is possible to recover the Fourier, Cattaneo and Coleman–Gurtin laws from the derived system by considering a scaled kernel instead of the original kernel through an appropriate singular limit method. The well-posedness of the derived problem is proved by means of the semigroups theory. Then, we show that the associated linear semigroup (without extensibility and with a constant symport term) is not differentiable by an approach based on the Gearhart–Herbst–Prüss–Huang theorem. The lack of analyticity and impossibility of localization of the solutions in time are immediate consequences. Then, by using a resolvent criterion developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the same associated linear semigroup under a condition on the physical coefficients. In particular, we show that the considered problem is not exponentially stable. Moreover, by following a result according to Arendt–Batty, we show that the linear semigroup is strongly stable. Full article
(This article belongs to the Special Issue Asymptotic Analysis and Applications)
8 pages, 255 KiB  
Article
Asymptotic Behaviours for an Index Whittaker Transform over E′(R+)
by Emilio R. Negrín, Jeetendrasingh Maan and Benito J. González
Mathematics 2025, 13(5), 689; https://doi.org/10.3390/math13050689 - 20 Feb 2025
Viewed by 322
Abstract
The objective of this research is to obtain the asymptotic behaviour of compactly supported distributions and generalized functions for a variant of the index Whittaker transform. Full article
(This article belongs to the Special Issue Asymptotic Analysis and Applications)
30 pages, 1121 KiB  
Article
Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks
by Rubén Darío Ortiz Ortiz, Oscar Martínez Núñez and Ana Magnolia Marín Ramírez
Mathematics 2024, 12(21), 3430; https://doi.org/10.3390/math12213430 - 1 Nov 2024
Cited by 1 | Viewed by 2464
Abstract
In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity [...] Read more.
In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations. Full article
(This article belongs to the Special Issue Asymptotic Analysis and Applications)
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