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Axioms
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Advances in Differential Equations and Its Applications
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Advances in Differential Equations and Its Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 4635

Special Issue Editors

Prof. Dr. Andrey Zahariev

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; functional-differential equations; impulsive differential equations; differential equations in Banach spaces; integral equations; integral inequalities; stability analysis; real and functional analysis; applied mathematics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hristo Kiskinov

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria
Interests: fractional differential equations; impulsive differential equations; functional-differential equations; differential equations in Banach spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The present Special Issue is devoted to new research in the area of differential equations (or systems) with fractional- or integer-order derivatives without delay or with a delayed argument (of a retarded or neutral type). Works devoted to stochastic differential equations with fractional derivatives are welcome too. The works can focus on aspects of fundamental theory such as initial value problems, boundary value problems, and different kinds of integral representations of the solutions and their continuous dependence on the given data. We also encourage works in the field of the qualitative theory, such as the asymptotic behaviour of solutions with various types of stability properties, namely Lyapunov’s type, finite time stability, Mittag–Leffler stability, robust stability, Ulam–Hyers–Rassias stability, etc. Works which contain results which are general to fractional and integer-based differential equations, as well as those which establish specific results for these classes, are also invited. Furthermore, works containing new models or including fractional variants of well-known classical models in different areas of sciences such as economics, physics, engineering, etc., will be met with increased interest.

We look forward to receiving your contributions.

Prof. Dr. Andrey Zahariev
Prof. Dr. Hristo Kiskinov
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. Axioms 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 2400 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

  • functional differential equations/systems
  • fractional differential equations/systems
  • fractional derivatives of a constant order
  • distributed-order fractional derivatives
  • variable-order fractional derivatives
  • stability analysis of fractional-order systems
  • asymptotic properties of fractional-order systems

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Published Papers (6 papers)

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Research

23 pages, 613 KiB  
Article
Application of Triple- and Quadruple-Generalized Laplace Transform to (2+1)- and (3+1)-Dimensional Time-Fractional Navier–Stokes Equation
by Hassan Eltayeb Gadain and Said Mesloub
Axioms 2024, 13(11), 780; https://doi.org/10.3390/axioms13110780 - 12 Nov 2024
Viewed by 328
Abstract
In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match [...] Read more.
In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match with the exact solutions of the problems, which proves that, as the terms of the series increase, the approximate solutions are closer to the exact solutions of each problem. To verify the appropriateness of these methods, some examples are offered. The TGLTDM and FGLTDM results indicate that the suggested methods have higher evaluation convergence as compared to the ADM and HPM. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
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14 pages, 317 KiB  
Article
Limit Property of an L2-Normalized Solution for an L2-Subcritical Kirchhoff-Type Equation with a Variable Exponent
by Xincai Zhu and Hanxiao Wu
Axioms 2024, 13(9), 571; https://doi.org/10.3390/axioms13090571 - 23 Aug 2024
Viewed by 449
Abstract
This paper is concerned with the following L2-subcritical Kirchhoff-type equation [...] Read more.
This paper is concerned with the following L2-subcritical Kirchhoff-type equation −a+b∫R2|∇u|2dxsΔu+V(x)u=μu+β|u|2u,x∈R2, with ∫R2|u|2dx=1. We give a detailed analysis of the limit property of the L2-normalized solution when exponent s tends toward 0 from the right (i.e., s↘0). Our research extends previous works, in which the authors have displayed the limit behavior of L2-normalized solutions when s=1 as a↘0 or b↘0. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
11 pages, 1330 KiB  
Article
Uncertainty Degradation Model for Initiating Explosive Devices Based on Uncertain Differential Equations
by Changli Ma, Li Jia and Meilin Wen
Axioms 2024, 13(7), 449; https://doi.org/10.3390/axioms13070449 - 3 Jul 2024
Viewed by 468
Abstract
The performance degradation of initiating explosive devices is influenced by various internal and external factors, leading to uncertainties in their reliability and lifetime predictions. This paper proposes an uncertain degradation model based on uncertain differential equations, utilizing the Liu process to characterize the [...] Read more.
The performance degradation of initiating explosive devices is influenced by various internal and external factors, leading to uncertainties in their reliability and lifetime predictions. This paper proposes an uncertain degradation model based on uncertain differential equations, utilizing the Liu process to characterize the volatility in degradation rates. The ignition delay time is selected as the primary performance parameter, and the uncertain distributions, expected values and confidence intervals are derived for the model. Moment estimation techniques are employed to estimate the unknown parameters within the model. A real data analysis of ignition delay times under accelerated storage conditions demonstrates the practical applicability of the proposed method. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
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11 pages, 273 KiB  
Article
A Study of Structural Stability on the Bidispersive Flow in a Semi-Infinite Cylinder
by Yuanfei Li
Axioms 2024, 13(7), 431; https://doi.org/10.3390/axioms13070431 - 27 Jun 2024
Viewed by 724
Abstract
We consider the bidispersive flow with nonlinear boundary conditions in a bounded region. By using the differential inequality technique, we get the bound for the L4-norm of the salinity which plays an important role. The continuous dependence and the convergence results [...] Read more.
We consider the bidispersive flow with nonlinear boundary conditions in a bounded region. By using the differential inequality technique, we get the bound for the L4-norm of the salinity which plays an important role. The continuous dependence and the convergence results on the Soret coefficient are established. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
16 pages, 310 KiB  
Article
Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term
by Xincai Zhu and Chunxia He
Axioms 2024, 13(5), 336; https://doi.org/10.3390/axioms13050336 - 20 May 2024
Viewed by 710
Abstract
This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function [...] Read more.
This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function V(x) fulfills an appropriate choice. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
23 pages, 5971 KiB  
Article
Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics
by Sydney Day, Zhidong Xiao, Ehtzaz Chaudhry, Matthew Hooker, Xiaoqiang Zhu, Jian Chang, Andrés Iglesias, Lihua You and Jianjun Zhang
Axioms 2024, 13(3), 185; https://doi.org/10.3390/axioms13030185 - 12 Mar 2024
Viewed by 1437
Abstract
How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this [...] Read more.
How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
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