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Axioms
Special Issues
Difference, Functional, and Related Equations
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Difference, Functional, and Related Equations

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

Deadline for manuscript submissions: 29 May 2024 | Viewed by 1978

Special Issue Editors

Dr. Ravi Shankar

E-Mail Website
Guest Editor
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Interests: fully nonlinear elliptic PDEs without uniform ellipticity (sigma-k and special Lagrangian equations); inverse problems of the lens rigidity and Calderón type; symmetries and conservation laws of fluid equations and general PDEs; applied mathematics, including numerical simulations of tsunami waves, singular perturbation theory of thin film PDEs, and nonlocal operators with integrable kernels
Dr. Qun Liu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Interests: stochastic differential equations and their applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to collect and showcase original and interesting results related to difference, functional, stochastic, and related equations with non-local character.  Articles are sought that deepen our understanding of non-local equations and their applicability.  The scope includes, but is not limited to, the following:

  1. Difference equations and related areas such as fractional difference equations, recursion relations, numerical and computational methods for equations, generating functions, and series;
  2. Functional equations and related topics including delay, functional differential, delay differential, fractional functional, and other equations;
  3. Stochastic equations and related topics;
  4. Applications of non-local equations to the natural and social sciences;
  5. Other new aspects and applications of non-local equations.

Dr. Ravi Shankar
Dr. Qun Liu
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

  • difference equations
  • functional equations
  • delay differential equations
  • fractional difference and other equations
  • numerical methods for equations
  • stochastic equation
  • stochastic analysis
  • applications to natural and social sciences

Published Papers (2 papers)

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Research

17 pages, 1459 KiB  
Article
On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach
by Nada A. M. Alshomrani, Abdelhalim Ebaid, Faten Aldosari and Mona D. Aljoufi
Axioms 2024, 13(2), 129; https://doi.org/10.3390/axioms13020129 - 19 Feb 2024
Viewed by 850
Abstract
The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind [...] Read more.
The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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15 pages, 330 KiB  
Article
A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations
by Tucker Hartland and Ravi Shankar
Axioms 2023, 12(11), 1059; https://doi.org/10.3390/axioms12111059 - 18 Nov 2023
Viewed by 840
Abstract
We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we [...] Read more.
We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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