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: 26 November 2024 | Viewed by 3162

Special Issue Editors


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
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

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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 (5 papers)

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Research

14 pages, 286 KiB  
Article
Some Results of Stochastic Differential Equations
by Shuai Guo, Wei Li and Guangying Lv
Axioms 2024, 13(6), 405; https://doi.org/10.3390/axioms13060405 - 16 Jun 2024
Viewed by 224
Abstract
In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to [...] Read more.
In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
12 pages, 713 KiB  
Article
Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation
by Faten Aldosari and Abdelhalim Ebaid
Axioms 2024, 13(6), 377; https://doi.org/10.3390/axioms13060377 - 4 Jun 2024
Viewed by 179
Abstract
This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be [...] Read more.
This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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21 pages, 354 KiB  
Article
Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities
by Rakhmatillo Djuraevich Aloev, Abdumauvlen Suleimanovich Berdyshev, Vasila Alimova and Kymbat Slamovna Bekenayeva
Axioms 2024, 13(5), 334; https://doi.org/10.3390/axioms13050334 - 17 May 2024
Viewed by 396
Abstract
In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is [...] Read more.
In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in 2-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in 2-norm with respect to a discrete perturbation is proved. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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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 981
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 928
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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