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Axioms
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Numerical and Analytical Methods for Partial Differential Equations with Integral...
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Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions

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

Deadline for manuscript submissions: 31 May 2026 | Viewed by 2637

Special Issue Editors

Prof. Dr. Geni Gupur

E-Mail Website
Guest Editor
College of Mathematics and Systems Science, Xinjiang University, Ürümqi, China
Interests: dynamics of queueing models; dynamics of reliability models; time-dependent solution; asymptotic behavior; operator semigroup; spectral theory
Dr. Zacharias A. Anastassi

E-Mail Website
Guest Editor
Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Interests: numerical analysis; numerical solutions of initial/boundary value problems; development and analysis of numerical algorithms
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue contains several relevant contributions to the study of partial differential equations with integral boundary conditions appearing in queueing theory, reliability theory, age-structured population dynamics, and age-structured epidemic dynamics. The papers are concerned with the existence and uniqueness of the time-dependent (dynamic) solution, the asymptotic behavior (asymptotic stability) of the time-dependent (dynamic) solution, the structure of the time-dependent solution, the asymptotic behavior of the system indices (time-dependent queueing length, time-dependent availability, etc.), and spectral analysis of the underlying operators which correspond to above partial-differential equations, as well as with numerical approaches.

Prof. Dr. Geni Gupur
Dr. Zacharias A. Anastassi
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 250 words) can be sent to the Editorial Office for assessment.

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

  • time-dependent solution
  • asymptotic behavior
  • operator semigroup
  • resolvent set
  • spectrum

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Published Papers (5 papers)

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Research

35 pages, 5499 KB  
Article
On the Complex Spectrum of the Underlying Operator of a Reliability Model
by Zhiyang Du and Geni Gupur
Axioms 2026, 15(4), 250; https://doi.org/10.3390/axioms15040250 - 26 Mar 2026
Viewed by 293
Abstract
We study the complex point spectral distribution of the underlying operator of the system consisting of a reliable machine, a storage buffer with infinite capacity and an unreliable machine. This system is described by infinitely many partial differential equations with integral boundary conditions. [...] Read more.
We study the complex point spectral distribution of the underlying operator of the system consisting of a reliable machine, a storage buffer with infinite capacity and an unreliable machine. This system is described by infinitely many partial differential equations with integral boundary conditions. The known literature proved that all points in a set in the left half of the complex plane are eigenvalues of the underlying operator and indicated that all points outside of the set remain undetermined. In this paper, we study the spectrum outside of the set and, under certain conditions, prove that some points outside the set are eigenvalues of the underlying operator, whereas other points are not. By combining our result with the results in the existing literature, we give a description of the point spectral distribution of the underlying operator on the whole complex plane. Our idea and method are suitable for studying point spectral distribution of the underlying operators of some queueing models described by infinitely many partial differential equations with integral boundary conditions. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)
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22 pages, 408 KB  
Article
Spectral Analysis and Asymptotic Behavior of an M/GB/1 Bulk Service Queueing System
by Nurehemaiti Yiming
Axioms 2026, 15(4), 243; https://doi.org/10.3390/axioms15040243 - 24 Mar 2026
Viewed by 354
Abstract
In this paper, we investigate the spectrum distribution and asymptotic behavior of an M/GB/1 bulk service queueing system. In this system, the server processes customers in batches of a fixed maximum capacity B, and the time required to serve [...] Read more.
In this paper, we investigate the spectrum distribution and asymptotic behavior of an M/GB/1 bulk service queueing system. In this system, the server processes customers in batches of a fixed maximum capacity B, and the time required to serve a batch is governed by a general distribution with a service rate function η(·), which determines the instantaneous probability of service completion. The system dynamics are described by an infinite set of partial integro-differential equations. First, by introducing the probability generating function and employing Greiner’s boundary perturbation method, we establish that the time-dependent solution (TDS) of the system converges strongly to its steady-state solution (SSS) in the natural Banach state space. To this end, when the service rate η(·) is a bounded function, we prove that zero is an eigenvalue of both the system operator and its adjoint operator, with geometric multiplicity one. Moreover, we show that every point on the imaginary axis except zero belongs to the resolvent set of the system operator. Second, we analyze the spectrum of the system operator on the left real axis. When the service rate η(·) is constant and the fixed maximum capacity B equals 2, we apply Jury’s stability criterion for cubic equations to demonstrate that the system operator possesses an uncountably infinite number of eigenvalues located on the negative real axis. Additionally, we prove that an open interval near zero on the left real axis is not part of the point spectrum of the system operator. Consequently, these results imply that the semigroup generated by the system operator is not compact, eventually compact, quasi-compact, or essentially compact. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)
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21 pages, 15067 KB  
Article
Fixed/Predefined-Time Synchronization for Delayed Memristive Reaction-Diffusion Neural Networks Subject to Stochastic Disturbances
by Gang Wang, Ikram Mamtimin and Abdujelil Abdurahman
Axioms 2026, 15(3), 209; https://doi.org/10.3390/axioms15030209 - 12 Mar 2026
Cited by 1 | Viewed by 490
Abstract
This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the [...] Read more.
This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the settling time bound regardless of initial conditions, and offers a more concise form than prior results. Second, by leveraging Green’s formula, integral inequality, and stochastic analysis, some sufficient conditions are derived to guarantee FXT and PDT synchronization of introduced stochastic MNNs with reaction-diffusion terms. Finally, numerical simulations are given to validate the effectiveness of the proposed synchronization scheme. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)
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53 pages, 473 KB  
Article
Analysis of a k/n(G) Retrial System with Multiple Working Vacations
by Changjiang Lai, Rena Eskar and Ehmet Kasim
Axioms 2025, 14(11), 853; https://doi.org/10.3390/axioms14110853 - 20 Nov 2025
Cited by 1 | Viewed by 404
Abstract
In this paper, a k/n(G) retrial system with multiple working vacations is considered, and a mathematical model of the system is established by supplementary variable method, and a dynamic analysis of the system is carried out. Firstly, the [...] Read more.
In this paper, a k/n(G) retrial system with multiple working vacations is considered, and a mathematical model of the system is established by supplementary variable method, and a dynamic analysis of the system is carried out. Firstly, the model is transformed into an abstract Cauchy problem in Banach space by introducing the state space, main operator and its definition domain. Secondly, the C0-semigroup theory in functional analysis and the spectral theory of linear operators are used to prove that the main operator of the model generates a positive contraction C0-semigroup, which leads to the existence of a unique, non-negative time-dependent solution of the system that satisfies the probabilistic properties. Finally, Greiner’s boundary perturbation idea and the spectral properties of the corresponding operators are used to show that the time-dependent solution strongly converges to its steady-state solution. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)
50 pages, 422 KB  
Article
Asymptotic Behavior of the Time-Dependent Solution of the M[X]/G/1 Queuing Model with Feedback and Optional Server Vacations Based on a Single Vacation Policy
by Nuraya Nurahmat and Geni Gupur
Axioms 2025, 14(11), 834; https://doi.org/10.3390/axioms14110834 - 12 Nov 2025
Viewed by 456
Abstract
By using the C0-semigroup theory, we study the asymptotic behavior of the time-dependent solution and the time-dependent indices of the M[X]/G/1 queuing model with feedback and optional server vacations based on a single vacation [...] Read more.
By using the C0-semigroup theory, we study the asymptotic behavior of the time-dependent solution and the time-dependent indices of the M[X]/G/1 queuing model with feedback and optional server vacations based on a single vacation policy. This queuing model is described by infinitely many partial differential equations with integral boundary conditions in an unbounded interval. Under certain conditions, by studying spectrum of the underlying operator of this queuing model on the imaginary axis, we prove that the time-dependent solution of this queuing model strongly converges to its steady-state solution. Next, we prove that the time-dependent queuing length of this queuing system converges to its steady-state queuing length and the time-dependent waiting time of this queuing system converges to its steady-state waiting time as time tends to infinity. Our results extend the steady-state results of this queuing system. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)
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