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Axioms
Special Issues
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 542

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 (2 papers)

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Research

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
Viewed by 146
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 179
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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