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Mathematics
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Q-differential/Difference Equations and Related Applications
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Q-differential/Difference Equations and Related Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (20 August 2023) | Viewed by 13322

Special Issue Editor

Prof. Dr. Jian Cao

E-Mail Website
Guest Editor
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, Zhejiang, China
Interests: q-seriesq-orthogonal polynomials; generating functions; q-difference equations; q-partial differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

All manuscripts should be written to be accessible to a broad scientific audience, who are interested in partial differential equations with their methods and applications in mathematical, physical, engineering sciences, etc. The covered topics include but are not limited to classical partial/difference equations, q-partial/difference equations, Nevanlinna theory, q-polynomials, generating functions, q-integrals, fractional calculus, and fractional q-calculus. 

Prof. Dr. Jian Cao
Guest Editor

Manuscript Submission Information

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Keywords

  • classical partial/difference equations
  • q-partial/difference equations
  • Nevanlinna theory
  • q-polynomials
  • generating functions
  • q-integrals
  • fractional calculus
  • and fractional q-calculus

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Published Papers (6 papers)

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Research

14 pages, 1528 KiB  
Article
Approximate Roots and Properties of Differential Equations for Degenerate q-Special Polynomials
by Jung-Yoog Kang and Cheon-Seoung Ryoo
Mathematics 2023, 11(13), 2803; https://doi.org/10.3390/math11132803 - 21 Jun 2023
Cited by 6 | Viewed by 956
Abstract
In this paper, we generate new degenerate quantum Euler polynomials (DQE polynomials), which are related to both degenerate Euler polynomials and q-Euler polynomials. We obtain several (q,h)-differential equations for DQE polynomials and find some relations of q [...] Read more.
In this paper, we generate new degenerate quantum Euler polynomials (DQE polynomials), which are related to both degenerate Euler polynomials and q-Euler polynomials. We obtain several (q,h)-differential equations for DQE polynomials and find some relations of q-differential and h-differential equations. By varying the values of q,η, and h, we observe the values of DQE numbers and approximate roots of DQE polynomials to obtain some properties and conjectures. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
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8 pages, 260 KiB  
Article
On Some Expansion Formulas for Products of Jacobi’s Theta Functions
by Hong-Cun Zhai, Jian Cao and Sama Arjika
Mathematics 2023, 11(3), 588; https://doi.org/10.3390/math11030588 - 22 Jan 2023
Viewed by 1285
Abstract
In this paper, we establish several expansion formulas for products of the Jacobi theta functions. As applications, we derive some expressions of the powers of (q;q)∞ by using these expansion formulas. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
18 pages, 1114 KiB  
Article
Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials
by Sunil Kumar Sharma, Waseem Ahmad Khan, Cheon-Seoung Ryoo and Ugur Duran
Mathematics 2022, 10(15), 2709; https://doi.org/10.3390/math10152709 - 31 Jul 2022
Cited by 3 | Viewed by 1339
Abstract
Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and [...] Read more.
Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p,q)-special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p,q)-cosine polynomials and (p,q)-sine polynomials, we consider a novel kinds of (p,q)-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p,q-integral representations and p,q-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
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16 pages, 416 KiB  
Article
A Novel Fractional-Order Discrete SIR Model for Predicting COVID-19 Behavior
by Noureddine Djenina, Adel Ouannas, Iqbal M. Batiha, Giuseppe Grassi, Taki-Eddine Oussaeif and Shaher Momani
Mathematics 2022, 10(13), 2224; https://doi.org/10.3390/math10132224 - 25 Jun 2022
Cited by 22 | Viewed by 2971
Abstract
During the broadcast of Coronavirus across the globe, many mathematicians made several mathematical models. This was, of course, in order to understand the forecast and behavior of this epidemic’s spread precisely. Nevertheless, due to the lack of much information about it, the application [...] Read more.
During the broadcast of Coronavirus across the globe, many mathematicians made several mathematical models. This was, of course, in order to understand the forecast and behavior of this epidemic’s spread precisely. Nevertheless, due to the lack of much information about it, the application of many models has become difficult in reality and sometimes impossible, unlike the simple SIR model. In this work, a simple, novel fractional-order discrete model is proposed in order to study the behavior of the COVID-19 epidemic. Such a model has shown its ability to adapt to the periodic change in the number of infections. The existence and uniqueness of the solution for the proposed model are examined with the help of the Picard Lindelöf method. Some theoretical results are established in view of the connection between the stability of the fixed points of this model and the basic reproduction number. Several numerical simulations are performed to verify the gained results. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
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14 pages, 289 KiB  
Article
Hilfer Fractional Quantum Derivative and Boundary Value Problems
by Phollakrit Wongsantisuk, Sotiris K. Ntouyas, Donny Passary and Jessada Tariboon
Mathematics 2022, 10(6), 878; https://doi.org/10.3390/math10060878 - 10 Mar 2022
Cited by 3 | Viewed by 2061
Abstract
In this paper, we introduce an extension of the Hilfer fractional derivative, the “Hilfer fractional quantum derivative”, and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a [...] Read more.
In this paper, we introduce an extension of the Hilfer fractional derivative, the “Hilfer fractional quantum derivative”, and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach’s contraction mapping principle. Examples illustrating the obtained results are also presented. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
15 pages, 277 KiB  
Article
Some Results on the q-Calculus and Fractional q-Differential Equations
by Ying Sheng and Tie Zhang
Mathematics 2022, 10(1), 64; https://doi.org/10.3390/math10010064 - 25 Dec 2021
Cited by 12 | Viewed by 3665
Abstract
In this paper, we first discuss some important properties of fractional q-calculus. Then, based on these properties and the q-Laplace transform, we translate a class of fractional q-differential equations into the equivalent q-differential equations with integer order. Thus, we [...] Read more.
In this paper, we first discuss some important properties of fractional q-calculus. Then, based on these properties and the q-Laplace transform, we translate a class of fractional q-differential equations into the equivalent q-differential equations with integer order. Thus, we propose a method for solving some linear fractional q-differential equations by means of solving the corresponding integer order equations. Several examples are provided to illustrate our solution method. Full article
(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)
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