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

Deadline for manuscript submissions: 20 August 2023 | Viewed by 6497

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Special Issue Editor

Prof. Dr. Jian Cao

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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
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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

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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. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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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

Published Papers (6 papers)

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Research

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Open AccessArticle

On Some Expansion Formulas for Products of Jacobi’s Theta Functions

Hong-Cun Zhai

Jian Cao

and

Sama Arjika

Mathematics 2023, 11(3), 588; https://doi.org/10.3390/math11030588 - 22 Jan 2023

Viewed by 526

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)}_{\infty}

by using these expansion formulas. Full article

(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)

Open AccessArticle

Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials

and

Mathematics 2022, 10(15), 2709; https://doi.org/10.3390/math10152709 - 31 Jul 2022

Cited by 1 | Viewed by 642

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)

► Show Figures

Figure 1

Open AccessArticle

A Novel Fractional-Order Discrete SIR Model for Predicting COVID-19 Behavior

and

Mathematics 2022, 10(13), 2224; https://doi.org/10.3390/math10132224 - 25 Jun 2022

Cited by 9 | Viewed by 1367

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 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)

► Show Figures

Figure 1

Open AccessArticle

Hilfer Fractional Quantum Derivative and Boundary Value Problems

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 1 | Viewed by 1121

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 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)

Open AccessArticle

Some Results on the q-Calculus and Fractional q-Differential Equations

Ying Sheng

and

Tie Zhang

Mathematics 2022, 10(1), 64; https://doi.org/10.3390/math10010064 - 25 Dec 2021

Cited by 4 | Viewed by 1738

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 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)

Review

Jump to: Research

Open AccessReview

A Review of q-Difference Equations for Al-Salam–Carlitz Polynomials and Applications to U(n + 1) Type Generating Functions and Ramanujan’s Integrals

and

Mathematics 2023, 11(7), 1655; https://doi.org/10.3390/math11071655 - 29 Mar 2023

Viewed by 374

Abstract

In this review paper, our aim is to study the current research progress of q-difference equations for generalized Al-Salam–Carlitz polynomials related to theta functions and to give an extension of q-difference equations for q-exponential operators and q-difference equations for Rogers–Szegö polynomials. Then, we continue to generalize certain generating functions for Al-Salam–Carlitz polynomials via q-difference equations. We provide a proof of Rogers formula for general Al-Salam–Carlitz polynomials and obtain transformational identities using q-difference equations. In addition, we gain

U (n + 1)

-type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials via q-difference equations. Finally, we derive two extensions of the Andrews–Askey integral via q-difference equations. Full article

(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)

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Special Issue "Q-differential/Difference Equations and Related Applications"

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