Q-difference Equations Research Articles

A note on the post quantum-Sheffer polynomial sequences

Abstract In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

The uniqueness of meromorphic function shared values with meromorphic solutions of a class of q-difference equations

<abstract><p>We first investigate the meromorphic solutions of a class of homogeneous second-order q-difference equations and the uniqueness problem for a meromorphic function with three shared values; then we discuss the uniqueness problem for the meromorphic solutions of a class of nonhomogeneous q-difference equations and a meromorphic function with four shared values.</p></abstract>

Quantum calculus with respect to another function

<abstract><p>In this paper, we studied the generalizations of quantum calculus on finite intervals. We presented the new definitions of the quantum derivative and quantum integral of a function with respect to another function and studied their basic properties. We gave an application of these newly defined quantum calculi by obtaining a new Hermite-Hadamard inequality for a convex function. Moreover, an impulsive boundary value problem involving quantum derivative, with respect to another function, was studied via the Banach contraction mapping principle.</p></abstract>

Qualitative analysis of neutral implicit fractional q-difference equations with delay

Qualitative analysis of neutral implicit fractional q-difference equations with delay

A Coupled System of Nonlinear Langevin Fractional q- Difference Equations Associated with Two Different Fractional Orders in Banach Space

In this research article, we study the coupled system of nonlinear Langevin fractional q-difference equations associated with two different fractional orders in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with Mönch fixed point theorem, the implementation Banach contraction principle fixed point theorem, and the employment of Urs’s stability approach. Two examples illustrating the effectiveness of the theoretical results are presented.

A new fixed point approach for solutions of a $ p $-Laplacian fractional $ q $-difference boundary value problem with an integral boundary condition

<p>We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.</p>

Symmetric quantum calculus in interval valued frame work: operators and applications

<p>The primary emphasis of the present study is to introduce some novel characterizations of the interval-valued $ (\mathcal{I}.\mathcal{V}) $ right symmetric quantum derivative and antiderivative operators relying on generalized Hukuhara difference. To continue the study, we start with the concept of symmetric differentiability in the interval-valued sense and explore some important properties. Furthermore, through the utilization of the $ (\mathcal{I}.\mathcal{V}) $ symmetric derivative operator, we develop the right-sided $ (\mathcal{I}.\mathcal{V}) $ integral operator and explore its key properties. Also, we establish various $ (\mathcal{I}.\mathcal{V}) $ trapezium-like inequalities by considering the newly proposed operators and support line. Later on, we deliver another proof of the trapezium inequality through an analytical approach. Also, we present the numerical and visual analysis for the verification of our results.</p>

Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus

<abstract><p>The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $ [\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $ \mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $ \mathrm{b_{1}} $, and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus.</p></abstract>

Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain

<abstract><p>In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass $ \mathcal{S} ^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike functions and the subclass $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $, as well as sharp bounds for the inverse function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $. We also investigated some results for the class of bi-univalent functions $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper.</p></abstract>

Sharp inequalities for $ q $-starlike functions associated with differential subordination and $ q $-calculus

<p>This paper employs differential subordination and quantum calculus to investigate a new class of $ q $-starlike functions associated with an eight-like image domain. Our study laid a foundational understanding of the behavior of these $ q $-starlike functions. We derived the results in first-order differential subordination. We established sharp inequalities for the initial Taylor coefficients and provided optimal estimates for solving the Fekete-Szegö problem and a second-order Hankel determinant applicable to all $ q $-starlike functions in this class. Furthermore, we presented a series of corollaries that demonstrate the broader implications of our findings in geometric function theory.</p>

Q-Hermite-Hadamard Integral Inequality for the Coordinated Convex Functions

The calculus without the notion of limits is quantum calculus. Its study dates back to L. Euler in the middle of the eighteenth century whereas the systematic initiation on it was done by F.H. Jackson in the beginning of the twentieth century. The rapid growth on q-calculus is due to its applications in various branches of mathematical and physical sciences. Of them, one of the most basic and important functions in the theory of geometric function is convexity having its wider applications in pure and applied mathematics. As it still lacks the intensive study on quantum estimates on the various types of integral inequalities, we focus our study on quantum estimates of Hermite-Hadamard type integral inequality especially on coordinated convex functions. In this paper, we have extended Hermite-Hadamard type integral inequality for coordinated convex function in terms of quantum framework.

Finite Difference Modeling of Time Fractal Impact on Unsteady Magneto-hydrodynamic Darcy–Forchheimer Flow in Non-Newtonian Nanofluids with the q-Derivative

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids.

Some properties of a class of generalized Janowski-type $q$-starlike functions associated with Opoola $q$-differential operator and $q$-differential subordination

Without qualms, studies show that quantum calculus has received great attention in recent times. This can be attributed to its wide range of applications in many science areas. In this exploration, we study a new qdifferential operator that generalized many known differential operators. The new q-operator and the concept of subordination were afterwards, used to define a new subclass of analytic-univalent functions that invariably consists of several known and new generalizations of starlike functions. Consequently, some geometric properties of the new class were investigated. The properties include coefficient inequality, growth, distortion and covering properties. In fact, we solved some radii problems for the class and also established its subordinating factor sequence property. Indeed, varying some of the involving parameters in our results led to some existing results.

Hyers–Ulam stability and existence of solutions for high-order fractional q-difference equations on infinite intervals

Hyers–Ulam stability and existence of solutions for high-order fractional q-difference equations on infinite intervals

Meromorphic solutions of linear q-difference equations

In this article, we construct explicit meromorphic solutions of first order linear q-difference equations in the complex domain and we describe the location of all their zeros and poles. The homogeneous case leans on the study of four fundamental equations, providing the previous informations in the framework of entire or meromorphic coefficients. The inhomogeneous situation, which stems from the homogeneous one and two fundamental equations, is also described in detail. We also address the case of higher-order linear q-difference equations, using a classical factorization argument. All these results are illustrated by several examples.

Positive periodic solutions for certain kinds of delayed q-difference equations with biological background

Positive periodic solutions for certain kinds of delayed q-difference equations with biological background

A new type of Szász–Mirakjan operators based on q-integers

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Strong Differential Subordinations and Superordinations for Riemann–Liouville Fractional Integral of Extended q-Hypergeometric Function

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries.

New Subclass of Close-to-Convex Functions Defined by Quantum Difference Operator and Related to Generalized Janowski Function

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex functions connected with generalized Janowski functions. We prove the necessary and sufficient conditions for functions that belong to newly defined classes, including the inclusion relations and estimations of the coefficients. The Fekete–Szegő problem for a more general class is also discussed. The results of this investigation expand upon those of the previous study.

Existence of solutions for fractional q-difference equations

"In this paper, we obtain some existence results for the integral boundary value problems of nonlinear fractional q-difference equations. The differential operator is taken in the Riemann-Liouville sense."