Q-difference Equations Research Articles

Some Milne’s rule type inequalities in quantum calculus

The main goal of the current study is to establish some new Milne?s rule type inequalities for single-time differentiable convex functions in the setting of quantum calculus. For this, we establish a quantum integral identity and then we prove some new inequalities of Milne?s rule type for quantum differentiable convex functions. These inequalities are very important in Open-Newton?s Cotes formulas because, with the help of these inequalities, we can find the bounds of Milne?s rule for differentiable convex functions in classical or quantum calculus. The method adopted in this work to prove these inequalities are very easy and less conditional compared to some existing results. Finally, we give some mathematical examples to show the validity of newly established inequalities.

Solution to Laplace’s Equation Using Quantum Calculus

The quantum calculus emerged as a new type of unconventional calculus relevant to both mathematics and physics. The study of quantum calculus or q-calculus has three hundred years of history of development since the era of Euler and Bernoulli, and was appeared as one of the most arduous techniques to use it in mathematics as well as physical science. At present, it is used in diverged mathematical areas like number theory, orthogonal polynomials, basic hypergeometric functions, etc. Furthermore, in order to get analytical approximate solutions to the ordinary as well as partial differential equations, q-reduced differential technique and quantum separation of variable technique are used in mathematics, Mechanics, and physics. In this paper, Laplace’s equation, a well-known equation in both Physical and Mathematical sciences, has been solved extensively based on the basics of calculus, transformation methods, and q-separation of variable method. In addition, solutions to the Laplace’s equation as obtained by using different boundary conditions are revisited and reviewed. Consequently, all the necessary basics of q-calculus are displayed one by one, and thereafter, the process of finding its solution in view of quantum calculus is described extensively. In order to find out the exact solutions the dimensionality of all the parameters related to the problem has been described. As an essential outcome, it is also found that, as q tends to 1, the solution takes the form as it is in general physics. Hence, this article presents a review and extension that describe the solution to Laplace’s equation in view of both Leibnitz and quantum calculus. Thus, it can add a pedagogical exercise for the students of both physical and mathematical sciences to understand the usefulness of quantum calculus.

On existence of meromorphic solutions for certain q-difference equation

On existence of meromorphic solutions for certain q-difference equation

Some new q-Hermite-Hadamard-Mercer inequalities and related estimates in quantum calculus

Some new q-Hermite-Hadamard-Mercer inequalities and related estimates in quantum calculus

Implicit Caputo fractional q-difference equations with non instantaneous impulses

Implicit Caputo fractional q-difference equations with non instantaneous impulses

Existence and uniqueness of solutions for fractional q-difference equations

Existence and uniqueness of solutions for fractional q-difference equations

On existence of meromorphic solutions for nonlinear q-difference equation

On existence of meromorphic solutions for nonlinear q-difference equation

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Existence result for fractional q-difference equations on the half-line

In this paper, we obtain an existence result for the integral boundary value problems of nonlinear fractional q-difference equations on the half-line using Schauder?s fixed point theorem.

Гиперболический объём 3-d многообразий, A-многочлены, численные проверки гипотез

We continue our study of the connections between the hyperbolic volume of the complement of a knot in the three dimensional sphere with topological invariants of this knot. This time we pay attention to A(M,L) parametrization for the affine variety with casp, produced by a knot (so-called A-polynomials). Then, using the known expressions of A-polynomials for number of knots we present results of the numerical tests for the conjectures on asymptotics of solutions of q-difference equations connected with the hyperbolic volume of these knots.

Fuzzy differential subordination and superordination results for $ q $ -analogue of multiplier transformation

<abstract><p>In this paper the authors combine the quantum calculus applications regarding the theories of differential subordination and superordination with fuzzy theory. These results are established by means of an operator defined as the $ q $-analogue of the multiplier transformation. Interesting fuzzy differential subordination and superordination results are derived by the authors involving the functions belonging to a new class of normalized analytic functions in the open unit disc $ U $ which is defined and investigated here by using this $ q $-operator.</p></abstract>

FRACTIONAL QUANTUM HERMITE–HADAMARD-TYPE INEQUALITIES FOR INTERVAL-VALUED FUNCTIONS

Based on the [Formula: see text]-shifting operator, we introduce the concept of Riemann–Liouville fractional quantum integration for interval-valued functions (IVFs), and establish new Riemann–Liouville fractional [Formula: see text]-Hermite–Hadamard ([Formula: see text]-HH) and [Formula: see text]-HH-Fejér inequalities for left and right [Formula: see text]-convex IVFs (LR-[Formula: see text]-convex-IVFs). The findings obtained generalize known results in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and fractional quantum calculus. The results are illustrated with examples.

A computational method for investigating a quantum integrodifferential inclusion with simulations and heatmaps

<abstract><p>We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.</p></abstract>

Some New Parameterized Quantum Fractional Integral Inequalities Involving s-Convex Functions and Applications

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behavior. Its significance is raised by the strong connection between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. By applying this identity, we obtain as main results some integral inequalities of trapezium, midpoint and Simpson’s type pertaining to s-convex functions. Moreover, we deduce several special cases, which are discussed in detail. To validate our theoretical findings, an example and application to special means of positive real numbers are presented. Numerical analysis investigation shows that the mixed fractional calculus with quantum calculus give better estimates compared with fractional calculus or quantum calculus separately.

New Variants of Quantum Midpoint-Type Inequalities

Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel quantum midpoint-type inequalities that are twice qξ2-differentiable for (α,m)-convex functions. Firstly, we obtain novel identity for qξ2-integral by employing quantum calculus tools. Then by using the auxiliary identity, we formulate new bounds by taking into account the known quantum Hölder and Power mean inequalities. An example is provided with a graphical representation to show the validity of obtaining results. The outcomes of this study clarify and expand earlier research on midpoint-type inequalities. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role.

Existence theory for implicit fractional q-difference equations in Banach spaces

"This paper deals with some existence results for a class of implicit fractional q-difference equations. The results are based on the fi xed point theory in Banach spaces and the concept of measure of noncompactness. An illustrative example is given in the last section."

Triangle inequality for quantum integral operator

We show that general integral triangle inequality does not hold for shifted q-integrals. Furthermore, we obtain a triangle inequality for shifted qintegrals. We also give an estimate for q-integrable product and use it to refine some recently obtained Ostrowski inequalities for quantum calculus.

Several Types of q-Differential Equations of Higher Order and Properties of Their Solutions

The purpose of this paper is to organize various types of higher order q-differential equations that are connected to q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the properties of q-sigmoid polynomials, we show the symmetric properties of q-differential equations of higher order. Moreover, we derive special properties for the approximate roots of q-sigmoid polynomials which are solutions of higher order q-differential equations.

On a system of ( p , q ) -analogues of the natural transform for solving ( p , q ) -differential equations

In this work, we apply the concept of \((p,q)\)-calculus or post quantum calculus to establish the definitions of \((p,q)\)-analogues of the natural transform of the first and second kind, which is a symmetric relation between \((p,q)\)-analogues of the natural, Laplace, and Sumudu transforms. Moreover, as a result of the convolution theorem, some properties and some functions present in the table of \((p,q)\)-analogues of the natural transform are discussed. Also, we apply them to solve higher order \((p,q)\)-IVP with constants and coefficients, and to show the performance and effectiveness of the proposed transform.