The q-derivative and differential equation

Abstract

The q-calculus appeared as a connection between mathematics and physics. It has several applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, quantum theory, and electronics. Recently, a great interest to its applications in differential transform methods, in order to get analytical approximate solutions to the ordinary as well as partial differential equations. In this paper, we present some of the interesting definitions of q-calculus and q-derivatives. By using q-calculus, solutions of some differential equations could be generated.

Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions

“…In addition, the q-calculus operator, the q-integral operator, and the q-derivative operator are used to build several classes of regular functions and play an intriguing role, since they are used and applied in many different branches of mathematics, including the theory of relativity, the calculus of variations, orthogonal polynomials, and basic hypergeometric functions. In [39], Akça et al used the q-derivative and generated solutions to some differential equations. Therefore, we have also made use of q-calculus and provide certain important new types of q-analogues of differential and integral operators, as mentioned in this paper.…”

Symmetric Difference Operator in Quantum Calculus

“…In this paper, we mainly concentrate on the q-derivative, and we utilize some concepts in the h-derivative. In [1], the authors implemented the q-derivative as f(qt) − f(t) qt − t . The q-derivative, or Jackson derivative, is a q-analog of the usual derivative established by F. H. Jackson in the fields of combinatorics and quantum calculus, and Jackson's q-integration is the inverse of this.…”

“…The operator ∆, defined as ∆ f(t) = f(t + 1) − f(t) is the foundation of difference equation theory, where f(t) is a sequence of numbers. The authors in [6,7,39,40] proposed the definition of generalized difference operator ∆ h , which is defined in Equation (1), and then developed the inverse theory concept (anti-difference operator (∆ −1 h ) for finding the closed-form solutions. In 1984, Jerzy Popenda and Szmanda [41] suggested a specific type of ∆ α operator defined as ∆ α f(t) = f(t + 1) − f(t), where α, t ∈ (−∞, ∞).…”

q-Bernoulli’s Equation

“…Adams (1928) solved transonic gas equation by using q-derivative and show that the huge different between classical solution and a new qanalogous solution of transonic gas equation. Akça et al (2019) solved many differential equations by using qcalculus. This article issued by follows.…”