q-Calculus and physics

Ernst, Thomas

doi:10.1007/978-3-0348-0431-8_12

Cited by 12 publications

(16 citation statements)

References 34 publications

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“…For the convenience of the reader, in this section, we recall some necessary concepts and definitions of the quantum calculus, for which we refer to [9,10,13] . Moreover, we give lemmas will be used to prove of our main results.…”

Section: Preliminariesmentioning

confidence: 99%

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Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

Ruzhansky

Torebek

2020

Fract Calc Appl Anal

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The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions Eα, β(i λ ϕ(x)), x ∈ ℝN and Eα, β(iαλ ϕ(x)), x ∈ ℝN for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

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Section: Preliminariesmentioning

confidence: 99%

“…A number of articles address the further developments and recent results in the q-deformed calculus (see e.g. [3], [6], [10], [26], [37], [38] and we refer to the books [11] and [13]).…”

Section: Introductionmentioning

confidence: 99%

Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

Ruzhansky

Torebek

2020

Fract Calc Appl Anal

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“…The study of q-difference equations, initiated in the beginning of the 20th century ( [1][2][3][4]), and, up to date, it has evolved into a multidisciplinary subject, (for example, see ( [5][6][7][8][9][10][11][12][13][14][15]) and references therein). For some recent work on q-difference equations, we refer the reader to the papers ( [16][17][18][19][20][21][22][23]).…”

Section: U(t) = F (T U(t)) T ∈ I U(0) = ηU(t) D Q U(0) = ηD Q U(t)mentioning

confidence: 99%

A study of second-order q-difference equations with boundary conditions

2012

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This paper studies a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the q-difference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of qdifference equations. The results corresponding to a second-order q-difference equation with anti-periodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical secondorder boundary value problems with non-separated boundary conditions in the limit q 1, which provides a useful check.

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“…Quantum calculus, or q−calculus, has had an important development in recent decades, both in pure mathematics and its applicability, for example in Physics [1]. The convexity of a function has played an important role as a tool in the development of inequalities.…”

Section: Introductionmentioning

confidence: 99%

Some New q—Integral Inequalities Using Generalized Quantum Montgomery Identity via Preinvex Functions

et al. 2020

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In this work the authors establish a new generalized version of Montgomery’s identity in the setting of quantum calculus. From this result, some new estimates of Ostrowski type inequalities are given using preinvex functions. Given the generality of preinvex functions, particular q —integral inequalities are established with appropriate choice of the parametric bifunction. Some new special cases from the main results are obtained and some known results are recaptured as well. At the end, a briefly conclusion is given.

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q-Calculus and physics

Cited by 12 publications

References 34 publications

Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

A study of second-order q-difference equations with boundary conditions

Some New q—Integral Inequalities Using Generalized Quantum Montgomery Identity via Preinvex Functions

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