Hilfer Fractional Quantum Derivative and Boundary Value Problems

Wongsantisuk, Phollakrit; Ntouyas, Sotiris K.; Passary, Donny; Tariboon, Jessada

doi:10.3390/math10060878

Cited by 3 publications

(2 citation statements)

References 30 publications

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“…The concepts of the definite q-integrals and the q-derivatives were initially introduced by Jackson. Numerous branches of mathematics, including orthogonal polynomials, hypergeometric functions, number theory, and combinatorics, as well as physics subjects including mechanics, relativity theory, and quantum theory, have found use for quantum calculus [29].…”

Section: New Q-fractional Derivatives Involving Different Functionsmentioning

confidence: 99%

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Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Tassaddiq,

Srivastava,

Kasmani

et al. 2023

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Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.

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Section: New Q-fractional Derivatives Involving Different Functionsmentioning

confidence: 99%

“…We can obtain a number of q-fractional derivatives and integrals involving the Laplace and Fourier transforms of the new representation by using [29,30]…”

Section: New Q-fractional Derivatives Involving Different Functionsmentioning

confidence: 99%

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Tassaddiq,

Srivastava,

Kasmani

et al. 2023

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A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

2023

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Hilfer fractional quantum system with Riemann-Liouville fractional derivatives and integrals in boundary conditions

Passary,

Ntouyas,

Tariboon

2024

MATH

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<abstract><p>In this paper, we initiate the study of existence and uniqueness of solutions for a coupled system involving Hilfer fractional quantum derivatives with nonlocal boundary value conditions containing $ q $-Riemann-Liouville fractional derivatives and integrals. Our results are supported by some well-known fixed-point theories, including the Banach contraction mapping principle, Leray-Schauder alternative and the Krasnosel'skiǐ fixed-point theorem. Examples of these systems are also given in the end.</p></abstract>

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Hilfer Fractional Quantum Derivative and Boundary Value Problems

Cited by 3 publications

References 30 publications

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

Hilfer fractional quantum system with Riemann-Liouville fractional derivatives and integrals in boundary conditions

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