On the Conformable Fractional Quantum Mechanics

Mozaffari, Fatemeh; Hassanabadi, H.; Sobhani, Hadi; Chung, Won Sang

doi:10.3938/jkps.72.980

Cited by 30 publications

(14 citation statements)

References 20 publications

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“…ough the concept of fractional derivative is more than 300 years old, in the last and present two centuries, the role of fractional calculus has been increasing because of its application area in various domains including the diffusion of biological population, signal processing, plasma physics, optical fiber, chemical kinematics, solid state physics, electrical network, finance, fluid flow, and control theory [1][2][3][4][5][6]. In particular, the noninteger order models have proven to be very useful to describe numerous scales, namely, nanoscale, microscale, and mesoscale.…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions for the Conformable Space-Time Fractional Zeldovich Equation with Time-Dependent Coefficients

Injrou

2020

International Journal of Differential Equations

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The aim of this paper is to improve a sub-equation method to solve the space-time fractional Zeldovich equation with time-dependent coefficients involving the conformable fractional derivative. As result, we obtain three families of solutions including the hyperbolic, trigonometric, and rational solutions. These solutions may be helpful to explain several phenomena in chemistry, including the combustion process. The study shows that the used method is effective and reliable and can be utilized as a substitution to construct new solutions of different types of nonlinear conformable fractional partial differential equations (NFPDEs) with variable coefficients.

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Section: Introductionmentioning

confidence: 99%

Exact Solutions for the Conformable Space-Time Fractional Zeldovich Equation with Time-Dependent Coefficients

Injrou

2020

International Journal of Differential Equations

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“…Zhao et al established the multivariate theory of GCFD and illustrated the conformable Maxwell equations, and theorems for Conformable Gauss's, Green's, and Stokes's Theorem, see Zhao et al 22 Though, this new local fractional derivative fails some properties as pointed out in previous studies, [23][24][25][26] it seems to account for many deficiencies of some of the earlier proposed definitions which are of great importance in applied sciences and therefore suitable for more applications. Some authors followed this work and explored potential applications in various fields such as the control theory of dynamical systems, [27][28][29][30] mathematical biology and epidemiology, 19,[31][32][33][34] mechanics, 16,35,36 systems of linear and nonlinear conformable fractional differential equations (CFDEs), [37][38][39] quantum mechanics, 40,41 variational calculus, 42,43 arbitrary time scale problems, [44][45][46] modelling of diffusion, 47,48 stochastic process, 49,50 and optics. 51 Some analytical and numerical methods have attracted great interest and became an important tool for differential equations with CFDs, (see previous studies ).…”

Section: Introductionmentioning

confidence: 99%

“…The eigenvalues of s-CFSLP1(40) and(41), considered on the C [a, b] space, are real and the eigenfunctions corresponding to distinct eigenvalues are orthogonal on the weighted space b…”

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confidence: 99%

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

Mortazaasl

Akbarfam

2020

Math Methods in App Sciences

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In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of r-CFSLPs, we discuss two types of CFSLPs which include left-and right-sided CFDs, each of order ∈ (n, n + 1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order ∈ (0, 1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and-II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. KEYWORDS conformable fractional derivatives and integrals, eigenvalues and eigenfunctions, fixed-point theorem, fractional diffusion, fractional Sturm-Liouville operators, transform methods JEL CLASSIFICATION

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“…Different definitions of fractional derivatives can be proposed, each with remarkable properties [22][23][24][25][26], all of them valid and mathematically acceptable.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in Ref. [23], in the framework of conformable fractional quantum mechanics, the threedimensional fractional harmonic oscillator is studied and by using an effective and efficient formalism, Schrödinger equation, probability density, probability flux and continuity equation have been investigated and in Ref. [24].…”

Section: Introductionmentioning

confidence: 99%

The investigation of classical particle in the presence of fractional calculus

Chung¹,

Zare²,

Hassanabadi³

et al. 2020

Rev. Mex. Fís.

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In this article, by applying a preliminary and comprehensive definition of the fractional calculus, the effect of this calculus on the fractional derivatives corresponding to different aspects of physics such as Laplace transforms, Riemann-Liouville and Caputo has been specified. Applications of the fractional calculus in studying the dynamics of particle motion in classical mechanics are investigated analytically. Furthermore, we compare our results with those given in the ordinary condition and we show that this condition is simply found by removing the fractional effects and the expected results are obtained.

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On the Conformable Fractional Quantum Mechanics

Cited by 30 publications

References 20 publications

Exact Solutions for the Conformable Space-Time Fractional Zeldovich Equation with Time-Dependent Coefficients

Exact Solutions for the Conformable Space-Time Fractional Zeldovich Equation with Time-Dependent Coefficients

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

The investigation of classical particle in the presence of fractional calculus

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