Solutions of Unified Fractional Schrödinger Equations

Chaurasia, V. B. L.; Kumar, Devendra

doi:10.5402/2012/935365

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…Many researchers have analyzed such concepts to achieve diverse objectives. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] Here, new transform formulae are computed, and their closed-form equations involve the special forms of basic Fox-Wright function denoted by p Ψ q and defined as (see Kilbas et al, 3, p. 56 ):…”

Section: Validity Of the New Generalized Representationmentioning

confidence: 99%

“…For the interest of readers, more comprehensive discussions and developments on the fractional kinetic equation can also be found in other studies. [4][5][6][7][8][9][10][11][12][13][14][15][16][17] Throughout in the article, ℜ denotes the real part of a complex number, R and C are used to symbolize the set of real and complex numbers respectively.…”

Section: Introductionmentioning

confidence: 99%

“…By following Saxena et al, 7 a considerable literature can be observed on the fractional kinetic equations involving special functions. [8][9][10][11][12][13][14][15][16][17] According to my review and investigation, fractional kinetic equations involving extended k-gamma function has not yet been focused on in the literature. Hence, the main objective is to investigate a new series representation for the extended k-gamma function in terms of the complex delta function.…”

Section: Introductionmentioning

confidence: 99%

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A new representation of the extended k‐gamma function with applications

Tassaddiq

2021

Math Methods in App Sciences

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In this study, a new series representation of the extended k‐gamma function is investigated. A new representation expresses this function as an infinite sum of delta functions. Several researchers have examined this family of functions, but no study has been conducted dealing with the fractional kinetic equation. A new representation proves useful to solve the fractional kinetic equation involving an extended k‐gamma function. Particular cases involving the original gamma function are discussed as corollaries. Such an application of the gamma function is not possible by using known representations; nevertheless, using this representation, new fractional transform formulae are evaluated. A new representation is also worthwhile to compute new integrals of products of the extended k‐gamma functions, proving to be reliable with known identities. Several distributional properties of the extended k‐gamma function are also discussed using Fourier transform.

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Section: Validity Of the New Generalized Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new representation of the extended k‐gamma function with applications

Tassaddiq

2021

Math Methods in App Sciences

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“…These considerations can be easily extended to the case of a particle moving in a potential field by incorporating a potential term ( , ) in the Lagrangian = − . This will necessitate incorporating an additional term − ( , ) /2 /Γ(1 + /2) in the right-hand side of (35) and an additional factor −( /ℎ )( ( , ) /2 /Γ(1+ /2)) in the exponential in (37). This results in changing (45) into…”

Section: Tfse For a Particle In A Potential Fieldmentioning

confidence: 99%

Time Fractional Schrodinger Equation Revisited

Achar

Yale

Hanneken

2013

Advances in Mathematical Physics

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The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [ 2 / 2 ]. The action is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced. The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber's work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained. The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given.

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“…Fractional analysis arises in much problems of physics [1,2], continuum mechanics [3], visco-elasticity [4,5], quantum mechanics [6][7][8], and other branches of applied mathematics [9][10][11][12][13]. However, these spherical defined fractional derivatives do not usually project the local geometric behaviours for a given function.…”

Section: Introductionmentioning

confidence: 99%