Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’s H -function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

[1]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[2]  Shyam L. Kalla,et al.  Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics , 2010 .

[3]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[4]  L. Boyadjiev,et al.  INTEGRAL TRANSFORMS METHOD TO SOLVE A TIME-SPACE FRACTIONAL DIFFUSION EQUATION , 2010 .

[5]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[6]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[7]  Nick Laskin,et al.  Fractals and quantum mechanics. , 2000, Chaos.

[8]  A. M. Mathai,et al.  Reaction-Diffusion Systems and Nonlinear Waves , 2006 .

[9]  A. M. Mathai,et al.  Solution of Generalized Fractional Reaction-Diffusion Equations , 2006 .

[10]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[11]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[12]  R. K. Saxena,et al.  Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics , 2010, Appl. Math. Comput..

[13]  N. Laskin.  Fractional Schrödinger equation. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[15]  Shyam L. Kalla,et al.  On fractional partial differential equations related to quantum mechanics , 2011 .

[16]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[17]  Jianfei Huang,et al.  The Grünwald-Letnikov method for fractional differential equations , 2011, Comput. Math. Appl..

[18]  Shyam L. Kalla,et al.  Numerical treatment of fractional heat equations , 2008 .

[19]  Xiaoyi Guo,et al.  Some physical applications of fractional Schrödinger equation , 2006 .

[20]  M. Naber.  DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION , 2003, math-ph/0311047.

[21]  R. Hilfer.  FRACTIONAL TIME EVOLUTION , 2000 .

[22]  A. M. Mathai,et al.  Fractional Reaction-Diffusion Equations , 2006, math/0604473.

[23]  Igor Sokolov,et al.  Lévy flights in external force fields: from models to equations , 2002 .

[24]  R. Gorenflo,et al.  Time-fractional derivatives in relaxation processes: a tutorial survey , 2008, 0801.4914.

[25]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[26]  Francesco Mainardi,et al.  Evolution equations for the probabilistic generalization of the Voigt profile function , 2007, J. Comput. Appl. Math..