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Solutions of Fractional Partial Differential Equations of Quantum Mechanics

Published online by Cambridge University Press:  03 June 2015

S. D. Purohit
S. D. Purohit*
Affiliation:
Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur-313001, India
*
*Corresponding author. Email: sunil_a_purohit@yahoo.com
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Abstract

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.

Keywords

26A3344A1033C6035J10Fractional Schrödinger equationLaplace transformFourier transformHilfer fractional derivativeFox’s H-function and Quantum mechanics
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 5 , October 2013 , pp. 639 - 651
Copyright
Copyright © Global-Science Press 2013

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References

[1]Brockmann, D. and Sokolov, I. M., Lévy flights in external force fields: from models to equations, Chem. Phys., 284 (2002), pp. 409–421.Google Scholar
[2]Caputo, M., Elasticitáe Dissipazione, Zanichelli, Bologna, 1969.Google Scholar
[3]Debnath, L., Fractional integral and fractional differential equations in fluid mechanics, Fract. Calc. Appl. Anal., 6(2) (2003), pp. 119–155.Google Scholar
[4]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York, 1955.Google Scholar
[5]Guo, X. and Xu, M., Some physical applications of Schrödinger equation, J. Math. Phys., 47(8) (2006), 082104.Google Scholar
[6]Haubold, H.J., Mathai, A.M. and Saxena, R.K., Solution of reaction-diffusion equations in terms of the H-function, Bull. Astro. Soc. India., 35(4)(2007), pp. 681–689.Google Scholar
[7]Hilfer, R., Fractional time evolution, in: Hilfer, R.(Editor), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130.Google Scholar
[8]Kilbas, A. A., Srivastava, H. M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[9]Laskin, N., Fractals and quantum mechanics, Chaos, 10(4) (2000), pp. 780–790.Google Scholar
[10]Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), pp. 298–305.Google Scholar
[11]Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66(5) (2002), 056108.CrossRefGoogle ScholarPubMed
[12]Mainardi, F. and Gorenflo, R., Time-fractional derivative in relaxation processes: a tutorial survey, Fract. Calc. Appl. Anal., 10(3) (2007), pp. 269–307.Google Scholar
[13]Mathai, A. M., Saxena, R. K and Haubold, H. J., The H-Function Theory and Applications, Springer, New York, 2010.Google Scholar
[14]Metzler, R. and Klafter, J., The random walk: a guide to anomalous diffusion: a fractional dynamics approach, Phy. Rep., 339 (2000), pp. 1–77.Google Scholar
[15]Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
[16]Naber, M., Distributed order fractional sub-diffusion, Fractals, 12(1) (2004), pp. 23–32.Google Scholar
[17]Nikolova, Y. and Boyadjiev, L., Integral transforms method to solve a time-space fractional diffusion equation, Fract. Calc. Appl. Anal., 13(1) (2010), pp. 57–67.Google Scholar
[18]Oldham, K. B. and Spanier, J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.Google Scholar
[19]Pagnini, R. and Mainardi, F., Evolution equations for the probabilistic generalization of Voigt profile function, Comput. Appl. Math., 233 (2010), pp. 1590–1595.Google Scholar
[20]Podlubny, I, Fractional Differential Equations (An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications), Academic Press, New York, 1999.Google Scholar
[21]Purohit, S. D. and Kalla, S. L., On fractional partial differential equations related to quantum mechanics, J. Phys. A Math. Theor., 44(4) (2011), 045202.Google Scholar
[22]Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishing, Yverdon, 1993.Google Scholar
[23]Saxena, R. K., Mathai, A. M. and Haubold, H. J., Fractional Reaction-Diffusion Equations, Astrophys. Space Sci., 305 (2006) Pp. 289–296.CrossRefGoogle Scholar
[24]Saxena, R. K., Mathai, A. M. and Haubold, H. J., Reaction-Diffusion Systems And Nonlinear Waves, Astrophys. Space Sci., 305 (2006), Pp. 297–303.Google Scholar
[25]Saxena, R. K., Mathai, A. M. and Haubold, H. J., Solution of generalized fractional reaction-diffusion equations, Astrophys. Space Sci., 305 (2006), pp. 305–313.Google Scholar
[26]Saxena, R. K., Saxena, R. and Kalla, S. L., Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics, Appl. Math. Comput., 216 (2010), pp. 1412–1417.Google Scholar
[27]Saxena, R. K., Saxena, R. and Kalla, S. L., Solution of space-time fractional Schrödinger equation occurring in quantum mechanics, Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190.Google Scholar
[28]Scherer, R., Kalla, S. L., Boyadjiev, L. and Al-Saqabi, B., Numerical treatment of fractional heat equations, Appl. Numer. Math., 58 (2008), pp. 1212–1223.Google Scholar
[29]Scherer, R., Kalla, S. L., Tang, Y. and Huang, J., The Grunwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62(3) (2011), pp. 902–917.CrossRefGoogle Scholar