THERMAL SCIENCE

International Scientific Journal

External Links

ON SOLUTIONS OF LOCAL FRACTIONAL SCHRODINGER EQUATION

ABSTRACT
In this study, we obtain the solution of a local fractional Schrödinger equation (LFSE). The solution is obtained by the implementation of the Laplace transform (LT) and Fourier transform (FT) in closed form in terms of the Mittag-Leffler function (MLF).
KEYWORDS
PAPER SUBMITTED: 2019-01-30
PAPER REVISED: 2019-05-25
PAPER ACCEPTED: 2019-06-26
PUBLISHED ONLINE: 2019-09-15
DOI REFERENCE: https://doi.org/10.2298/TSCI190130353Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 6, PAGES [S1929 - S1934]
REFERENCES
  1. Hilfer, R., Applications of fractional calculus in physics, World Scientific, Singapore, 2000
  2. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J. A., Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, New York, 2007
  3. Carpinteri A., Mainardi F., Fractals and fractional calculus in continuum mechanics, Springer, New York, 1997
  4. Koeller, R.C., Applications of fractional calculus to the theory of visco-elasticity. J. Appl. Mech. 51 (1984), 2, pp. 299-307
  5. Mainardi, F., Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, Singapore, 2009
  6. Laskin, N., Fractional quantum mechanics. Phys. Rev. E, 62 (2000), pp. 3135-3145
  7. Tofight, A., Probability structure of time fractional Schrödinger equation. Acta Physica Polonica A, 116 (2009), 2, pp. 111-118
  8. Chaurasia, V.B.L, Kumar, D., Solutions of Unified Fractional Schrödinger Equations, ISRN Mathematical Physics, 2012 (2012), 7 pages, doi:10.5402
  9. Oldham, K.B., Spanier, J., The Fractional Calculus, Academic Press, New York, 1974
  10. Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993
  11. Podlubny, I., Fractional Differential Equations, Academic Press, New York, New York, 1999
  12. Samko, S.G., Kilbas, A.A, Marichev, O.I, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993
  13. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006
  14. Ben-Adda, F., Cresson, J., About non-differentiable functions, J. Math. Anal. Appl. 263 (2001), pp. 721-737
  15. Babakhani, A., Daftardar-Gejji, V., On calculus of local fractional derivatives, J. Math. Anal. Appl. 270 (2002), pp. 66-79
  16. Kolwankar, K.M., Gangal, A.D., Fractional differentiability of nowhere differentiable functions and dimension, Chaos, 6 (1996), pp. 505-513
  17. Kolwankar, K.M., Gangal, A.D., Hölder exponents of irregular signals and local fractional derivatives, Pramana, 48 (1997), pp. 49-68
  18. Chen, Y., Yan, Y., Zhang, K., On the local fractional derivative, J. Math. Anal. Appl., 362 (2010), pp. 17-33
  19. Kolwankar, K.M., Lévy Véhel, J., Measuring functions smoothness with local fractional derivatives, Fract. Calc. Appl. Anal. 4 (2001), pp. 49-68
  20. Yang, X.J., Local Fractional Integral Transforms. Progress in Nonlinear Science, 4 (2011), pp. 1-225
  21. Yang, X.J., Local Fractional Functional Analysis and Its Applications, Asian Academic publisher, Hong Kong, 2011
  22. Yang, X.J., Advanced local fractional calculus and its applications, World Science Publisher, New York, 2012
  23. Yang, X.J., Local fractional Fourier analysis, Advances in Mechanical Engineering and its Applications, 1 (2012), 1, pp. 12-16
  24. Yang, X.J., Liao, M.K., and Chen, J.W., A novel approach to processing fractal signals using the Yang-Fourier transforms. Procedia Engineering, 29 (2012), pp. 2950-2954
  25. Guo, S.M., Mei, L. Q., Li Y., and Sun., Y.F., The improved fractional sub-equation method and its applications to the spacetime fractional differential equations in fluid mechanics. Phys. Lett. A., 376 (2011), 4, pp. 407-411
  26. Yang, X.J., A short introduction to Yang-Laplace transforms in fractal space, Advances in Information Technology and Management, 1 (2012), 2, pp. 38-43
  27. Yang, X.J., The discrete Yang-Fourier transforms in fractal space, Advances in Electrical Engineering Systems, 1 (2012), 2, pp. 78-81
  28. Yang, X.J., Machado, J.A.T., Nieto, J.J., A new family of the local fractional PDEs. Fundamenta Informaticae, 151 (2017), pp. 63-75
  29. Yang, X.J., Machado, J.A.T., Srivastava, H.M., A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Applied Mathematics and Computation, 274 (2016), pp. 143-151
  30. Singh, J., Kumar, D., Baleanu, D., Rathore, S., On the local fractional wave equation in fractal strings, Mathematical Methods in the Applied Sciences, 42 (2019), 5, pp. 1588-1595
  31. Kumar, D., Singh, J., Purohit, S.D. and Swroop, R., A hybrid analytical algorithm for nonlinear fractional wave-like equations, Mathematical Modelling of Natural Phenomena, 14 (2019), 3, 304
  32. Yang, X.J., Baleanu, D. and Srivastava, H.M., Local Fractional Integral Transforms and Their Applications, Elsevier, New York, Academic Press., 2016
  33. Compte, A., Stochastic foundations of fractional dynamics, Physical Review E, vol. 53 (1996), 4, pp. 4191-4193
  34. Metzler, R., Klafter, J., The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Report, 339 (2000), 1, pp. 1-77

© 2020 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence