### Analytic and Numerical Solutions of Time-Fractional Linear Schrödinger Equation

#### Abstract

Fractional Schrödinger Equation is a basic equation in fractional quantum mechanics. In this paper, we consider both analytic and numerical solutions of time-fractional linear Schrödinger Equations. This is done via a proposed semi-analytical method upon the modification of the classical Differential Transformation Method (DTM). Some illustrative examples are used; the results obtained converge faster to their exact forms. This shows that this modified version is very efficient, and reliable, as less computational work is involved, even without given up accuracy. Therefore, it is strongly recommended for both linear and nonlinear time-fractional partial differential equations (PDEs) with applications in other areas of applied sciences, management, and finance.

#### Keywords

#### Full Text:

PDF#### References

E. Schrödinger, "An Undulatory Theory of the Mechanics of Atoms and Molecules", Physical Review 28 (6): (1926), 1049–1070.

M.M. Mousa, S. F. Ragab, and Z. Nturforsch, “Application of the homotopy perturbation method to linear and nonlinear Schrödinger equations,” Zeitschrift Fur Naturforschung A, 63, no. 3-4, (2008): 140–144.

N. Laskin, “Fractional Quantum Mechanics and Lévy Path Integrals”, Physics Letters 268A, (2000): 298-304.

N. Laskin, “Fractional Schrödinger equation”, Physical Review E66, 056108, (2000): 7 pages. available online: http://arxiv.org/abs/quant-ph/0206098).

M. Naber, “Time fractional Schrodinger equation” J. Math. Phys. 45, (2004) 3339-3352. arXiv:math-ph/0410028.

S. Wang, M. Xu, “Generalized fractional Schrödinger equation with space-time fractional derivatives”, J. Math. Phys. 48 (2007) 043502.

I. Podlubny, “Fractional Differential Equations”, Academic Press, (1999).

M. Dalir, “Applications of Fractional Calculus”, Applied Mathematical Sciences, 4,(21), (2010): 1021-1032.

F. Mainardi, “On the Initial Value Problem for the Fractional Diffusion-Wave Equation, in: S. Rionero, T. Ruggeeri, Waves and Stability in continuous media”, World Scientific, Singapore (1994):246-251.

O. Abu Arqub, A. EI-Ajou, Z. Al Zhour, S. Momani, “Multiple solutions of nonlinear boundary value problems of fractional order: A new analytic iterative technique”, Entropy (16), (2014): 471–493.

J. Song, F. Yin, X. Cao and F. Lu, “Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations”, Journal of Apllied Mathematics, 2013, Article ID 392567, 10 pages.

S.A. Khuri, “A new approach to the cubic Schrödinger equation: an application of the decomposition technique”, Appl Math Comput., 97 (1988) 251-254.

J.H. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng., 178, (1999) 257–262.

H. Wang, “Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations”, Appl Math Comput., 170 (2005) 17-35.

S.T. Mohyud-Din, M. A. Noor and K. I. Noor, “Modified Variational Iteration Method for Schrödinger Equations”, Mathematical and Computational Applications, 15 (3), (2010), 309-317.

B. Jang, “Solving linear and nonlinear initial value problems by the projected differential transform method”, Computer Physics Communications 181 (2010), 848-854.

S.O. Edeki, G.O. Akinlabi, S.A. Adeosun, “On a modified transformation method for exact and approximate solutions of linear Schrödinger equations”, 2015 Progress in Applied Mathematics in Science and Engineering (PIAMSE), Conference proceedings, (in-press, 2015).

DOI: http://dx.doi.org/10.26713%2Fcma.v7i1.327

### Refbacks

- There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905