Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements

Esen, Alaattin; Tasbozan, Orkun

doi:10.1515/amsil-2016-0015

Abstract

In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms L₂ and L_∞ are presented in tables.

References

[1] Dehghan M., Jalil M., Abbas S., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Meth. Part. D. E. 26 (2010), 448-479.10.1002/num.20460
Search in Google Scholar Back to article
[2] Ertürk V.S., Momani S., Solving systems of fractional differential equations using differential transform method, J. Comput. Appl. Math. 215 (2008), 142-151.10.1016/j.cam.2007.03.029
Search in Google Scholar Back to article
[3] Esen A., Tasbozan O., An approach to time fractional gas dynamics equation: Quadratic B-spline Galerkin method, Appl. Math. Comput. 261 (2015), 330-336.
Search in Google Scholar Back to article
[4] Esen A., Tasbozan O., Cubic B-spline collocation method for solving time fractional gas dynamics equation, Tbilisi Math. J. 8 (2015), 221-231.
Search in Google Scholar Back to article
[5] Esen A., Tasbozan O., Numerical solution of time fractional Burgers equation, Acta Univ. Sapientiae Math. 7 (2015), 167-185.
Search in Google Scholar Back to article
[6] Esen A., Tasbozan O., Numerical solution of time fractional Burgers equation by cubic B-spline finite elements, Mediterr. J. Math. 13 (20016), 1325-1337.10.1007/s00009-015-0555-x
Search in Google Scholar Back to article
[7] Esen A., Tasbozan O., Ucar Y., Yagmurlu N.M., A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math. J. 8 (2015), 181-193.
Search in Google Scholar Back to article
[8] Esen A., Ucar Y., Yagmurlu N.M., Tasbozan O., A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations, Math. Model. Anal. 18 (2013), 260-273.10.3846/13926292.2013.783884
Search in Google Scholar Back to article
[9] Esen A., Yagmurlu N.M., Tasbozan O., Approximate analytical solution to timefractional damped Burger and Cahn-Allen equations, Appl. Math. Inf. Sci. 7 (2013), 1951-1956.10.12785/amis/070533
Search in Google Scholar Back to article
[10] Jafari H., Momani S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A 370 (2007), 388-396.10.1016/j.physleta.2007.05.118
Search in Google Scholar Back to article
[11] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
Search in Google Scholar Back to article
[12] Logan D.L., A first course in the finite element method, Fourth edition, Thomson, Toronto 2007.
Search in Google Scholar Back to article
[13] Machado J.A.T., Silva M.F., Barbosa R.S., Jesus I.S., Reis C.M., Marcos M.G., Galhano A.F., Some applications of fractional calculus in engineering, Math. Probl. Eng. 2010, Article ID 639801, 34 pp., http://dx.doi.org/10.1155/2010/639801.10.1155/2010/639801
Search in Google Scholar Back to article
[14] Miller K.S., Ross B., An introduction to the fractional calculus and fractional differantial equations, John Wiley, New York, 1993.
Search in Google Scholar Back to article
[15] Mohebbi A., Abbaszadeh M., Dehghan M., The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem. 37 (2013), 475-485.10.1016/j.enganabound.2012.12.002
Search in Google Scholar Back to article
[16] Momani S., Odibat Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006), 271-279.10.1016/j.physleta.2006.02.048
Search in Google Scholar Back to article
[17] Momani S., Odibat Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton Fractals 31 (2007), 1248-1255. 10.1016/j.chaos.2005.10.068
Search in Google Scholar Back to article
[18] Odibat Z., Momani S., A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett. 21 (2008), 194-199.10.1016/j.aml.2007.02.022
Search in Google Scholar Back to article
[19] Oldham K.B., Spainer J., The fractional calculus, Academic Press, New York, 1974.
Search in Google Scholar Back to article
[20] Otteson N., Pettorson H., Introduction to the finite element method, Prentice Hall, London, 1992.
Search in Google Scholar Back to article
[21] Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999.
Search in Google Scholar Back to article
[22] Prenter P.M., Splines and variasyonel methods, John Wiley, New York, 1975.
Search in Google Scholar Back to article
[23] Shawagfeh N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002), 517-529.
Search in Google Scholar Back to article
[24] Tasbozan O., Esen A., Yagmurlu N.M., Ucar Y., A numerical solution to fractional diffusion equation for force-free case, Abstr. Appl. Anal. 2013, Article ID 187383, 6 pp., http://dx.doi.org/10.1155/2013/187383.10.1155/2013/187383
Search in Google Scholar Back to article
[25] Yuste S.B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), 264-274.10.1016/j.jcp.2005.12.006
Search in Google Scholar Back to article
[26] Yuste S.B., Acedo L., An explicit finite difference method and a new von Neumanntype stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005), 1862-1874.10.1137/030602666
Search in Google Scholar Back to article
[27] Wang Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Soliton Fractals 35 (2008), 843-850.10.1016/j.chaos.2006.05.074
Search in Google Scholar Back to article
[28] Wei L., He Y., Zhang X., Wang S., Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation, Finite Elem. Anal. Des. 59 (2012), 28-34.10.1016/j.finel.2012.03.008
Search in Google Scholar Back to article
[29] Wei L., Zhang X., Kumar S., Yildirim A., A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system, Comput. Math. Appl. 64 (2012), 2603-2615.10.1016/j.camwa.2012.07.004
Search in Google Scholar Back to article
[30] Wua G.C., Baleanu D., Variational iteration method for the Burgers flow with fractional derivatives-New Lagrange multipliers, Appl. Math. Model. 37 (2013), 6183-6190.10.1016/j.apm.2012.12.018
Search in Google Scholar Back to article

Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements

Abstract

Paradigm

My account