Numerical solution of time fractional Schrödinger equation by using quadratic B-spline finite elements
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.
Keywords
finite element method; Galerkin method; time fractional Schrödinger equation; quadratic B-spline
References
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2. Ertürk V.S., Momani S., Solving systems of fractional differential equations using differential transform method, J. Comput. Appl. Math. 215 (2008), 142–151.
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.
4. Esen A., Tasbozan O., Cubic B-spline collocation method for solving time fractional gas dynamics equation, Tbilisi Math. J. 8 (2015), 221–231.
5. Esen A., Tasbozan O., Numerical solution of time fractional Burgers equation, Acta Univ. Sapientiae Math. 7 (2015), 167–185.
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.
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.
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.
9. Esen A., Yagmurlu N.M., Tasbozan O., Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations, Appl. Math. Inf. Sci. 7 (2013), 1951–1956.
10. Jafari H., Momani S., Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A 370 (2007), 388–396.
11. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
12. Logan D.L., A first course in the finite element method, Fourth edition, Thomson, Toronto 2007.
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.
14. Miller K.S., Ross B., An introduction to the fractional calculus and fractional differantial equations, John Wiley, New York, 1993.
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.
16. Momani S., Odibat Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006), 271–279.
17. Momani S., Odibat Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton Fractals 31 (2007), 1248–1255.
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.
19. Oldham K.B., Spainer J., The fractional calculus, Academic Press, New York, 1974.
20. Otteson N., Pettorson H., Introduction to the finite element method, Prentice Hall, London, 1992.
21. Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999.
22. Prenter P.M., Splines and variasyonel methods, John Wiley, New York, 1975.
23. Shawagfeh N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002), 517–529.
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.
25. Yuste S.B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), 264–274.
26. Yuste S.B., Acedo L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42 (2005), 1862–1874.
27. Wang Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Soliton Fractals 35 (2008), 843–850.
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.
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.
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.
EsenA., & TasbozanO. (2016). Numerical solution of time fractional Schrödinger equation by using quadratic B-spline finite elements. Annales Mathematicae Silesianae, 31, 83-98. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13941
Alaattin Esen
Department of Mathematics, Inönü University, Turkey Turkey
Department of Mathematics, Inönü University, Turkey Turkey
Orkun Tasbozan
orkun.tasbozan@inonu.edu.tr
Department of Mathematics, Mustafa Kemal University, Turkey Turkey
Department of Mathematics, Mustafa Kemal University, Turkey Turkey
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