Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition
In this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form...
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EJAAM
2020-12-01
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| Series: | E-Journal of Analysis and Applied Mathematics |
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| Online Access: | https://ejaam.org/articles/2020/10.2478-ejaam-2020-0006.pdf |
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| author | Betul Hicdurmaz |
| author_facet | Betul Hicdurmaz |
| author_sort | Betul Hicdurmaz |
| collection | DOAJ |
| description | In this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form of the nonlinear fractional Schrödinger equation are revisited to derive new applications of these theorems on the nonlinear fractional Schrödinger problems with Neumann boundary condition. Consequently, first and second order of accuracy difference schemes are constructed for the nonlinear fractional Schrödinger problem with Neumann boundary condition. Stability analysis show that the constructed difference schemes are stable. Stability theorems for the stability of the nonlinear fractional Schrödinger problem with Neumann boundary condition are presented. Additionally, applications of the new theoretical results are presented on a one dimensional nonlinear fractional Schrödinger problem and a multidimensional nonlinear fractional Schrödinger problem with Neumann boundary conditions. Numerical results are presented on one and multidimensional nonlinear fractional Schrödinger problems with Neumann boundary conditions and different orders of derivatives in fractional derivative term. Numerical results support the validity and applicability of the theoretical results. Numerical results present the convergence rates are appropriate with the theoretical findings and construction of the difference schemes for the nonlinear fractional Schrödinger problem with Neumann boundary condition. |
| format | Article |
| id | doaj-art-b58ffa1ef06c479e8a61261e72edea18 |
| institution | Kabale University |
| issn | 2544-9990 |
| language | English |
| publishDate | 2020-12-01 |
| publisher | EJAAM |
| record_format | Article |
| series | E-Journal of Analysis and Applied Mathematics |
| spelling | doaj-art-b58ffa1ef06c479e8a61261e72edea182025-02-08T18:35:22ZengEJAAME-Journal of Analysis and Applied Mathematics2544-99902020-12-01202010.2478/ejaam-2020-0006Finite difference method for a nonlinear fractional Schrödinger equation with Neumann conditionBetul Hicdurmaz0Department of Mathematics, Faculty of Engineering and Natural Sciences, Istanbul Medeniyet University, Istanbul, TurkeyIn this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form of the nonlinear fractional Schrödinger equation are revisited to derive new applications of these theorems on the nonlinear fractional Schrödinger problems with Neumann boundary condition. Consequently, first and second order of accuracy difference schemes are constructed for the nonlinear fractional Schrödinger problem with Neumann boundary condition. Stability analysis show that the constructed difference schemes are stable. Stability theorems for the stability of the nonlinear fractional Schrödinger problem with Neumann boundary condition are presented. Additionally, applications of the new theoretical results are presented on a one dimensional nonlinear fractional Schrödinger problem and a multidimensional nonlinear fractional Schrödinger problem with Neumann boundary conditions. Numerical results are presented on one and multidimensional nonlinear fractional Schrödinger problems with Neumann boundary conditions and different orders of derivatives in fractional derivative term. Numerical results support the validity and applicability of the theoretical results. Numerical results present the convergence rates are appropriate with the theoretical findings and construction of the difference schemes for the nonlinear fractional Schrödinger problem with Neumann boundary condition.https://ejaam.org/articles/2020/10.2478-ejaam-2020-0006.pdffractional derivativeconvergenceneumann boundary condition |
| spellingShingle | Betul Hicdurmaz Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition E-Journal of Analysis and Applied Mathematics fractional derivative convergence neumann boundary condition |
| title | Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition |
| title_full | Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition |
| title_fullStr | Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition |
| title_full_unstemmed | Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition |
| title_short | Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition |
| title_sort | finite difference method for a nonlinear fractional schrodinger equation with neumann condition |
| topic | fractional derivative convergence neumann boundary condition |
| url | https://ejaam.org/articles/2020/10.2478-ejaam-2020-0006.pdf |
| work_keys_str_mv | AT betulhicdurmaz finitedifferencemethodforanonlinearfractionalschrodingerequationwithneumanncondition |