{"title":"Quintic non-polynomial spline for time-fractional nonlinear Schrödinger equation.","authors":"Qinxu Ding, Patricia J Y Wong","doi":"10.1186/s13662-020-03021-0","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the <i>L</i>1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and <math><mo>(</mo> <mn>2</mn> <mo>-</mo> <mi>γ</mi> <mo>)</mo></math> th order accurate in the temporal dimension, where <i>γ</i> is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.</p>","PeriodicalId":53311,"journal":{"name":"Advances in Difference Equations","volume":null,"pages":null},"PeriodicalIF":4.1000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7565234/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Difference Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13662-020-03021-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2020/10/16 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.
期刊介绍：
The theory of difference equations, the methods used, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last 15 years, the proliferation of the subject has been witnessed by hundreds of research articles, several monographs, many international conferences, and numerous special sessions.
The theory of differential and difference equations forms two extreme representations of real world problems. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. The actual behavior of the population is somewhere in between.
The aim of Advances in Difference Equations is to report mainly the new developments in the field of difference equations, and their applications in all fields. We will also consider research articles emphasizing the qualitative behavior of solutions of ordinary, partial, delay, fractional, abstract, stochastic, fuzzy, and set-valued differential equations.
Advances in Difference Equations will accept high-quality articles containing original research results and survey articles of exceptional merit.
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