文章编号:1673-5196(2023)01-0158-06时间分数阶Cable方程修正格式的误差分析吴晓蕾*1,杨艳1,闫玉斌2(1.吕梁学院数学系,山西吕梁033000;2.彻斯特大学数学系,英国CH24BJ)摘要:考虑时间分数阶Cable方程在修正的二阶向后差分格式下的误差分析.利用连续Laplace变换、反Laplace变换方法得到方程的准确解,类似得到空间有限元半离散解;运用Lubich的修正方法引入此分数阶微分方程的修正格式,离散的Laplace变换、反Laplace变换法得到Cable方程的时间离散解,进而讨论了时间离散下L2范数的误差估计,得到二阶收敛阶,并用数值算例验证了定理的结论.这个结论比不修正的情形下一阶收敛阶要高.关键词:分数阶Cable方程;Riemann-Liouville分数阶导;Laplace变换;非光滑数据误差估计中图分类号:O241.8文献标志码:AErroranalysisofmodifiedschemesfortime-fractionalCableequationWUXiao-lei1,YANGYan1,YANYu-bin2(1.DepartmentofMathematics,LüliangUniversity,Lüliang033000,China;2.SchoolofMathematicsandStatistics,UniversityofChes-ter,Chester,CH24BJ,UK)Abstract:Erroranalysisofthemodifiedsecond-orderbackwarddifferenceschemeforthetime-fractionalCableequa-tioniscarriedout.ByusingcontinuousLaplacetransformandinverseLaplacetransform,theexactsolutionofthee-quationisobtained,andthefiniteelementsemidiscretesolutionisobtainedsimilarly.ThenLubich’scorrectionmeth-odisusedtogetthemodifiedformofthefractionaldifferentialequation.ThediscretesolutionsoftheCableequationareobtainedbymeansofthediscreteLaplacetransformandtheinverseLaplacetransform.Finally,theer-rorestimatesunderthenormarediscussedandthesecondorderofconvergenceisobtained.Numericalre-sultsverificationisfinallyperformedtovalidatethetheoreticalfindingsdiscussedhere,whichprovedthatitisbetterthanthefirstorderconvergencewithoutmodification.Keywords:fractionalCableequation;Riemann-Liouvillefractionalderivative;Laplacetransform;Nons-moothdataerrorestimation在希尔伯特空间H=L2(D)上考虑如下模型:u'(t)+R0DαtAu(t)+γR0Dβtu(t)=f(t)00,u0∈H,f(t)∈H,R0Dstu(t)代表s∈(0,1)阶Riemann-Liouville分数阶导,定义为[1-2]R0Dstu(t)=|Γ1-s()ddt∫t0t-σ()-sdσ收稿日期:2021-12-31基金项目:国家自...
