时间分数阶扩散方程的高阶Diethelm方法  

Higher order Diethelm method for the time fractional diffusion equation

作　　者：杨艳 王希云[2] YANG Yan;WANG Xi-yun(Department of mathematics,Luliang University,Luliang033000,China;School of Applied Sciences,Taiyuan University of Science and Technology,Taiyuan030024,China)

机构地区：[1]吕梁学院数学系,山西吕梁033000 [2]太原科技大学应用科学学院,山西太原030024

出　　处：《兰州理工大学学报》2020年第4期157-163,共7页Journal of Lanzhou University of Technology

基　　金：国家自然科学基金(11771184);山西省自然科学基金(201801D121010);山西省高等学校科技创新项目(2020L0700)。

摘　　要：针对时间分数阶扩散方程,提出了一种新的隐式差分方法,其中空间导数采用中心差分方法离散.对于时间分数阶导数,将Caputo分数阶导数转化为Riemman-Liouville分数阶导数后,写成Hadamard有限部分积分,再用分段二次多项式对该有限积分部分逼近,由此推导出Caputo分数阶导数的3-α阶离散方法,从而得到无条件稳定的和收敛的分数阶扩散方程的隐式差分格式.数值实验验证该隐式差分格式的有效性.A new implicit difference method is proposed for a time fractional diffusion equation,in which space derivatives are discretized by the central difference method.For a time fractional derivative,the Caputo fractional derivative is transformed into the Riemman-Liouville fractional derivative,and further forming it in the form of the Hadamard finite part integral.The finite part integral is then approximated by piecewise quadratic polynomials.A new 3-αorder approximation scheme to the Riemman-Liouville fractional derivative can be derived as result of the approximation.Consequently,an implicit difference scheme for fractional diffusion equations,which is unconditionally stable and convergent,can be obtained.Our numerical experiments verify effectiveness of the implicit difference scheme.

关 键 词：分数阶导数 Hadamard有限部分积分 分段二次插值多项式 稳定性 

分 类 号：O241.8[理学—计算数学]