How to empower Grünwald–Letnikov fractional difference equations with available initial condition?

Nonlinear Analysis

Chuang Li¹  Jinde Cao²  Yiheng Wei²  Yangquan Chen³ 

[1] Hainan University;Southeast University;University of California;

关键词: fractional calculus;    independence;    initial condition;    Grünwald–Letnikov definition;    dynamic properties;   

DOI  :  10.15388/namc.2022.27.26623

来源: DOAJ

【 摘 要 】

In this paper, the initial condition independence property of Grünwald–Letnikov fractional difference is revealed for the first time. For example, the solution x(k) of equation aG∇kαx(k) = f(x(k)),  k > a + 1, cannot be calculated with initial condition x(a). First, the initial condition independence property is carefully investigated in both time domain and frequency domain. Afterwards, some possible schemes are formulated to make the considered system connect to initial condition. Armed with this information, the concerned property is examined on three modified Grünwald–Letnikov definitions. Finally, results from illustrative examples demonstrate that the developed schemes are sharp.

【 授权许可】

Unknown