How to empower Grünwald–Letnikov fractional difference equations with available initial condition?
Articles
Yiheng Wei
Southeast University
Jinde Cao
Southeast University
https://orcid.org/0000-0003-3133-7119
Chuang Li
Hainan University
https://orcid.org/0000-0001-5104-4589
Yangquan Chen
University of California
Published 2022-04-05
https://doi.org/10.15388/namc.2022.27.26623
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Keywords

fractional calculus
independence
initial condition
Grünwald–Letnikov definition
dynamic properties

How to Cite

Wei Y., Cao J., Li C. and Chen Y. (2022) “How to empower Grünwald–Letnikov fractional difference equations with available initial condition?”, Nonlinear Analysis: Modelling and Control, 27(4), pp. 650-668. doi: 10.15388/namc.2022.27.26623.

Abstract

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 aGkαx(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.

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