How to empower Grünwald–Letnikov fractional difference equations with available initial condition?
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 indepen...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2022-04-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://www.journals.vu.lt/nonlinear-analysis/article/view/26623 |
| _version_ | 1818160425667133440 |
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| author | Yiheng Wei Jinde Cao Chuang Li Yangquan Chen |
| author_facet | Yiheng Wei Jinde Cao Chuang Li Yangquan Chen |
| author_sort | Yiheng Wei |
| collection | DOAJ |
| description |
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.
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| first_indexed | 2024-12-11T16:01:41Z |
| format | Article |
| id | doaj.art-bce5efc7d8b643b6b855c87066fbe843 |
| institution | Directory Open Access Journal |
| issn | 1392-5113 2335-8963 |
| language | English |
| last_indexed | 2024-12-11T16:01:41Z |
| publishDate | 2022-04-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj.art-bce5efc7d8b643b6b855c87066fbe8432022-12-22T00:59:18ZengVilnius University PressNonlinear Analysis1392-51132335-89632022-04-012710.15388/namc.2022.27.26623How to empower Grünwald–Letnikov fractional difference equations with available initial condition?Yiheng Wei0Jinde Cao1Chuang Li2Yangquan Chen3Southeast UniversitySoutheast UniversityHainan UniversityUniversity of California 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. https://www.journals.vu.lt/nonlinear-analysis/article/view/26623fractional calculusindependenceinitial conditionGrünwald–Letnikov definitiondynamic properties |
| spellingShingle | Yiheng Wei Jinde Cao Chuang Li Yangquan Chen How to empower Grünwald–Letnikov fractional difference equations with available initial condition? Nonlinear Analysis fractional calculus independence initial condition Grünwald–Letnikov definition dynamic properties |
| title | How to empower Grünwald–Letnikov fractional difference equations with available initial condition? |
| title_full | How to empower Grünwald–Letnikov fractional difference equations with available initial condition? |
| title_fullStr | How to empower Grünwald–Letnikov fractional difference equations with available initial condition? |
| title_full_unstemmed | How to empower Grünwald–Letnikov fractional difference equations with available initial condition? |
| title_short | How to empower Grünwald–Letnikov fractional difference equations with available initial condition? |
| title_sort | how to empower grunwald letnikov fractional difference equations with available initial condition |
| topic | fractional calculus independence initial condition Grünwald–Letnikov definition dynamic properties |
| url | https://www.journals.vu.lt/nonlinear-analysis/article/view/26623 |
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