Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2019
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| _version_ | 1866909334738305024 |
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| author | Yin, BaoLi Liu, Yang Li, Hong Zhang, Zhimin |
| author_facet | Yin, BaoLi Liu, Yang Li, Hong Zhang, Zhimin |
| contents | In this article, we introduce two families of novel fractional $θ$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^α$ with a second order convergence rate. A new fractional BT-$θ$ method connects the fractional BDF2 (when $θ=0$) with fractional trapezoidal rule (when $θ=1/2$), and another novel fractional BN-$θ$ method joins the fractional BDF2 (when $θ=0$) with the second order fractional Newton-Gregory formula (when $θ=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $θ$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $θ$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_01242 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations Yin, BaoLi Liu, Yang Li, Hong Zhang, Zhimin Numerical Analysis In this article, we introduce two families of novel fractional $θ$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^α$ with a second order convergence rate. A new fractional BT-$θ$ method connects the fractional BDF2 (when $θ=0$) with fractional trapezoidal rule (when $θ=1/2$), and another novel fractional BN-$θ$ method joins the fractional BDF2 (when $θ=0$) with the second order fractional Newton-Gregory formula (when $θ=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $θ$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $θ$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis. |
| title | Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/1906.01242 |