一种用于多项式时间分数阶微分方程的扩展三次B样条Galerkin有限元方法 Sobia Ashraf, Muhammad Yaseen, Khidir Shaib Mohamed, Alawia Adam, Muntasir Suhail

《Axioms》：An Extended Cubic B-Spline Galerkin Finite Element Method for Multi-Term Time-Fractional Differential Equations Sobia Ashraf, Muhammad Yaseen, Khidir Shaib Mohamed, Alawia Adam and Muntasir Suhail

　　

摘要

本研究提出了一种扩展的三次B样条Galerkin方法，用于数值求解多项式时间分数阶微分方程。该方法结合了扩展的三次B样条和Caputo分数阶导数来模拟时间分数阶算子，并使用高斯求积法精确计算积分。文中对算法的稳定性进行了分析，并通过以下公式评估了其精度：


L

sub 2'
data-speech-attached="true" data-semantic-braille="???" data-braille-attached="true" has-speech="true">



L

sub 2'
data-semantic-summary=" subscript'
data-semantic-braille="???">

L
'
data-semantic-prefix=" Base'
data-semantic-summary=" identifier'
data-semantic-braille="??">
??

2'
data-semantic-prefix=" Subscript'
data-semantic-summary=" number"
data-semantic-braille="?">
2








L

sub infinity'
data-speech-attached="true" data-semantic-braille="?????" data-braille-attached="true" has-speech="true">



L

sub infinity'
data-semantic-summary=" subscript'
data-semantic-braille="?????">

L
'
data-semantic-prefix=" Base'
data-semantic-summary=" identifier'
data-semantic-braille="??">
??

infinity'
data-semantic-prefix=" Subscript'
data-semantic-summary=" identifier"
data-semantic-braille="??">
∞





该算法在不同空间节点和网格细化情况下的误差范数表明，数值结果与精确解非常吻合，这一点通过表格和图表得到了验证。这些发现证实了所提出方法在求解多项式时间分数阶微分方程时的鲁棒性、高效性和可靠性。