In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply $C^1$-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about $O\left(h^{min\{4-\alpha ,p\}}\right)$ while the interpolating function belongs to $C^p(p\ge 1)$, where h is the mesh size and $\alpha$ the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases.

中文翻译：

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变阶非线性分数阶微分方程的厄米三次样条配点法

在本文中，我们开发了一种Hermite三次样条搭配方法（HCSCM），用于求解变阶非线性分数阶微分方程，该方法适用于 $C^{\mathrm{1个}}$-连续的节点基础作用于一个近似问题。我们还验证了HCSCM的收敛顺序大约是$Ø（H^{分\{4-\alpha ，p\}}）$ 而插值函数属于 $C^p（p\ge \mathrm{1个}）$，其中h是网格尺寸，$\alpha$分数导数的顺序。为了验证HCSCM对于分数阶微分方程的有效性，进行了许多数值测试，其中包括Helmholtz方程以及具有Riemann-Liouville，Caputo和Patie-Simon感以及二阶常阶和变阶分数阶Burgers方程。案件。