The Hilfer fractional derivative interpolates the commonly used Riemann-Liouville and Caputo fractional derivative. In general, solutions to Hilfer fractional differential equations are singular for $t \downarrow 0$ and are difficult to approximate with high numerical accuracy. We propose a numerical Bernstein splines technique to approximate solutions to generalized nonlinear initial values problems with Hilfer fractional derivatives. Convergent approximations are obtained using an efficient vectorized solution setup with few convergence requirements for a wide range of nonlinear fractional differential equations. We demonstrate efficiency of the developed method by applying it to the fractional Van der Pol oscillator, a system with applications in control systems and electronic circuits.

翻译：Hilfer分数阶导数插值于常用的Riemann-Liouville和Caputo分数阶导数之间。一般而言，Hilfer分数阶微分方程的解在$t \downarrow 0$时具有奇异性，难以用高数值精度进行逼近。我们提出了一种数值Bernstein样条技术，用于逼近具有Hilfer分数阶导数的广义非线性初值问题的解。通过采用高效的向量化解法设置，在较少的收敛要求下，为广泛的非线性分数阶微分方程获得了收敛的逼近解。我们将所开发的方法应用于分数阶Van der Pol振荡器（一个在控制系统和电子电路中具有应用价值的系统），从而证明了该方法的有效性。