A new method for approximating fractional derivatives/ integrals as a series of higher-integer-order derivatives - examples and results of applying the method to initial/boundary value problems

The posted research data includes examples of the application of the author's fractional derivative/integral approximation method using the sum of higher integer derivatives. The attached text files contain the numerical solutions of the presented examples, recorded as a set of numerical values obtained from the performed computations. Example 4.1 \(\begin{cases} \displaystyle ^{C}D^{\alpha}_{a+}\sin (x), \\ x \in \langle a, 3\pi \rangle \quad \hbox{and} \quad \alpha = \{1.0,\ 0.8,\ 0.6,\ 0.4,\ 0.2\}, \end{cases}\) Example 4.2 \( \begin{cases} \displaystyle I^{\alpha}_{0+} e^{-x}\cos 7x, \\ x \in \langle 0,1\rangle \quad \hbox{and} \quad \alpha =\{1.0,\ 1.2,\ 1.4,\ 1.6,\ 1.8,\ 2.0 \}, \end{cases} \) Example 5.1 \(\begin{cases} ^{C} D_{0+}y(x)+2y(x)=x+ \frac{2x^{\alpha+1}}{\Gamma(\alpha+2)},\\ x\in\langle0,1\rangle, \\ y(0) = 0; \quad y(1) = \frac{1}{\Gamma(\alpha+2)}, \\ \alpha = \{1.2,\ 1.4,\ 1.6,\ 1.8,\ 2.0\}. \end{cases}\) Example 5.2 \(\begin{cases} ^{C}D_{0+}^{\alpha}y(x)+1.8 y(x)=0,\\ x\in \langle 0,2\rangle \quad \hbox{and} \quad \alpha=\{1.0,\ 0.8,\ 0.6,\ 0.4,\ 0.2\},\\ y(0)=1.\end{cases}\)本次公开的研究数据集包含作者提出的基于高阶整数导数和的分数阶微分与积分近似方法的应用实例。附带的文本文件收录了各实例的数值解，即通过计算得到的离散数值结果集。 例4.1 $egin{cases} displaystyle {}^{C}D^{alpha}_{a+}sin(x), \ xinlangle a, 3pi angle quad ext{且} quad alpha={1.0, 0.8, 0.6, 0.4, 0.2}, end{cases}$ 其中${}^{C}D^{alpha}_{a+}$为$a$处的$alpha$阶Caputo分数阶导数（Caputo fractional derivative）。 例4.2 $egin{cases} displaystyle I^{alpha}_{0+} e^{-x}cos7x, \ xinlangle0,1 angle quad ext{且} quad alpha={1.0, 1.2, 1.4, 1.6, 1.8, 2.0}, end{cases}$ 其中$I^{alpha}_{0+}$为$0$处的$alpha$阶Riemann-Liouville分数阶积分（Riemann-Liouville fractional integral）。 例5.1 $egin{cases} {}^{C}D_{0+}y(x)+2y(x)=x+frac{2x^{alpha+1}}{Gamma(alpha+2)}, \ xinlangle0,1 angle, \ y(0) = 0; quad y(1) = frac{1}{Gamma(alpha+2)}, \ alpha = {1.2, 1.4, 1.6, 1.8, 2.0}. end{cases}$ 其中$Gamma(cdot)$为伽马函数（Gamma function），${}^{C}D_{0+}y(x)$为$0$处的Caputo分数阶导数。 例5.2 $egin{cases} {}^{C}D^{alpha}_{0+}y(x)+1.8 y(x)=0, \ xinlangle0,2 angle quad ext{且} quad alpha={1.0, 0.8, 0.6, 0.4, 0.2}, \ y(0)=1.end{cases}$