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 \(\begin{cases} ^{C}D_{0+}^{\alpha}y(x)+1.8 y(x)=0,\\ x\in \langle 0,2\rangle \quad \hbox{and} \quad \alpha=\{1.0,\ 0.8,\ 0.6,\ 0.4,\ 0.2\},\\ y(0)=1.\end{cases}\)本公开的研究数据包含作者提出的基于高阶整数导数和的分数阶微分/积分近似方法的应用实例。附件文本文件存储了上述实例的数值解，即以计算所得的一组数值形式记录的结果。 实例4.1： $$ egin{cases} displaystyle {}_{a+}^C D^alpha sin(x), \ x in langle a, 3pi angle, quad alpha = {1.0, 0.8, 0.6, 0.4, 0.2} end{cases} $$ 其中${}_{a+}^C D^alpha$为Caputo分数阶导数（Caputo fractional derivative）。 实例4.2： $$ egin{cases} displaystyle {}_{0+}I^alpha left( e^{-x}cos7x ight), \ x in langle 0,1 angle, quad alpha = {1.0, 1.2, 1.4, 1.6, 1.8, 2.0} end{cases} $$ 其中${}_{0+}I^alpha$为Riemann-Liouville分数阶积分（Riemann-Liouville fractional integral）。 实例5.1： $$ egin{cases} {}_{0+}^C D^alpha y(x) + 2y(x) = x + frac{2x^{alpha+1}}{Gamma(alpha+2)}, \ xinlangle0,1 angle, \ y(0) = 0, quad y(1) = frac{1}{Gamma(alpha+2)}, \ alpha = {1.2, 1.4, 1.6, 1.8, 2.0} end{cases} $$ 其中$Gamma(cdot)$为伽马函数（Gamma function）。 实例5.2： $$ egin{cases} {}_{0+}^C D^alpha y(x) + 1.8y(x) = 0, \ xinlangle0,2 angle, quad alpha = {1.0, 0.8, 0.6, 0.4, 0.2}, \ y(0) = 1 end{cases} $$