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 \(\begin{cases}    ^{C}D_{0+}^{\alpha}y(x)+1.8 y(x)=0,\\    x\in \langle 0,2\rangle \quad \hbox{and} \quad \alpha=\{1.0,\ 0.8,\ 0.6,\ 0.4,\ 0.2\},\\    y(0)=1.\end{cases}\)