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The following arose in a physical problem:

Solve the two equations \begin{eqnarray} p(t)+D^\alpha x(t)+\omega x(t)=0\\ _T^- D^\alpha p(t)-\omega p(t)+k x(t)=0 \end{eqnarray} subject to the conditions: $x(0)=x_i$ and $x(T)=x_f$. Given: $0<\alpha\leq 1$ and $\omega$ and $k$ are positive constants. In the above $D^\alpha$ is the Caputo fractional derivative $$D^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^tds \frac{ds}{(t-s)^{1-\alpha}}\frac{df(s)}{ds}$$ and $$_T^-D^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_t^T ds \frac{ds}{(s-t)^{1-\alpha}}f(s).$$

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  • $\begingroup$ Don't you mean $D^\alpha f(t) = \frac{1}{\Gamma(1-\alpha)}\int_0^t ds \frac{1}{(t-s)^{\alpha}}\frac{d}{ds}f(s)$ for the Caputo derivative? $\endgroup$ – Ton Feb 13 '18 at 22:57

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