Two-point boundary value problem of a non-linear differential equation with fractional derivatives, having exponential growth by solution

Sufficient conditions for the existence and uniqueness of the positive solution of a two-point boundary value problem for a differential equation with fractional derivatives of order $5/4 \leq\alpha\leq2$,
\begin{equation}\label{eq0} D_{0+}^\alpha u(t) + f(t,u(t)) = 0, \ 0 < t < 1, \end{equation}

$$u(0) = u(1) = 0$$
in the case when $f(t,u)$ has exponential growth with respect to $u$. Moreover, a numerical method for constructing this solution is indicated, and the dependence of the solution on the order of differentiation on a particular example is investigated. In the equation \eqref{eq0} the derivative is understood in the sense of Riemann-Liouville.