Geodesic
Two-point boundary value problem of a non-linear differential equation with fractional derivatives, having exponential growth by solution
Daghestan Electronic Mathematical Reports, Tome 8 (2017), pp. 61-69

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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.
Keywords: two-point boundary value problem, fractional derivative, positive solution, numerical method. 
@article{DEMR_2017_8_a6,
     author = {E. I. Abduragimov and R. A. Omarova},
     title = {Two-point boundary value problem of a non-linear differential equation with fractional derivatives, having exponential growth by solution},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {61--69},
     publisher = {mathdoc},
     volume = {8},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2017_8_a6/}
}
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E. I. Abduragimov; R. A. Omarova. Two-point boundary value problem of a non-linear differential equation with fractional derivatives, having exponential growth by solution. Daghestan Electronic Mathematical Reports, Tome 8 (2017), pp. 61-69. http://geodesic.mathdoc.fr/item/DEMR_2017_8_a6/