### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 8 (2017)

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

### UDK: 517.958

### Pages: 61 - 69

### DOI: 10.31029/demr.8.7

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 $5/4 \leq\alpha\leq2$, $D_{0+}^\alpha u(t) + f(t,u(t)) = 0, \ 0 < t < 1,$ $u(0) = u(1) = 0$ 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 the derivative is understood in the sense of Riemann-Liouville.

**Keywords: **
Two-point boundary value problem, fractional derivative, positive solution, numerical method.