On a boundary value problem for a high order mixed type equation

Obtaining a representation of the solution of the Cauchy problem for one equation with a fractional derivative and applying it to the equation of forced beam vibrations

“…The Cauchy problem for equations with fractional derivatives of Dzhrbashyan‐Nersesyan and Riemann‐Liouville was considered in Pskhu 8,9 and Karasheva, 10 respectively. We will use the ideas from these works.…”

“…Following Karasheva, 10 consider the function $$$$ {\Gamma}_b\left(x-\xi, y\right)=\frac{y^b}{2n}\sum \limits_{k=0}^{n-1}{\lambda}_k\phi \left(-\frac{\alpha }{2n},b+1;-{\lambda}_k\left|x-\xi \right|{y}^{-\frac{\alpha }{2n}}\right), $$$$ where $$$$ \phi \left(-\delta, \varepsilon; z\right)=\sum \limits_{k=0}^{\infty}\frac{z^k}{k!\Gamma \left(-\delta k+\varepsilon \right)} $$$$ is the Wright function, 15 $$$$ b\in R,{\lambda}_k^{2n}={\left(-1\right)}^{n-1}, $$$$ $$$$ {\lambda}_k={e}^{\frac{2k-n+1}{2n} i\pi},k=\overline{0,n-1}; $$$$ it is not hard to check that …”