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Symmetric differential operators of fractional order and their extensions. (English. Russian original) Zbl 1420.45004
Trans. Mosc. Math. Soc. 2018, 177-185 (2018); translation from Tr. Mosk. Mat. O.-va 79, No. 2, 209-219 (2018).
Fractional differential operators are nowadays subject of intensive study. The authors aim here to produce a self-adjoint problem in this context. They consider first the fractional operator of right-hand Riemann-Liouville type $D^\lambda f(t)=- (d/dt)\,\Gamma(1-\lambda)^{-1} \int^1_t(t-s)^{-\lambda} f(s)\,ds,$ which is obviously non-symmetric. The left-hand fractional Caputo operator $$\mathcal{D}^\lambda$$ is then defined similarly. By assuming $$1/2<\lambda<1$$ and $$0<x<1$$, the authors draw their attention to the $$2\lambda$$-fractional order operator $Lu(x)=\mathcal{D}^\lambda D^\lambda u(x)$ in $$L^2([0,1])$$. Under suitable boundary conditions, they prove self-adjointness and discuss the spectral properties of the problem. Such an investigation in fractional calculus is new and interesting.

##### MSC:
 45J05 Integro-ordinary differential equations 34A08 Fractional ordinary differential equations 35P05 General topics in linear spectral theory for PDEs
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