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Русская версия

**Alexander Fedorovich Tedeev**

PhD in phisics and mathematics

Assoc.Professor

Dept. of functional analysis and differential equations

K. Khetagurov North-Osetian State University

Vatutin str. 44-46,

362025, Vladikaukaz, RSO-A, Russia

In this paper we consider a mixed Dirichlet problem for a linear homogeneous fractional diffusion equation. The existence and uniqueness of the solution of the problem using the harmonic continuation of the solution is proved. As a result, the solution of the differential equation is defined as a trace of a harmonic function. This approach allows us to introduce an integral identity whose solutions are pairs of functions such that the first coordinate of this pair is the desired solution, and the second one is its harmonic continuation. The continuity of the first coordinate in the integral norm is proved as well.

- a pair of functions
- fractional diffusion
- mixed Dirichlet problem
- weak solution

- Lions, ZH. L. Nekotorye metody resheniya nelinejnykh kraevykh zadach. Moscow. 1972. 586 p. (in Russ.)
- Tedeev, Al. F. On an inequality for solving the differential diffusion equation. Differencial'nye uravnenija i processy upravlenija. 2017. Vol. 4. P. 1-13. (in Russ.)
- Tedeev, Al. F. [Properties of solutions of the Cauchy problem for a second-order nonlinear parabolic equation with degeneration in an independent variable] Vestnik VGU. Seriya: Fizika. Matematika. 2018. Vol. 3. P. 185-196. (in Russ.)
- Tedeev, Al. F. [Finiteness of the support of the solution of the Dirichlet problem of the diffusion equation with an inhomogeneous source in regions of the octant type] VGU. Seriya: Fizika. Matematika. 2014. Vol. 4. P. 180-192
- Andreu, F., Mazon, J. M., Toledo, J., Igbida, N. A degenerate elliptic-parabolic problem with nonlinear dinamical boundary conditions. Interfaces Free Bound. 2006. № 8 (4). P. 447-479.
- Athanasopoulos, J., Caffarelli, L. A. Continuity of the temperature in boundary heat control problems. Adv. Math. 2010. № 224 (1). P. 293-315.