The boundary value problemfor one-dimensional fractional differential equationof advection-diffusion

The use of fractional derivatives for describing and studying the physical processes of stochastic transport has become one of the most popular fields of physics in the recent years, many of the problems of fluid flow in highly-porous (fractal) environments also lead to the need to study boundary value problems for the equations of fractional order.The paper considers one of the boundary value problems for one-dimensional differential equation of fractional order. Using the Fourier method, the solution to this problem was explicitly written. The author also studied the qualitative properties of the solutions of the boundary value problem. It was proved that, in the case of going to infinity, the limit of the decisions recorded in the form of the function and the limit of the derivative of this solution tend to zero.The results can find application in the theory of fluid flow in a fractal environment and in order to simulate changes in temperature.Fractional integrals and derivatives of fractional integral-differential equations find wide application in contemporary studies of theoretical physics, mechanics and applied mathematics. Fractional calculus is a very powerful tool for describing the physical systems, which have memory and are non-local. Many processes in complex systems are non-locality and have long-term memory. The fractional integral operators and the fractional differential operators allow describing some of these properties. The use of fractional calculus will be useful for obtaining the dynamic models, in which integraldifferential operators describe the power of long-term memory and time coordinate and three-dimensional nonlocality for medium and complex processes.