Boundary value problems for discontinuously loaded parabolic equations

The article deals with boundary value problems for a discontinuously loaded parabolic equation with a Riemann – Liouville fractional integro-differentiation operator with variable coefficients. The unambiguous solvability of the Cauchy – Dirichlet problem for a discontinuously loaded parabolic equation of fractional order is proved. The paper also examines the existence and uniqueness of the solution of the first boundary value problem for a discontinuously loaded parabolic equation. Using the method of the Green function, using the properties of the fundamental solution of the corresponding homogeneous equation, as well as assuming that the coefficients of the equation are bounded, continuous and satisfy the Helder condition, while remaining non-negative, it is shown that the solution of the problem is reduced to a system of Volterra integral equations of the second kind.