Abstract. The concept of double-order fractional derivative generalizing the well-known Hilfer’s derivative is introduced. The formula is given for the Laplace transform of double-order fractional derivative, which is used to solve the problem of Cauchy type for equations of fractional order with this derivative. The closed solutions of some boundary-value problems for the equation of anomalous diffusion with double-order fractional derivative in time are obtained.

Keywords: anomalous diffusion, differential equations of fractional order, boundary-value problems, Hilfer’s fractional derivative, closed solutions.