TWO-PHASE HEAT CONDUCTION PROBLEMS WITH STURM-TYPE BOUNDARY CONDITIONS AND WITH A FRACTIONAL TIME DERIVATIVE

The paper investigates an initial-boundary value problem of the Sturm type for the heat equation with a piecewise-constant coefficient and a Caputo derivative of order 0 < a < 1 with respect to time. The domain under consideration is an interval (0, l) containing a strictly internal point of discontinuity x = x₀. From a physical point of view, this problem models the process of temperature field propagation in a thin rod of length l. The rod is composite and consists of two sections (0, x₀) and (x₀, l), possessing different thermophysical characteristics. At the contact point of the two media x = x₀ , perfect contact conditions are prescribed, implying continuity of temperature and continuity of heat flux when passing from one medium to another. The main goal of the work is to substantiate the solution of the posed initial-boundary value problem by the method of separation of variables (Fourier method). Application of this method leads to the necessity of investigating the corresponding spectral problem for an ordinary differential operator with a discontinuous coefficient at the highest derivative.

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