Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, MAI, STAW and GAR). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ∈(0,1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law.
Materials whose mechanical and/or thermodynamical behaviour is determined not only by their present status but also by their past history can be termed materials with memory. A well known example of material with memory is a rigid heat conductor with memory. This model, according to [8] and [1], describes a body, assumed rigid, in which the memory affects its thermodynamical behaviour. Specifically, the heat flux relaxation function depends only on the time variable through the present time as well as the whole past history.
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