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Barenblatt's book contains many examples of analytic solutions of various problems. The list includes heat propagation from a source in the linear case (for constant thermal conductivity) and in the nonlinear case, and also in the presence of heat loss. The problem of the hydrodynamic propagation of energy from a localized explosion is also considered. In both cases, the problem in its ordinary formulation - without loss -
was solved many years ago in these problems the dimensions of the constants that characterize the medium (its density, equations of state,
and thermal conductivity) and the dimensions of energy uniquely dictate the exponents of self-similar solutions