Recovering Memory Kernels in Heat Flow

Materials with memory enable to model the
propagation of heat by finite speed. Models with memory are
also used in phase transition problems and
thermoelasticity. In the present book inverse problems to
determine kernels of heat flux and internal energy in the
one- dimensional non-homogeneous heat flow are studied. We
consider the case when these kernels are degenerate, i.e.
representable as sums of the known space-dependent
functions times the unknown time- dependent coefficients. We
proved existence and uniqueness of two problems of such
kind. The first one is a problem with purely temperature
observations. Then the kernel of internal energy is
determined with higher smoothness than the kernel of heat
flux. The second one is te problem with purely flux
observations. Then the kernels of internal energy and heat
flux are determined with the same level of accuracy.
Moreover, it has been shown that the homogeneous inverse
problem with flux observations is severely il-posed. The
method involves application of the Laplace transform and
reduction of the transformed problem to a fixed-point form.