This project deals with a selection of theoretical and numerical aspects for certain classes of parabolic partial differential equations. Determination of surface temperature and heat flux from a finite set of interior temperature measurements, stable under noisy data will be investigated. Theoretical and numerical analysis of methods involving eigenfunction expansions of initial data and Whittaker Cardinal expansions of initial and boundary data will be studied. In this part of the work, the investigator will employ more standard regularization techniques for recovering the Fourier coefficients of initial data. However, the stability of the numerical inversion based on Whittaker Cardinal expansions is novel. In this approach, the problem of choosing a regularizing parameter as in standard approaches no longer exists. However, the "regularizing" necessary for this method takes the form of an appropriate choice of the number of spatial sensors and the sampling time, both of which have been completely determined in recent work for special cases. A further regularization is obtained by a suitable apriori restriction of the class of initial data appropriate to Sinc Expansions.
