Parameter estimation and inference in dynamic systems described by linear partial differential equations

Differential equations (DEs) are commonly used to describe dynamic sys-
tems evolving in one (ordinary differential equations or ODEs) or in more than one
dimensions (partial differential equations or PDEs). In real data applications, the para-
meters involved in the DE models are usually unknown and need to be estimated from
the available measurements together with the state function. In this paper, we present
frequentist and Bayesian approaches for the joint estimation of the parameters and of
the state functions involved in linear PDEs. We also propose two strategies to include
state (initial and/or boundary) conditions in the estimation procedure. We evaluate the
performances of the proposed strategy through simulated examples and a real data
analysis involving (known and necessary) state conditions.