When thermal-shock is an issue, the underlying thermal- and stress-states are often difficult to determine because the boundary conditions must be known. For direct problems where the boundary conditions such as temperature or flux are known a priori, the procedure is usually mathematically tractable and can therefore be solved analytically. On the other hand, the inverse problem where the boundary conditions must be determined remotely is inherently ill-posed and therefore sensitive to errors. Moreover, there are only limited numbers of analytical solutions and they are usually restricted to timeframes before the thermal wave reaches the natural boundaries of the structure. Fortunately, generalized solutions based on either measured temperature- and/or strain-histories can be used to determine the underlying thermal excitation via a least-squares determination of coefficients that require the direct solution to enforce the data. Once the inverse problem is solved and the unknown boundary-condition determined, the resulting polynomial can then be used with the generalized Direct solution to determine the thermal- and stress-states as a function of time and position. For the two geometries explored (thick-walled cylinder under an internal transient with external convection and a slab with one adiabatic surface), excellent agreement was seen with various test cases for histories based on either temperature or strain. However, the strain-based solutions may be preferable since they do not suffer from time-lags associated with thermally thick structures. The derived solutions appear to be well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties are independent of temperature.