Real-time estimation/control of Partial Differential Equation (PDE) systems, especially for large-scale applications, generally involves high computational burdens. In this paper, we propose a distributed computation scheme, which can leverage available and otherwise idle computing resources to cooperatively solve the high-dimensional controller/estimator implementations for fine-grained management of such PDE systems. Such a real-time distributed computation scheme requires communication among the computing resources which is subject to uncertainties due to imperfections of the communication network. Given this scenario, the proposed approach: 1) includes a modeling framework in the controller/estimator implementation that explicitly addresses network uncertainties, 2) uses a diagonalization-based scheme where the approximated ODE form is transformed into the diagonal form before implementation in order to minimize the communication requirement, and 3) includes a filtering solution to suppress the effect of communication uncertainties. The proposed scheme is illustrated via a real-time state estimation of individual battery cells in vehicle battery packs using a network of vehicular computing units. Simulation results are included to illustrate the effectiveness of the scheme.