Electrochemical oxidation of hydrogen and hydrocarbon fuels over solid oxide fuel cell (SOFC) anodes is crucial to the performance of SOFCs. Although SOFC technology has been evolving over the past few decades, not much is known about elementary electrochemical mechanisms that govern their performance. In this work we have employed quantum Density Functional Theory (DFT) and statistical thermodynamic calculations to investigate the underlying molecular mechanisms that govern the electrochemical oxidation of hydrogen and methane over Ni-based SOFC anodes. Aside from presenting the results of our calculations we will also discuss in detail the approach that was utilized to address the issue of electrostatic potential and electric field which are present in electrochemical systems.