The governing equations of a mathematical model describe how the values of the unknown variables (i.e. the dependent variables) will change. The change of the value of a variable with respect to time may be explicit, in that a governing equation includes a derivative with respect to time, or implicit, such as when a governing equation has velocity or flux as an unknown variable.
Examples
The classic governing equations in continuum mechanics are those for the
For isolated systems the first four equations are the familiar conservation equations in physics. A governing equation may also take the form of a flux equation such as the diffusion equation or the heat conduction equation. In these cases the flux itself is a variable describing change of the unknown variable or property (e.g., mole concentration or internal energy or temperature).
A governing equation may also be an approximation and adaptation of the above basic equations to the situation or model in question. A governing equation may also be derived directly from experimental results and therefore be an empirical equation. A governing equation may also be a state equation, an equation describing the state of the system, and thus actually be a constitutive equation that has "stepped up the ranks" because the model in question was not meant to include a time-dependent term in the equation. This is the case for a model of a petroleum processing plant. Results from one thermodynamic equilibrium calculation are input data to the next equilibrium calculation together with some new state parameters, and so on. In this case the algorithm and sequence of input data form a chain of actions, or calculations, that describes change of states from the first state (based solely on input data) to the last state that finally comes out of the calculation sequence.
Some examples using differential equations are