Clarke's equation

In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978.[1][2][3][4] The equation describes the thermal explosion process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal sound speeds.[5] The equation reads as

${\displaystyle (\theta _{t}-\gamma e^{\theta })_{tt}=(\theta _{t}-e^{\theta })_{xx}}$

where ${\displaystyle \theta }$ is the non-dimensional temperature perturbation and ${\displaystyle \gamma }$ is the specific heat ratio. The term ${\displaystyle \theta _{t}-e^{\theta }}$ describes the explosion at constant pressure and the term ${\displaystyle \theta _{t}-\gamma e^{\theta }}$ describes the explosion at constant volume. Similarly, the term ${\displaystyle (\ )_{tt}-(\ )_{xx}}$ describes the wave propagation at adiabatic sound speed and the term ${\displaystyle \gamma (\ )_{tt}-(\ )_{xx}}$ describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.