This paper investigates the transient solution for a non-linear heat conduction problem in automotive clutch system. The frictional heat generated during clutch disks engagement is known to cause thermoelastic distortion which in turn modifies the contact pressure distribution. The contact boundary condition that defines the area over which the heat is to be generated is also known to change and shift if the engagement process is run long enough. The thermal boundary in this case is time dependent and the problem becomes non-linear. To account for such boundary condition, a solution methodology is proposed in this paper and tested in the context of a clutch disk sliding between two rigid bodies.