Abstract
In this paper the linear oscillator problem in fractal steady heat-transfer is studied within the local fractional theory. In particular, the local fractional Sumudu transform will be used to solve both the homogeneous and the non-homogeneous local fractional oscillator equations under fractal steady heat-transfer. It will be shown that the obtained non-dfferentiable solutions characterize the fractal phenomena with and without the driving force in fractal steady heat transfer at low excess temperatures.