In this paper, we study the solution of fractal energy integral equation for one-dimensional compressible flows without body force using measure of noncompactness. We also discuss the solution of the local fractal equation of losing energy system using the notion of local fractal differential idea. For this, a new notion of χ-Δ-set contraction condition under simulation function is defined and two main fixed point and coupled fixed point results are obtained. © 2020 Faculty of Engineering, Alexandria University