In this paper, we present some our ongoing researches related to a collocation method based on a double projection scheme via regularization procedure, to numerically solve Cauchy integral equations of the second kind. For the collocation approach, we use spline quasi-interpolating projectors of degree $d$. We prove the existence of the solution for a double projection scheme. Moreover, we give new error bounds.

In this paper, heat transfer analysis during cutting 2024-T351 aluminium alloy with carbur tungsten (25° rake angle) and especially in 2, 3D orthogonal cutting operation numerically was performed. The assembly was modelled, meshed and analysed using commercial computational tool ABAQUS EXPLICIT. The parameters to be varied are depth of cut and impact speed, the thermal boundary conditions are adiabatic, the thermal contact between tool and workpiece is imperfect and the following method is Lagrangian, the results were well visualized and compared with literature, [23] and are in good agreement with reality of cutting. It was found that maximum temperature turn around 400°C when cutting and it is located in chip formation on the other hand, heat conduction with impact speed were two antagonist phenomenon, in other terms, increase in impact speed has less effect on heat transfer and in conduction espescialy.