In this paper, we presented an integro differential equation (IDE), in compelx plane, with singular kernel. The IDE is investigated from the first and second boundary value problem (BVP) of a weakened infinite plate by a curvilinear hole (C,) in the presence of heat. The hole is conformally mapped inside a unit circle by a special transformation with complex constants coefficients\(z=\omega (\zeta ),\,\zeta =\xi +i\eta ,\,i=\sqrt { - 1} .\) This mapping will conform the curvilinear hole in the infinite elastic plate into a unit circle \(\gamma ,\left| \zeta \right|\;<\,\,1,\) such that inside the circle. The stress and strain compounds were calculated, in the presence of heat effect, and this was clarified by numerical results. Many applications for the problem are discussed. Maple 2019 is used for computations results. MSC (2010): 74B10, 30C20.