The article deals with integral equations that are widely used in various sections of physics (theory of waves on the surface of liquids, quantum mechanics, problems of spectroscopy, crystallography, acoustics, analysis and diagnostics of plasma, etc.), Geophysics (problems of gravimetry, kinematic problems of seismics), mechanics (vibrations of structures), etc. When the physics introduced aftereffect, it is not enough ordinary differential equations or partial differential equations, otherwise the initial data would determine the future state. To take into account the continuous sequence of previous States, we need to use integral and integro-differential equations, where the sign of the integral appears functions of parameters that characterize the system, which depend on time for some period preceding the moment under consideration. In this article we have considered the solution of Fredholm integral equations of the second kind by the method of successive approximations and the method of iterated nuclei.