Nonlinear partial integro-differential equations (PIDEs) are applied to present the various practical phenomena in a multitude of sectors of modern science and engineering, especially in optic fiber, quantum electronics, modern physics, and the special field of nonlinear wave motion. Basically, our research demonstrates a way to generate a significant quantity of solutions to these types of two PIDEs. In this research, we have used an efficient mathematical tool namely the generalized \(\left(G{\prime}/G\right)\) -expansion method to acquire the closed form soliton solutions for the (2 + 1)-dimensional first integro-differential Kadomtsev–Petviashivili (KP) hierarchy equation and the (2 + 1)-dimensional second integro-differential KP hierarchy equation utilizing a code likely Mathematica. The explicit closed form soliton solutions of these two PIDEs are found in the pattern of trigonometric, hyperbolic, and rational functions which are compared to all the well-known results that are yielded in the paper. We attain solutions that are graphically displayed in addition to physically described in 3D structure, contour, and 2D, such as a bell-shaped soliton, a singular bell-shaped soliton, and some kink-shaped solitons. As far as the authors’ wisdom, the outcomes of these problems gained by the offered expansion method are renewed closed form solitary wave and investigated here for the first time. The analysis of obtained results will be able to provide a constructive explanation of the physical phenomena in optical physics and engineering.
