Exact solutions of the time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation

The extended (3+1)-dimensional Kadomtsev–Petviashvili equation is widely used in such domains as fluid mechanics, optics and so on. In this paper, we derive a new time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation based on the conformable fractional derivative for the first time. With the help of the Hirota bilinear method, we obtain the N-soliton, breather and lump solutions of the time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation. In addition, the semi-inverse variational method and the advanced \(exp(\phi (-\xi ))\) $exp(\varphi (-\xi ))$ -expansion method are introduced to construct the exact solutions of this equation. By choosing suitable parameters, these solutions which can help solve issues in the marine science, fluctuation theory and other fields are presented through 3D graphics, contour and density plots. The findings of this work can further extend the study of fractional partial differential equations.