Soliton solutions for nonlinear variable-order fractional Korteweg–de Vries (KdV) equation arising in shallow water waves

Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space. The present study considers the advanced and broad spectrum of the nonlinear (NL) variable-order fractional differential equation (VO-FDE) in sense of VO Caputo fractional derivative (CFD) to describe the physical models. The VO-FDE transforms into an ordinary differential equation (ODE) and then solving by the modified (G′/G)-expansion method. For accuracy, the space-time VO fractional Korteweg-de Vries (KdV) equation is solved by the proposed method and obtained some new types of periodic wave, singular, and Kink exact solutions. The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations. The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation. The results revealed that the NL VO-FDEs are highly active, functional and convenient in explaining the problems in scientific physics.