This research paper explores the practical application of three fractional mathematical models: the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models are investigated using an expansion method to extract new soliton solutions. The solutions are categorized into two types: trigonometric and exponential forms. Graphical representations of the results are provided in 2D, 3D, and contour plots to explain the physical phenomena of the models under different parameter conditions. The numerical investigation reveals various types of solutions, including smooth kink-shaped solitons, ant-kink-shaped solitons, bright kink-shaped solitons, singular periodic solutions, and multiple singular periodic solutions. The amplitude of the waves increases with pulsation in time, and the order of the time fractional coefficient significantly enhances wave propagation and influences nonlinearity impacts. The study highlights the effectiveness of the $(\frac{G}{G+F+A})$-expansion method in solving nonlinear fractional PDEs and its potential for modeling complex natural phenomena.This research paper explores the practical application of three fractional mathematical models: the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models are investigated using an expansion method to extract new soliton solutions. The solutions are categorized into two types: trigonometric and exponential forms. Graphical representations of the results are provided in 2D, 3D, and contour plots to explain the physical phenomena of the models under different parameter conditions. The numerical investigation reveals various types of solutions, including smooth kink-shaped solitons, ant-kink-shaped solitons, bright kink-shaped solitons, singular periodic solutions, and multiple singular periodic solutions. The amplitude of the waves increases with pulsation in time, and the order of the time fractional coefficient significantly enhances wave propagation and influences nonlinearity impacts. The study highlights the effectiveness of the $(\frac{G}{G+F+A})$-expansion method in solving nonlinear fractional PDEs and its potential for modeling complex natural phenomena.