Advanced Mathematical Models & Applications

This paper presents a comparative numerical study of nonlinear time-fractional partial differential equations using the Laplace–Adomian Decomposition Method (L-ADM). We consider three definitions of time-fractional derivatives: Caputo, Caputo–Fabrizio, and Atangana–Baleanu in the Caputo sense. Unlike earlier works focused on a single operator, our study provides a unified framework to examine how different kernels influence accuracy, convergence, and stability. The approach combines the Laplace transform, which handles fractional operators and initial conditions effectively, with Adomian decomposition, which addresses nonlinear terms systematically. The obtained series solutions converge rapidly and are easy to compute. Two benchmark models, the Kudryashov–Sinelshchikov and Kaup–Kupershmidt equations, are used for validation. Numerical results—including error tables and 2D/3D plots—show higher accuracy than other methods. The analysis also illustrates how the choice of fractional derivative affects error distribution and the smoothness of solutions. These findings guide researchers in selecting suitable operators for modeling phenomena such as anomalous diffusion and viscoelasticity.