On solutions of space-time fractional nonlinear Fokker–Planck equation. II

Abstract This paper extends our previous study [Singla, K. and Leonenko, N.: Fract. Calc. Appl. Anal. 28 , 1094-1105 (2025)] on the time-fractional nonlinear Fokker-Planck equation to a more general nonlinear space-time fractional formulation involving both temporal and spatial Riemann-Liouville fractional derivatives. The considered model incorporates long-memory effects through the time-fractional derivative and nonlocal spatial transport through the space-fractional derivative, allowing the description of anomalous diffusion processes with both temporal memory and Lévy-type spatial interactions. By applying Lie symmetry analysis for space-time fractional partial differential equations, the governing equation is reduced to a nonlinear fractional ordinary differential equation involving Erdélyi-Kober fractional operators. Exact analytical solutions are then constructed using the power series method together with convergence analysis based on a majorant series approach and the implicit function theorem. The obtained results generalize the corresponding solutions of the previously studied time-fractional model and demonstrate how the interplay between temporal memory and spatial nonlocality influences the qualitative behavior of the solutions. Graphical illustrations are also presented to visualize the effects of different fractional orders on the evolution of the solution profiles.