Analytical solution of generalized Bratu type equations with the Le Roy-type Mittag-Leffler function

This work generalized the differential equation by incorporating the three-parameter Le Roy-type Mittag-Leffler function and generalized fractional derivative operators into the traditional Bratu equation. Furthermore, using the homotopy perturbation transform method to solve nonlinear fractional differential equations. In all cases, the HPTM series solutions demonstrated rapid convergence, with the second- or third-order approximations often nearly coinciding, numerically validating the convergence analysis. The existence, uniqueness, and convergence of the solution are thoroughly established. Particular extraordinary circumstances are thoroughly examined. The proposed method offers a comprehensive approach to solving generalised Bratu-type equations and provides new insights into the behaviour of fractional systems exhibiting intricate nonlinear dynamics.