Laplace Transform-Adomian Decomposition Method for Solving Bratu-Type Differential Equations

This study applies a coupled Laplace Transform-Adomian Decomposition Technique (LT-ADT) to obtain approximate analytical solutions of Bratu-type differential equations. The approach combines the operational features of the Laplace transform with the recursive treatment of nonlinear terms provided by the Adomian decomposition method. By transforming the governing nonlinear differential equations into integral forms and decomposing the exponential nonlinearities through Adomian polynomials, the method generates series solutions without linearisation, discretisation, perturbation assumptions, or restrictive approximations. Three standard Bratu-type test problems with different parameter choices were considered to assess the applicability and accuracy of the proposed procedure. The obtained approximate solutions were evaluated against the corresponding exact solutions and compared with perturbation-based results reported in the literature. For all selected computational points in the three test problems, the LT-ADT approximations coincided with the exact solutions to the precision presented in the tables, giving absolute errors recorded as zero. In contrast, the perturbation and regular perturbation methods used for comparison produced non-zero errors that increased across the computational interval. The results indicate that the coupled method provides rapidly convergent series approximations for the considered Bratu-type problems and that only a limited number of recursive components is required to reproduce the reported exact expansions. The implementation in Maple 2022 further demonstrates the computational straightforwardness of the procedure and facilitates symbolic evaluation of the derived series. Within the scope of the selected examples, the proposed LT-ADT framework offers an efficient semi-analytical approach for treating nonlinear Bratu-type differential equations involving exponential source terms. The findings also support its use as a basis for further numerical comparison in related nonlinear models.