This study is devoted to constructing an approximate analytic solution of the fractional form of a strongly nonlinear boundary value problem with multi-fractional derivatives that comes in chemical reactor theory. We construct the solution algorithm based on the generalized differential transform technique in four simple steps. The fractional derivative is defined in the sense of Caputo. We also mathematically prove the convergence of the algorithm. The applicability and effectiveness of the given scheme are justified by simulating the equation for given parameter values presented in the system and compared with existing published results in the case of standard derivatives. In addition, residual error computation is used to check the algorithm's correctness. The results are presented in several tables and figures. The goal of this study is to justify the effects and importance of the proposed fractional derivative on the given nonlinear problem. The generalization of the adopted integer-order problem into a fractional-order sense which includes the memory in the system is the main novelty of this research.