A Parameterized Generalized Transform Framework for Nonlinear Differential Models

This paper develops a parameterized generalized transform framework for nonlinear differential models. The method combines a generalized Laplace-type transform, Adomian decomposition, Chebyshev–Padé rational reconstruction, and a μ-scaled generalized transform to construct admissible semi-analytical approximations. The framework treats the transform geometry as part of the approximation process, allowing the transformed domain to be adjusted while preserving an explicit analytical structure. The theoretical analysis establishes admissibility conditions, existence of admissible minimizers, characterization of the admissible region for the parametric kernel, interior optimality conditions, inverse and residue inversion formulas, and fixed-point consistency with a first-order truncation estimate. The method is illustrated on a logistic–Allee tumor-growth model using experimental data. The resulting compact representations remain real-valued and admissible on the full data interval and produce errors comparable to standard numerical reference solutions.