The Operational Calculus of the Generalized Laplace Transform: A Unified Solution Method for Anomalous Decay Models

This paper presents a generalized Laplace transform, denoted by Lϕ, defined through a strictly increasing kernel function ϕt. Unlike prior works that focus on formal definitions, our framework unifies classical, Gaussian, and Mellin-type transforms while providing a systematic operational calculus. In particular, we rigorously derive second-derivative identities, resolving ambiguities and inconsistencies that appear in the existing literature for singular and nonstandard kernels. Beyond formal generalization, we introduce a unified method to solve first- and second-order differential equations with variable coefficients. By leveraging a distinctive cancellation phenomenon, complex variable-coefficient equations, including Hermite-type models, can be algebraically simplified in the transform domain. We further illustrate the practical utility of the approach with step-by-step examples in anomalous diffusion and viscoelastic decay, demonstrating that the Lϕ framework provides a robust and analytically tractable tool even in situations where classical exponential-based methods fail.