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 , defined through a strictly increasing kernel function ϕ ( t ). 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 framework provides a robust and analytically tractable tool even in situations where classical exponential‐based methods fail.