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

Rubayyi T. Alqahtani and Mehmet Zeki Sarikaya

Journal of Mathematics, 2026, vol. 2026, 1-11

Abstract: This paper presents a generalized Laplace transform, denoted by LÏ•, defined through a strictly increasing kernel function Ï•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 LÏ• framework provides a robust and analytically tractable tool even in situations where classical exponential-based methods fail.

Date: 2026
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/jmath/2026/1677097.pdf (application/pdf)
http://downloads.hindawi.com/journals/jmath/2026/1677097.xml (application/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:1677097

DOI: 10.1155/jom/1677097

Access Statistics for this article

More articles in Journal of Mathematics from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().