A Rigorous Review of Recent Integral Transforms for Solving Differential Equations: Analysis, Properties and Applications

This paper presents a rigorous comparative review of five pivotal integral transforms: Laplace, Sumudu, Elzaki, Aboodh, and the recently introduced RAHMOH transform. We establish a unified theoretical framework to analyze kernel structures, derive the complex inversion formula for the RAHMOH operator, and formally prove its mathematical equivalence to the Laplace transform. Unlike previous surveys, we derive analytical solutions for integer-order ODEs and PDEs, as well as fractional differential equations (FDEs) using all methods, confirming that while they are mathematically isomorphic, they differ significantly in algebraic pathways. Specifically, the analysis identifies the RAHMOH transform as a generalized ''bridge'' operator, encapsulating the scaling properties of Sumudu and the decay properties of Laplace through its dual-variable kernel. Furthermore, numerical simulations via MATLAB validate the consistency of the RAHMOH transform, demonstrating its dimensional stability and accuracy in modeling both dissipative and fractional systems.